The influence of cluster emission and the symmetry energy on neutron-proton spectral double ratios

Transcription

1 The influence of cluster emission and the symmetry energy on neutron-roton sectral double ratios Yingxun Zhang 1,,3, P. Danielewicz 1,,4, M. Famiano 5, Zhuxia Li 3, W.G. Lynch 1,,4, M.B. Tsang 1,,4* 1 Joint Institute of Nuclear Astrohysics, Michigan State University, East Lansing, MI 4884, USA, National Suerconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 4884, USA, 3 China Institute of Atomic Energy, P.O. Box 75 (18), Beijing 1413, P.R. China, 4 Physics and Astronomy Deartment, Michigan State University, East Lansing, MI 4884, USA. 5 Physics Deartment, Western Michigan University, Kalamazoo, MI, USA. Abstract The emissions of neutrons, rotons and bound clusters from central 14 Sn+ 14 Sn and 11 Sn+ 11 Sn collisions are simulated using the Imroved Quantum Molecular Dynamics model for two different density-deendent symmetry-energy functions. The calculated neutron-roton sectral double ratios for these two systems are sensitive to the density deendence of the symmetry energy, consistent with revious work. Cluster emission increases the double ratios in the low energy region relative to values calculated in a coalescence-invariant aroach. To circumvent uncertainties in cluster roduction and secondary decays, it is imortant to have more accurate measurements of the neutron-roton ratios at higher energies in the center of mass system, where the influence of such effects is reduced. Corresonding author: tsang@nscl.msu.edu 1

2 Information about the Equation of State (EOS) of asymmetric nuclear matter can imrove our understanding of the radii and moments of inertia, maximum masses [1-3], crustal vibration frequencies [4], and cooling rates [3,5] of neutron stars, which are currently being investigated at ground-based and satellite observatories. Recent X-ray observations have been interreted as requiring an unusually reulsive equation of state for neutron matter [6]. It is imortant to determine whether such interretations are suorted by laboratory measurements. Measurements of isoscalar collective vibrations, collective flow and kaon roduction in energetic nucleus-nucleus collisions have constrained the equation of state for symmetric matter for densities ranging from normal saturation density to five times saturation density [7-9]. On the other hand, the extraolation of the EOS to neutron rich matter deends on the density deendence of the nuclear symmetry energy, for which there are comaratively few exerimental constraints [1]. Various robes in reaction exeriments have been found to be sensitive to the symmetry energy term of the equation of state. These include isoscaling [11-13], isosin diffusion [14], neutron to roton (n-) ratios (R n/ ) [15-17], neutron and roton flow [18], π + /π - ratios, and π + and π - flow [19, ]. In this aer, we focus on the ratio R n/ of re-equilibrium neutron over roton sectra. The ratio R n/ is enhanced by the reulsion of neutrons and attraction of rotons roduced by the symmetry mean field otential, which changes over time with the evolving density and asymmetry of the system [16,17,1]. Exerimentally, neutrons and rotons are usually measured utilizing two different detection systems with different energy calibrations and efficiencies. An accurate determination of absolute detection efficiencies for neutrons is rather difficult. For these reasons, the first comarison [15] of neutron to roton sectra used the double ratio, DR(n/) = R n/ (A)/ R n/ (B) = dm dm n ( A) / de ( A) / de c. m. c. m. dm ( B) / dec. m., dm ( B) / de constructed by measuring the energy sectra, dm/de C.M, of neutrons and rotons for two systems A and B characterized by different isosin asymmetries. The sensitivity of R n/ to the symmetry energy has been studied in the ast decade using the Boltzmann Uhling Uhlenbeck equation [16, 17, 1], which does not redict cluster formation. Conservation laws dictate that the inclusion of nucleons from α articles and other relatively symmetric clusters can significantly modify the values for R n/ []. Thus, it is imortant to examine n c. m.

