Constraining the symmetry energy with heavy-ion collisions and Bayesian analyses
P. Morfouace, C.Y. Tsang, Y. Zhang, W.G. Lynch, M.B. Tsang, D.D.S Coupland, M. Youngs, Z. Chajecki, M.A. Famiano, T.K. Ghosh, G. Jhang, Jenny Lee, H. Liu, A. Sanetullaev, R. Showalter, J. Winkelbauer
CConstraining the symmetry energy with heavy-ion collisions andBayesian analyses
P. Morfouace a , ∗ , C.Y. Tsang a , Y. Zhang b , W. G. Lynch a , M.B. Tsang a , D.D.S Coupland a ,M. Youngs a , Z. Chajecki c , M.A. Famiano c , T.K. Ghosh e , G. Jhang a , Jenny Lee d , H. Liu f ,A. Sanetullaev a , R. Showalter a and J. Winkelbauer a a National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA b China Institute of Atomic Energy; Beijing, 102413, PRC c Department of Physics, Western Michigan University, Kalamazoo, Michigan 49008, USA d Department of Physics, The University of Hong Kong, Hong Kong, China e Variable Energy Cyclotron Centre, 1/AF Bidhannagar, Kolkata 700064, India f Texas Advanced Computing Center, University of Texas, Austin, Texas 78758, USA
A R T I C L E I N F O
Keywords :Symmetry energyHeavy-ion collisionsBayesian analysis
A B S T R A C T
Efficiency corrected single ratios of neutron and proton spectra in central
Sn+
Sn and
Sn+
Sn collisions at 120 MeV/u are combined with double ratios to provide constraints on thedensity and momentum dependencies of the isovector mean-field potential. Bayesian analyses of thesedata reveal that the isoscalar and isovector nucleon effective masses, 𝑚 ∗ 𝑠 − 𝑚 ∗ 𝑣 are strongly correlated.The linear correlation observed in 𝑚 ∗ 𝑠 − 𝑚 ∗ 𝑣 yields a nearly independent constraint on the effectivemass splitting Δ 𝑚 ∗ 𝑛𝑝 = ( 𝑚 ∗ 𝑛 − 𝑚 ∗ 𝑝 )∕ 𝑚 𝑁 = −0 . +0 . . 𝛿 . The correlated constraint on the standardsymmetry energy, 𝑆 and the slope, 𝐿 at saturation density yields the values of symmetry energy 𝑆 ( 𝜌 𝑠 ) = 16 . +1 . . MeV at a sensitive density of 𝜌 𝑠 ∕ 𝜌 = 0 . +0 . . . Connecting the properties of matter within neutron starsto the properties of nuclei on earth presents both opportu-nities and challenges. The ability to study the symmetryenergy by nuclear measurements in the laboratory presentsa definite opportunity. On the other hand, the large differ-ence between the asymmetry of matter within nuclei andthat of neutron stars presents a definite challenge. In nu-clei, the Coulomb forces shift the energy minimum to moreneutron-rich isotopes in heavy nuclei, but the symmetry en-ergy shifts the energy minimum to more symmetric isotopes.Consequently, the interplay of Coulomb and symmetry en-ergies limit the neutron number 𝑁 available for elements ofproton number 𝑍 to a narrow range about 𝑁 and the asym-metries 𝛿 = ( 𝑁 − 𝑍 )∕( 𝑁 + 𝑍 ) of nuclei remain less than0.25. However, inside neutron stars, the neutron fraction canreach above 90% at normal density under 𝛽 equilibrium con-ditions. This vastly increases the importance of probing thesymmetry energy and understanding its effects in the labo-ratory over wide range of densities and asymmetries.Recent observation of a neutron star-merger event [1]yields the first glimpse of neutron-star properties such astidal deformability that are governed by the nuclear equa-tion of state (EoS). The EoS at zero temperature is the sumof the symmetry energy and the energy for symmetric mat-ter with equal neutron and proton density, 𝜌 𝑛 = 𝜌 𝑝 . Forneutron stars, the density dependence of the symmetry en-ergy, 𝐸 𝑠𝑦𝑚 = 𝑆 ( 𝜌 ) 𝛿 , strongly influences the relationshipbetween pressure and the density, 𝜌 = 𝜌 𝑛 + 𝜌 𝑝 , of stellar mat-ter and thus, the neutron star mass-radius relationship [2, 3] ∗ Corresponding author [email protected] (P. Morfouace)