3 the effect of clusters on n- ratios constructed in dynamical models. To understand this issue, we have erformed simulations with the Imroved Quantum Molecular Dynamics transort model [3, 4] using two equations of state that differ in their symmetry energy terms. Within the ImQMD model, nucleons are reresented by Gaussian waveackets. The mean fields acting on these waveackets are derived from an energy functional with the otential energy U that includes the full Skyrme otential energy with just the sin-orbit term omitted:. (1) In the above, is the Coulomb energy. The nuclear contributions are reresented in a local form with 3 U, md = u, mdd r () and the density otential of symmetry energy C s γ i ( ) δ is included in u η + 1 α β g g sur C = + + η + 1 η 8 / 3 sur, iso s γ i ( ) + [ ( n )] + ( ) δ + g τ 5 / 3 where, the asymmetry is defined as δ = ( ) ( + ) (3) n / n, and n and are the neutron and roton densities, resectively. The energy density associated with the mean-field momentum deendence is given by u md = 1 N, N = n, π 6 d 3 d 1 3 f r r [ ( )] 4 ( ) f ( ) 1.57 ln ( Δ ) N1 1 N, (4) r r where f are nucleon Wigner functions, Δ = 1. The energy is in MeV and momenta are in N MeV/c. The associated mean fields acting on the waveackets can be found in Ref. [5]. In this work, the values of α=-356 MeV, β=33 MeV and η=7/6 are emloyed, corresonding to an isoscaler comressibility constant of K= MeV. Other arameter values are g sur =19.47 MeVfm, g suriso = MeVfm, C s =35.19 MeV, and g τ = MeV. These calculations use isosin-deendent in-medium nucleon-nucleon scattering cross sections in the collision term and emloy Pauli blocking effects that are described in [3,4, 6]. Cluster yields are calculated by means of the coalescence model, widely used in QMD calculations, in which articles with relative momenta. 3

4 smaller than and relative distances smaller than R are coalesced into one cluster. In the resent P work, the values of R = 3. 5 fm and P = 5MeV / c are emloyed. As a consequence of the above assumtions, the symmetry energy er nucleon emloyed in the simulations is a sum of kinetic and interaction terms: E sym ( ) / A = 1 3 h m / 3 3π / 3 Cs + γ i, (5) where m is the nucleon mass. The symmetry energy for the resent ImQMD calculations (solid lines) is lotted as a function of density in Figure 1 for γ i =.5 and. The symmetry energy values increase with decreasing γ i at subsaturation densities while the oosite is true at surasaturation densities. At any density, higher values of symmetry energy tend to drive systems more raidly towards isosin symmetry, resulting in higher values of R n/ for neutron rich systems. We have erformed calculations of central collisions at an imact arameter of b= fm and an incident energy of 5 MeV er nucleon for two systems: A= 14 Sn+ 14 Sn and B= 11 Sn+ 11 Sn. In central collisions at this incident energy, articles are mostly emitted when the system exands and breaks u at sub-saturation densities. While the ratios of total emitted neutron over total emitted roton numbers are fixed by conservation laws, the imortant symmetry energy information is contained in the re-equilibrium emission of nucleons from the early asymmetric system, which dominates at high center of mass (C.M.) energies at θ C.M. 9. The right and left anels of Figure show R n/ (14) and R n/ (11) for the 14 Sn+ 14 Sn and 11 Sn+ 11 Sn collisions, resectively, as a function of the C.M. energy of nucleons emitted at 7 θ C.M. 11. The oen and solid symbols reresent R n/ values calculated using the softer (γ i =.5) and stiffer (γ i =) density-deendent symmetry terms, resectively. As exected, more re-equilibrium neutrons get emitted from the neutron rich 14 Sn+ 14 Sn system. Relatively more re-equilibrium neutrons are also emitted in the calculations with the softer density-deendent symmetry energy because the emission occurs redominantly at sub-saturation density. The uncertainties for these calculations in Figure and in the subsequent figures are statistical. To facilitate comarisons to existing and future transort model calculations, we restrict our calculations to b= fm in this letter. (Calculations with exerimental multilicity gates imosed on imact arameter averaged events yield results which are consistent with those for b= fm, within 4

5 statistical uncertainties.) While the various uncertainties of the calculations are too large to allow for rigorous comarisons with data, (see the discussion below), some comarisons with data [15] will be shown to rovide context for the discussion. The shaded regions in the left anel of Figure 3 reresent the range, determined by uncertainties in the simulations, of redicted double ratios DR(n/)=R n/ (14)/ R n/ (11), as a function of the nucleon center of mass energy, for two different density deendencies of the symmetry otential: for γ i =.5 (uer shaded region) and for γ i = (lower shaded region). All calculated results exceed the no-sensitivity limit of DR(n/)=N A Z B /( N B Z A )=1. (dotted lines) given by conservation laws. As exected, the double ratios DR(n/) are higher for γ i =.5, which yields weaker deendence of symmetry energy on density. The measured [15] double ratios DR(n/) are lotted as solid stars for comarison. Both calculations yield results, which increase in values with kinetic energy as observed in the data. To examine the influence of sequential decays, we have simulated decays of fragments created in the collisions using the Gemini code [7]. Sequential decays mainly enhance the single ratios for low energy rotons and neutrons, but such effects are largely suressed in the double ratios. This underscores the utility of double ratios for comarisons of calculated and measured neutron and roton sectra at energies where the secondary decay contributions may be small but uncertain. For models that do not include clusters such as the BUU calculations discussed below, coalescence-invariant DR(n/) are used. These double ratios are constructed by including all neutrons and rotons emitted at a given velocity, regardless of whether they are emitted free or within a cluster. The data, shown as solid stars in the right anel of Figure 3, increase monotonically from the no-sensitivity limit DR(n/) 1. and attain values at large E C.M. consistent with those shown in the left anel of Figure 3 for free nucleons. The corresonding coalescence-invariant n- double ratios using the fragments roduced in the ImQMD simulations are lotted as shaded regions in the right anel in Figure 3. Here, the measured fragments with Z> mainly contribute to the low energy sectra and do not affect the high-energy sectra very much. The redicted ImQMD coalescence-invariant double-ratios for γ i = change only slightly at low E C.M.. On the other hand, the coalescence-invariant double-ratios for γ i =.5 decrease by nearly a factor of two at low E C.M. and aroach the no-sensitivity limit of DR(n/) 1. as E C.M. decreases. In both cases, the ImQMD calculations at E C.M. /A>4 MeV retain sensitivity to the density deendence. Over the whole energy 5

6 range for both free and coalescence-invariant DR(n/), the data seem closer to the γ i =.5 calculation but the uncertainties in the measured values are rather large at E C.M. >4 MeV, where the effects of cluster emission and secondary decays turn out to be small, as discussed below. More accurate measurements would be needed to distinguish between the γ i =.5 and γ i = calculations; such measurements should be feasible with a well-designed and dedicated setu. The revious theoretical studies of R n/ utilized two BUU models, BUU97 [16] and IBUU4 [17] that make no redictions for comlex fragment formation. The density deendencies of the symmetry energies emloyed in IBUU4 (x= and x=-1) and BUU97 (F1 and F3) are shown with the dot-dashed and dotted lines in the right anel of Fig. 1. The symmetry energy density deendence of F1 is very similar to that for x=-1 and the symmetry energy density deendence of F3 is softer than that for x=. More imortantly, the IBUU4 code includes: mean field momentum deendencies consistent with the Lane otential, in-medium nucleon-nucleon cross-sections either coinciding with those in free sace or incororating density-deendent modifications that are not included in BUU97. The ublished BUU97 calculations were erformed over a range of imact arameters of b=-5 fm [16] while the IBUU4 calculations were carried out at b= fm [17], as the resent calculations. The solid and dashed lines in Figure 4 reresent the latest IBUU4 calculations with arameters (x= and x=-1) from ref. [17]. Those lines bracket the isosin diffusion data of ref. [14]. The shaded regions reresent redictions from BUU97 calculations erformed in ref. [16] for two symmetry energy functions, F1 and F3. Irresectively of the large uncertainties for the BUU97 calculations, it is aarent that the BUU97 results are well in excess of the nosensitivity limit of DR(n/)=1.. Furthermore, it is aarent that far more sensitivity to the symmetry energy is observed for the BUU97 calculations than for the IBUU4 calculations. We do not know the origins of these differences. At high nucleon energies, the results of the resent ImQMD calculations for γ i =.5 are similar to the results of momentum indeendent BUU97 calculations from ref. [16] for the iso-soft (F3) symmetry energy (uer shaded region in Fig. 4). However, the uncertainties in the BUU97 calculations are too large to allow making definitive conclusions. Lacking clusters, the BUU results must be comared to coalescence-invariant n- double ratio, DR(n/), shown as solid stars in Figure 4. The stiffer density-deendent (F1) BUU97 results overla the data at E C.M. /A<4 MeV while the softer density-deendent (F3) results overla the data 6