ORCID (s): as well as the nuclear lattice and nucleonic gas within the in-ner crust, the boundary between the core and the inner crustand the nature of lattice or pasta structures of nuclei. Labo-ratory constraints have been obtained on the EoS [4] and onthe momentum dependence of the mean-field potentials forsymmetric matter [5, 6]. Present efforts to constrain 𝑆 ( 𝜌 ) have focused on the first two coefficients 𝑆 and 𝐿 in theTaylor expansion of 𝑆 ( 𝜌 ) around the saturation density 𝜌 , 𝑆 ( 𝜌 ) = 𝑆 + 𝐿 𝜌 ( 𝜌 − 𝜌 ) + 𝑂 (( 𝜌 − 𝜌 ) ) (1)Information about 𝑆 and 𝐿 have been obtained from anal-yses of the masses [7, 8], Pygmy Dipole Resonances (PDR)[9, 10, 11], Electric dipole polarizability [12, 13], neutronskin thickness [14], Isobaric Analog States (IAS) [15] andisospin diffusion in heavy-ion collisions [16]. These analy-ses result in positively-correlated constraints on 𝑆 and 𝐿 .Depending on the experimental condition, the slope of thecorrelation between 𝑆 and 𝐿 are different. That is becausethe slope is a signature of the sensitive density being probedby a given laboratory experiment [17].In addition to the density dependence of the symmetrypotential, the nuclear mean-field potential has momentumdependencies from the Fock exchange term, finite range andcorrelation effects [18, 19, 20, 21, 25, 22, 23, 24, 26]. Theneutron and proton effective masses associated with these ef-fects influence many of the thermal properties of hot proto-neutron stars formed in core-collapse supernovae [2, 3, 27,28]. The mean-field potential contains an isoscalar effectivemass 𝑚 ∗ 𝑠 that is reduced in nuclei from the nucleon mass 𝑚 𝑁 by approximately 𝑚 ∗ 𝑠 𝑚 𝑁 ≈ 0 .
65 − 0 . [18, 20, 21]. Further-more, momentum dependencies in the isovector (symmetry) Morfouace et al.:
Preprint submitted to Elsevier
Page 1 of 6 a r X i v : . [ nu c l - e x ] M a y (MeV) c.m. E n / p R Sn Sn+
Corrected Exp. resultsUncorrected Exp. resultsPosterior mean value region s Posterior 2 region s Prior 2
40 60 80 (MeV) c.m. E n / p R Sn Sn+
20 40 60 80 (MeV) c.m. E n / p DR Figure 1:
Single neutron over proton ratio for
Sn+
Sn and
Sn+
Sn at 120 MeV/u. The open blue circles show the ratiobefore correction and the full red points show the ratio after correction due to multiple scattering and nuclear reaction losses.The brown shaded area corresponds to the prior 𝜎 region of the 49 sets of ImQMD calculation spanning the parameters spacewhile the blue colored area correspond to the posterior 𝜎 region. mean-field potential will cause the neutron and proton effec-tive masses to differ [19]. This effect strongly modifies thecooling of neutron stars via neutrino emission [29]. Parameter range . ≤ 𝑆 ≤ (MeV) ≤ 𝐿 ≤ (MeV) . ≤ 𝑚 ∗ 𝑠 ∕ 𝑚 𝑁 ≤ . . ≤ 𝑚 ∗ 𝑣 ∕ 𝑚 𝑁 ≤ . Table 1
Model parameter values for prior distribution. 49 sets of calcu-lation have been performed within this 4D model space usinga Latin hyper-cube sampling.
Theoretical calculations and commonly used effective in-teractions differ regarding the sign and magnitude of the effective-mass splitting Δ 𝑚 ∗ 𝑛𝑝 = 𝑚 ∗ 𝑛 − 𝑚 ∗ 𝑝 𝑚 𝑁 . Positive values for the masssplitting are expected from Landau Fermi liquid theory [30]and this sign appears to be consistent with recent fits to theenergy dependence of the nucleon elastic scattering that ob-tain Δ 𝑚 ∗ 𝑛𝑝 = (0 .