7 at higher energies. In contrast, the IBUU4 calculations lie far below the data near the nosensitivity limit. Calculations with IBUU4 and a weaker density deendence than (x=) might be somewhat larger, but the other differences between the transort quantities used in the calculations, such as the cross sections or the effective masses, might contribute more to the differences between the IBUU4 and BUU97 calculations, than do the differences in the symmetry energies. In both calculations, the results from softer density deendence on symmetry energy (x= and F3) increase with E C.M., but results from stiffer density deendence (x=-1 and F1) do not. The emission of T= alha clusters enhances the asymmetry of the free nucleons [], and it seems likely that modeling light clusters would lead to larger values for DR(n/) in either BUU aroach. In any case, the resent ImQMD calculations aear to be caable (but both BUU97 and IBUU4 are incaable) of reroducing the energy deendence of the double ratios from low energies, where clusters dominate, to high energies, E/A>4 MeV, where the cluster yields can be neglected. Until this is understood and more accurate data are obtained, the discreancy with IBUU4 results raises concerns about the extraction of constraints on the symmetry energy from the isosin diffusion data [8]. Additional studies are needed to resolve these issues. To better understand how the sensitivity of DR(n/) to the symmetry energy changes with incident energy, we have extracted the calculated excitation function of the double ratios in Figure 5 for high-energy neutrons and rotons at incident energies of 35 to 15 MeV er nucleon. Consistent with the forgoing analyses, we show values for DR(n/) in Fig. 5 for high energy nucleons emitted at 7 θ C.M. 11 with E C.M. >4 MeV. At all incident energies, the double ratios are larger for γ i =.5 than for γ i =; the largest difference is found at E/A=5 MeV. The values for DR(n/) for both γ i =.5 and γ i = and their difference decrease with increasing incident energy. As the imortance of collisions and the mean field momentum deendence increases with incident energy, the incident energy deendence of DR(n/) could rovide a useful test of the descrition of other transort quantities such as effective masses of neutron and rotons and the isosin deendence of the inmedium cross-sections. In summary, we have erformed ImQMD transort equation simulations for the systems 14 Sn+ 14 Sn and 11 Sn+ 11 Sn. Cluster roduction modifies the sectral double ratios at E C.M <4 MeV. The ImQMD model relicates the difference between sectral ratios obtained for free nucleons and those obtained from a coalescence-invariant aroach. It also redicts sectral double 7