27 ± 0 . 𝛿 [5]. Calculations predict theeffective-mass splitting to increase strongly with density, aneffect that becomes increasingly important in astrophysicalenvironments such as neutron stars and in central heavy-ioncollisions [31, 33, 34].In this paper, we probe both the density and the mo-mentum dependence of the symmetry energy via measure-ments of neutron and light charged particle spectra in cen-tral Sn+
Sn and
Sn+
Sn collisions at E/A=120MeV. Details about this experiment can be found in ref. [35].Light-charged particles were measured in the Large-Area Sil-icon Strip Array (LASSA) [38] placed 20 cm from the targetcovering the polar angle range of ◦ < 𝜃 𝑙𝑎𝑏 < ◦ with a0.9 ◦ angular resolution. Neutrons were measured by the twowalls of the MSU Large-Area Neutron Array (LANA) [39]at 5 and 6 m from the reaction target. The LANA spannedpolar angles of ◦ < 𝜃 𝑙𝑎𝑏 < ◦ with an angular resolution of . ◦ to . ◦ . Neutrons were distinguished from 𝛾 raysby pulse shape discrimination and from charged particles byuse of a charged-particle veto array of BC-408 plastic scin-tillator detectors placed between the target and the neutronwalls. In order to avoid systematic uncertainties associatedwith the single ratios, ref. [35] reported only spectral doubleratios described below. In this work, we carefully correct thecharged-particle spectra for reaction and scattering losses inthe CsI(Tl) crystals causing a mis-identification of chargedparticles in the LASSA with an estimated accuracy of ±2% [40], while the neutron efficiency has been carefully esti-mated in Ref. [35] by using the SCINFULQMD Monte Carlocode [41]. In general, most transport models have difficultyreproducing the relative abundances of light isotopes pro-duced as the system expands and disassembles. Followingref. [35] we calculated the coalescence invariant (CI) neu-tron and proton spectra by combining the free nucleons withthose bound in light isotopes with < 𝐴 < .The open blue circles in the left and middle panel ofFig. 1 show the uncorrected ratios of the coalescence-invariantneutron spectra divided by the coalescence-invariant protonspectra for the Sn+
Sn and
Sn+
Sn. The solidred points show the corresponding efficiency-corrected sin-gle ratios. Both spectra have been transformed to the centerof mass as described in ref. [35]. The right panel shows thedouble ratios obtained by taking the single ratios in the leftpanel and dividing by the corresponding ratios in the middlepanel. The efficiency corrected double ratios are consistentlylower than the uncorrected double ratios but still within theexperimental uncertainties. The residual differences in thedouble ratios come from different admixtures of the variousisotopes, each with its own detection efficiency in each reac-tion.The availability of the new single ratio data allows multi-parameter analysis to extract both the density and momen-tum dependencies of the mean-field potentials [31, 32]. Weperform these evaluations with the ImQMD transport model
Morfouace et al.:
Preprint submitted to Elsevier
Page 2 of 6 S L S *s m S *v m S
30 35 S L
40 60 80 100 120 L *s m L *v m L
30 35 S * s m
40 60 80 100 120 L * s m *s m *v m * s m
30 35 S * v m
40 60 80 100 120 L * v m *s m * v m *v m Figure 2:
The posterior likelihood for two parameters showing the constrains on those parameters. The projections of thoseplots correspond to the one-dimensional spectrum illustrating how a given parameter is constrained by the data. of ref. [31], which parameterizes the mean fields in terms ofstandard Skyrme parameterizations. We focus on four quan-tities: 𝑆 and 𝐿 , which describe the density dependence ofthe mean-field potential, the isoscalar effective mass 𝑚 ∗ 𝑠 andthe isovector effective mass 𝑚 ∗ 𝑣 . Using a Bayesian MarkovChain Monte Carlo statistical analysis software, we exploredthe four-dimensional parameter space as listed in Tab. 1. Otherparameters in the Skyrme interactions, the in-medium nucleon-nucleon cross section and Pauli blocking algorithm were keptat the default values given in Ref. [31, 42, 43, 44].All calculations were performed at impact parameter 𝑏 =2 fm, corresponding to central collisions. That choice is jus- tified because the calculated values for the single ratio 𝑅 𝑛 ∕ 𝑝 are relatively insensitive to impact parameter, changing neg-ligibly ( < %) over the range 𝑏 = 2 − 6 fm. For each system( Sn+
Sn and
Sn+
Sn at 120 MeV/u), 49 parame-ter sets have been selected on a Latin hyper-cube to span theparameter space listed in Tab. 1. The 𝑖 𝑡ℎ set of parametervalues in our parameter space can be represented by a 4Dvector ⃗𝑥 𝑖 = { 𝑆 , 𝐿, 𝑚 ∗ 𝑠 , 𝑚 ∗ 𝑣 } . For each of these 49 sets werun the full ImQMD model and the results of those calcula-tion will serve to train the emulator that models the ImQMDcalculations [47]. Partly due to the steep decrease in high en-ergy particle, our empirical studies indicate that a minimum Morfouace et al.:
Preprint submitted to Elsevier
Page 3 of 6 .4 - - I f n/p Only DR=0.037 m =0.164 s - - I f n/p Only SR=0.025 m =0.076 s - - I f n/p and SR n/p DR=0.063 m =0.057 s Figure 3:
Prior of the 𝑓 𝐼 distribution when using only the double ratio 𝐷𝑅 𝑛 ∕ 𝑝 in the Bayesian analysis (left), only the single ratio 𝑅 𝑛 ∕ 𝑝 (center) and the double and single ratio combined (right). of 200000 events per set is needed to stabilize the currentresults.From Bayes theorem the probability 𝑝𝑜𝑠𝑡 ( ⃗𝑥 𝑖 , 𝑦 𝑒𝑥𝑝 ) , fortheoretical values ⃗𝑥 𝑖 to be correct is given by 𝑝𝑜𝑠𝑡 ( ⃗𝑥 𝑖 , 𝑦 𝑒𝑥𝑝 ) ∝ 𝑙𝑖𝑘𝑒𝑙𝑖ℎ𝑜𝑜𝑑 ( 𝑦 𝑒𝑥𝑝 , ⃗𝑥 𝑖 ) 𝑝𝑟𝑖𝑜𝑟 ( ⃗𝑥 𝑖 ) , where 𝑝𝑟𝑖𝑜𝑟 is the assumed prob-ability distribution of the parameter set ⃗𝑥 𝑖 determined fromother information prior to comparing to the experiment. Wetake the conditional probability of getting a measured set ofdata given ⃗𝑥 𝑖 to be the likelihood function ( ⃗𝑥 𝑖 ) : ( ⃗𝑥 𝑖 ) ≈ 𝑒𝑥𝑝 ( − ∑ 𝑎 ( 𝑦 𝑀𝑎 ( ⃗𝑥 𝑖 ) − 𝑦 𝑒𝑥𝑝𝑎 ) 𝜎 𝑎 ) . (2)In this approach, we compare the model value 𝑦 𝑀𝑎 ( ⃗𝑥 𝑖 ) for theexperimental measurement 𝑦 𝑒𝑥𝑝𝑎 , and the uncertainties incor-porate both the experimental and the theoretical ones.The prior probability distributions for 𝑆 , 𝐿 and 𝑚 ∗ 𝑣 areassumed to be uniform within the model space. For 𝑚 ∗ 𝑠 whichhas been shown to have a value close to 0.7, we assume aGaussian distribution centered as 0.7 with a width of 0.05.We then use the efficiency corrected experimental data shownin Fig. 1 to evaluate the post probability distribution usingMarkov Chain Monte Carlo sampling (MCMC) which is im-plemented in the PyMC library [45]. The AutoGrad package[46] is used in order to train the emulator by minimizing theemulated error. The brown shaded area in the three pan-els of Fig. 1 show the extreme values for the 49 sets of theImQMD calculation corresponding to the prior distributionof the theoretical parameters, while the blue area show theposterior 𝜎 region for the four fitted parameters. The diag-onal panels in Fig. 2 show the posterior distribution for thefour parameters. Some of the parameters are highly corre-lated. For example, the two-dimensional plot in the upperleft panels show that there is a strong correlation between 𝑆 and 𝐿 , that has been observed in previous studies [48].There is also a strong correlation between 𝑚 ∗ 𝑠 and 𝑚 ∗ 𝑣 with 𝑚 ∗ 𝑠 ∕ 𝑚 𝑁 = 0 .
67 ± 0 . and 𝑚 ∗ 𝑣 ∕ 𝑚 𝑁 = 0 .
69 ± 0 . . Using the following relationship for the effective mass splitting Δ 𝑚 ∗ 𝑛𝑝 𝑓 𝐼 = ( 𝑚 𝑁 𝑚 ∗ 𝑠 − 𝑚 𝑁 𝑚 ∗ 𝑣 ) = 𝛿 ( 𝑚 𝑁 𝑚 ∗ 𝑛 − 𝑚 𝑁 𝑚 ∗ 𝑝 ) 𝑓 𝐼 ≈ − 𝛿 Δ 𝑚 ∗ 𝑛𝑝 ( 𝑚 𝑁 𝑚 ∗ 𝑠 ) , (3)one gets 𝑓 𝐼 = 0 .