8 ratios comarable to the data. However, both the data and calculations at high center of mass energies are not sufficiently accurate to lace significant constraints on the density deendence of the symmetry energy. Significant differences are observed between the ImQMD and two BUU models, which need to be resolved before definitive extractions of the density deendence of the symmetry energy from such calculations can be made. This work has been suorted by the U.S. National Science Foundation under Grants PHY , PHY-667, PHY (Joint Institute for Nuclear Astrohysics), the High Performance Comuting Center (HPCC) at Michigan State University, the Chinese National Science Foundation of China under Grants , , 1353, and the Major State Basic Research develoment rogram under contract No. G774. References: [1] J.M. Lattimer, M. Prakash, A. J. 55 (1) 46. [] J.M. Lattimer, M. Prakash, Science 34 (4) 536. [3] A.W. Steiner et al., Phys. Re. 411 (5) 35. [4] Anna L. Watts, and Tod E. Strohmayer, Astrohy. J637 (6) L117. [5] D.G. Yakovlev and C.J. Pethick, Annu. Rev. Astron. Astrohys. 4 (4) 169. [6] F. Özel, Nature 441 (6) [7] P. Danielewicz, R. Lacey, W.G. Lynch, Science 98 () 159. [8] C.Fuch and H.Wolter, Eur.Phys.J.A3 (6) 5. [9] U. Garg, Nucl. Phys. A731 (4) 3 and references therein. [1] B.A. Brown, Phys. Rev. C 43 (1991) R1513. [11] H.S. Xu, et. al., Phys. Rev. Lett. 85 () 716. [1] M.B. Tsang, et al., Phys. Rev. Lett. 86 (1) 53. [13] D.V. Shetty, et al., Phys.Rev. C7 (4) 1161 [14] M.B. Tsang, et al., Phys. Rev. Lett. 9, 671 (4) [15] M.A.Famiano, T.Liu, W.G.Lynch, et al., Phys.Rev.Lett.97 (6) 571. [16] B.A. Li, C.M. Ko, Z. Ren, Phys. Rev. Lett. 78 (1997) [17] Bao-An Li, Lie-Wen Chen, Gao-Chan Yong and Wei Zuo, Phys.Lett.B634 (6) 378. [18] B.A.Li, Phys. Rev. Lett. 85 () 41. [19] B.A. Li, Nucl. Phys. A 734 (4) 593c. [] Gao-Chan Yong, Bao-An Li, Lie-Wen Chen, Phys.Rev.C73 (6) [1] Bao-An Li, Pawel Danielewicz and William G. Lynch, Phys. Rev. C71 (5) [] L. G. Sobotka, J. F. Demsey, and R. J. Charity and P. Danielewicz, Phys. Rev. C 55 (1997) 19. [3] Yingxun Zhang, Zhuxia Li, Phys. Rev. C 71 (5) 464. [4] Yingxun Zhang, Zhuxia Li, Phys. Rev. C 74 (6) 146. [5] J.Aichelin, A.Rosenhauer, G.Peilert, H.Stocker,W.Greiner, Phys. Rev. Lett. 58 (1987) 196. [6] Yingxun Zhang, Zhuxia Li, P.Danielewicz, Phys.Rev.C75 (7) [7] R. J. Charity et al., Nucl. Phys. A 483 (1988) 371. [8] Lie-Wen Chen, Che Ming Ko and Bao-An Li, Phys. Rev. Lett. 94 (5)

9 Fig.1: (Color online) Symmetry energy er nucleon lotted as a function of density, The dot-dashed lines labeled x = and x = -1 in both anels reresent the symmetry energies used in IBUU4 calculations [17]. The solid lines labeled γ i =.5, and γ i =. in the left anel reresent the symmetry energies used in the current ImQMD simulations as defined in Equation 5l. The dotted lines in the right anel labeled F1 and F3 reresent the symmetry energies used in BUU97 calculations [16]. 9

10 Fig.: (Color online) The ratio of neutron to roton yields for the 11 Sn+ 11 Sn reaction (left anel) and the 14 Sn+ 14 Sn reaction (right anel) as a function of the kinetic energy, for free nucleons emitted at 7 θ C.M 11. The oen (solid) symbols reresent results from simulations using γ i =.5 (γ i =.) as defined in Equation 5. 1

11 Fig.3: (Color online) The free neutron-roton double-ratio (left anel), and the coalescenceinvariant neutron-roton double-ratios (right anel) lotted as a function of kinetic energy of the nucleons. The shaded regions reresent calculated results from the ImQMD simulations at b= fm. More details are given in the text. The data (solid star oints) are taken from Ref [15]. 11

12 Fig.4: (Color online) Coalescence-invariant neutron-roton double ratios lotted as a function of kinetic energy of the nucleons. The shaded regions reresent calculations from the BUU97 simulations taken from ref [16]. The solid and dashed lines reresent the results of IBUU4 calculations at b= fm, from ref. [17]. The solid stars reresent data of Ref [15]. 1

13 Fig.5: (Color online) Excitation function for neutron- roton double ratios, constructed from high energy (E C.M. >4MeV) neutrons and rotons, for γ i =.5 (oen symbols) and γ i =. (solid symbols). 13