06 ± 0 . or Δ 𝑚 ∗ 𝑛𝑝 = (−0 .
05 ± 0 . 𝛿 . Ifthe single ratio data are removed from the Bayesian anal-ysis, only the correlation between 𝑚 ∗ 𝑠 and 𝑚 ∗ 𝑣 remains andthe 𝑓 𝐼 distribution becomes broader but still centers closerto 0 ( 𝜇 = 0 . and 𝜎 = 0 . ) as shown in the left panelof Fig. 3. Within the uncertainties, this is consistent withthe double ratio analysis [35]. Inclusion of the single ratiodata allows the extraction of the 𝑆 and 𝐿 correlation andthe tightening of the effective mass splitting constraint for 𝑓 𝐼 ( 𝜇 = 0 . and 𝜎 = 0 . ) as shown in the right panel ofFig. 3. This uncertainty reflects the relatively small influenceof the effective mass splitting on the single and double ratiosfor nucleon energies less than 𝐸 𝑐.𝑚. = 100 MeV. We notethat a much larger sensitivity is expected for nucleons emit-ted at higher incident energies; measurements such higherenergies should lead to more definite result [31, 32, 33]. Thecurrent results are consistent with the lower bound on Δ 𝑚 ∗ 𝑛𝑝 obtained from elastic scattering in ref. [6], but lower than theresult obtained by a statistical analysis of values publishedfor 𝑆 and 𝐿 in ref. [5]. As discuss below, such 𝑆 and 𝐿 values require model dependent extrapolations of 𝑆 and 𝐿 from measurements that are sensitive for 𝑆 ( 𝜌 ) at much lowerdensities. It is unlikely that the theoretical uncertainties ofthose extrapolations are fully reflected in the error bars ofref. [6].We use the overlap method described in [17] to deter-mine the sensitive density and symmetry energy from ouranalysis. First we choose three points (black crosses in Fig-ure 2) along the best-fit linear correlation between 𝑆 and 𝐿 . The three black curves plotted in Fig. 4 represent 𝑆 ( 𝜌 ) calculated as a function of density. 𝑆 ( 𝜌 ) corresponds to thehomogenous hadron EoS that require the symmetry energyto be zero at zero density. The value of the parameters tocalculate those three curves are listed in Table. 2. The three Morfouace et al.:
Preprint submitted to Elsevier
Page 4 of 6 lack curves cross over at 𝜌 𝑠 ∕ 𝜌 = 0 . +0 . . with 𝑆 ( 𝜌 𝑠 ) =16 . +1 . . MeV which is plotted as the open red star in Fig. 4.This may indicates that even though we can obtain the den-sity dependence of the symmetry energy over a very largerange of density regions using the Bayesian analysis, the re-gion best explored by the experiment is limited. The othersymbols are the extracted symmetry energy obtained in ref.[17]from masses [7, 8] and isobaric analog sates [15] around . 𝜌 . The dipole polarizability measurements [13] and theisospin diffusion data sit at . 𝜌 and . 𝜌 , respectively.To exploit the full potential of the Bayesian analysis toextract multi-parameters in the Equation of state, both thedata and the theoretical model need to be improved. TheBayesian analysis depends highly on the theoretical modelused. Specifically, the current ImQMD model and most trans-port models have trouble reproducing the shape of the singleratio especially for low energy particles less than 40 MeV/u.There is also a discrepancies in the single ratios at high nu-cleon energy. Data with better quality and extended to highenergy explore the region with higher sensitivity to the ef-fective mass [35]. While this analysis considers uncertain-ties in the predictions for the double and single ratios due touncertainties in values for 𝑆 , 𝐿 , 𝑚 ∗ 𝑠 and 𝑚 ∗ 𝑣 , other modeluncertainties in the functional form for the effective massterms [36, 37] are more difficult to define and therefore arenot explored in this paper. We note that if any effect leadsto a renormalization of the calculations of the order of 5%,the correlation between 𝑚 ∗ 𝑣 and 𝑚 ∗ 𝑠 is significantly alteredwhile the correlation between 𝑆 and 𝐿 and their associatedconstraints remain. This is partly because the effects of ef-fective mass splitting are much smaller than the effect of thedensity dependence of the symmetry energy. Obtaining ac-curate data at higher nucleon energies are the main goals ofa series of recent experiments [50]. 𝑆 (MeV) 𝐿 (MeV) 𝑚 ∗ 𝑠 ∕ 𝑚 𝑁 𝑚 ∗ 𝑣 ∕ 𝑚 𝑁 𝑓 𝐼
28 48.0 0.67 0.72 0.09830 61.8 0.65 0.69 0.08032 75.6 0.63 0.66 0.076
Table 2
Parameter value used to calculate the three symmetry energyrepresented by the three black curves in Fig. 4, that correspondto the three black crosses in Fig. 2.
In summary, we have presented new results for the singleratios of coalescence invariant neutron/proton spectra fromcentral
Sn+
Sn and
Sn+
Sn collisions at 120 MeV/u.We have shown that the Bayesian analyses can be used formultivariable analysis. However, the results from the anal-ysis is model dependent. Nonetheless, the analysis show astrong correlation between the values for 𝑆 and 𝐿 , whichis absent if the single ratio data is not included in the analy-sis. Together with the double ratio, these data also providesignificant constraints on the effective-mass splitting aroundhalf saturation density which is near the crust-core transi-tion density in neutron star [49]. This region also serves as abridge between the density regions investigated with nuclear r / r ) ( M e V ) r S ( DFT mass [7]A. Brown [8]Zhen Zhang [14]IAS [15] [13] D a HIC [17]Current work
Figure 4:
The three black lines correspond to the symmetryenergy 𝑆 ( 𝜌 ) versus the density for different values of 𝑆 and 𝐿 following the slope shown by the black crosses in Fig. 2.The open red star corresponds to the cross-over point of theblack lines shown in the left plot corresponding to the sensitivedensity 𝜌 𝑠 ∕ 𝜌 = 0 . +0 . . with 𝑆 ( 𝜌 𝑠 ) = 16 . . MeV. structure experiments ( ≈ 0 . 𝜌 ) and very-low-energy heavy-ion collisions ( ≤ . 𝜌 ) which is important for the questionof clustering at very low density that corresponds to densityrelevant for the neutrino sphere physics.We thank Scott Pratt, Earl Lawrence and Michael Grosskopffor many helpful discussions on the Bayesian analysis. Thisresearch is supported by the National Science Foundationunder Grant No. PHY-1565546. We acknowledge the com-putational resources provided by the Austin Advance Com-puter Center and the Institute for Cyber-Enabled Research atMichigan State University. References [1] B.P. Abbott et al. Phys. Rev. Lett.
Science
536 (2004).[3] A.W. Steiner et al. Phys. Rep.
325 (2005).[4] Pawel. Danielewicz et al. Science
Phys. Lett. B et al. Phys. Lett. B et al. Phys. Rev. C Phys. Rev. Lett. et al. Phys. Rev. C Progress in Particle and Nuclear Physics et al. Phys. Rev. C et al. Phys. Rev. Lett. at al. Phys. Rev. Lett. Phys. Lett. B
Nucl. Phys. A
147 (2017).[16] M.B. Tsang et al. Phys. Rev. Lett.
Phys. Rev. C et al. Phys. Rev. C Morfouace et al.:
Preprint submitted to Elsevier
Page 5 of 6
20] K.A. Brueckner
Phys. Rev. et al. Physics Reports et al. Eur. Phys. J. A
469 (2002)[23] F. Hofmann et al. Phys. Rev. C et al. Phys. Rev. C et. al Nucl. Phys. A et al. Phys. Rev. Lett. Annual Review of Nuclear and Particle Science
407 (2012).[28] H.A. Bethe
Rev. Mod. Phys.
801 (1990).[29] M. Baldo et al. Phys. Rev. C Nucl. Phys. A
511 (1976).[31] Yingxun Zhang et al. Phys. Lett. B et al. Phys. Lett. B et al. Phys. Rev. C et al. J. Phys. G et al. Phys. Rev. C et al. Phys. Rev. C et al. Phys. Rev. C et al. Nucl. Instr. and Meth. in Phys. Res. A et al. Nucl. Instr. and Meth. in Phys. Res. A et al. Nucl. Instr. and Meth. in Phys. Res. A Technical report,Japan Atomic Energy Agency (2006) [jAEAData/Code 2006-023].[42] Yingxun Zhang and Zhuxia Li
Phys. Rev. C Phys. Rev. C et al. Phys. Rev. C et al. Phys. Rev. C et al. J. Phys. G: Nucl. Part. Phys. Physics Report et al.
NSCL14030 & 15190
Morfouace et al.: