- Email: [email protected]
Transverse flows in charge asymmetric collisions
19 August 1999
Physics Letters B 461 Ž1999. 9–14
Transverse flows in charge asymmetric collisions L. Scalone, M. Colonna, M. Di Toro Laboratorio Nazionale del Sud, and UniÕersity of Catania, Via S. Sofia 44, I-95123 Catania, Italy Received 2 April 1999 Editor: J.-P. Blaizot
Abstract Collective flows in heavy ion collisions are shown to be largely dependent on the symmetry term of the nuclear Equation of State ŽEOS.. A comparison with data leads to a clear evidence for a strong increase of the repulsion of the symmetry term above normal density. This will also imply noticeable differences in the neutron and proton components to be seen in the transverse flows of light ions. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 21.30.q Fe; 25.70.-z; 25.75.Ld; 26.60.q c Keywords: Asymmetric nuclear matter; Collective flows; Isospin
The availability of exotic Žradioactive. beams has driven a large interest on nuclear structure studies of b-unstable nuclei. However, it is clear that essential complementary information will come from charge asymmetry effects on non-equilibrium nuclear dynamics. In particular we can show that it would be possible to extract some unique information on the symmetry term of the nuclear EOS in regions away from normal density. Quite stimulating predictions exist on new phases of asymmetric nuclear matter ŽNM. that eventually could be reached during heavy ion reaction dynamics with radioactive beams. The onset of coupled mechanical and chemical instabilities is envisaged w1–4x, that should lead to clear experimental signatures. All these effects are expected to be very sensitive to the behaviour of the symmetry potential in low density regions. In this letter we shall focus the
attention on collective flows in medium energy heavy ion collisions showing the dependence on the symmetry term of the EOS, this time in high density regions. In particular we will show how contributions from mean field and nucleon-nucleon ŽNN. cross sections can be separated allowing a direct insight into the EOS of asymmetric NM. We can put then some quite general experimental constraints on the theoretical predictions used in astrophysical contexts w5x, where such information is essential in understanding the lives of neutron stars w6–8x. In the last years some first data have appeared on isospin effects in reaction dynamics, with a few theoretical analyses. See the recent reviews w9–12x. Although the data are mostly of inclusive type and the theoretical studies are not always focussing the effect of different charge symmetry terms, it emerges quite clearly a noticeable dependence of the reaction mechanisms on charge asymmetry.
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 8 3 5 - 7
L. Scalone et al.r Physics Letters B 461 (1999) 9–14
10
In the nuclear equation of state we have a symmetry term in the form: E A
Ž r,I. s
E A
Ž r. q
Esym A
Ž r. I2 ,
Ž 1.
with I s Ž N y Z .rA asymmetry parameter, given by kinetic and potential contributions: Esym A
Ž r. s
eF Ž r . 3
q
CŽ r . 2 r0
r.
Ž 2.
In Fig. 1a we report the density dependence of the potential symmetry contribution Žsecond term of Eq. Ž1.. for three different Skyrme effective interactions, SIII, SKM ) ŽRef. w13x. and Skyrme–Bonn, obtained to reproduce the EOS of microscopic nuclear matter calculations using the Bonn A potential w3x. While all curves obviously cross at normal density, quite large differences are present in low density and particularly in high density regions. Collective flows in heavy ion collisions give important information on the dynamical response of excited nuclear matter w14–16x, and references therein. Particularly interesting for isospin effects is the directed transverse flow, measured on the reaction plane, which provides a direct signature of the expected transition from mean field to hydro-dynamics in the evolution of dissipative heavy ion collisions at intermediate energies w9,16x. Since the transition takes place from a delicate balance between EOS properties Žmean field and compressibil-
ity.. Coulomb repulsion and nucleon-nucleon cross sections in the medium, we will see a noticeable isospin dependence. In particular we expect a large sensitivity to the form of the symmetry term in regions above normal density Žup to 1.5 r 0 . which are explored in this energy range Žfrom 40 to 120 A MeV.. We solve microscopic transport equations w14,17– 22x where asymmetry effects are suitably accounted for. A density dependent symmetry term is used also in the ground state construction of the initial conditions, isospin effects on nucleon cross section and Pauli blocking are consistently evaluated. Medium effects on NN cross sections are included from the work of Refs. w23x, based on a Dirac–Brueckner approach with the Bonn NN interaction. In general we have a reduction with respect to the free case, partially due to the effective mass partially to a weakening of the force. The latter is particularly clear for the np channel due to the disappearance of the deuteron correlation at high density. This will be important for the isospin dependence of two-body collisions and the related influence on flows. In the energy range of interest here it is essential an appropriate treatment of the momentum dependence of the mean field: with local fields the transition energy can be reproduced only with a non realistic high stiffness w16x. In our calculation we use the momentum dependence suggested by Gale– Bertsch–DasGupta ŽGBD. in a Skyrme-like effective mean field w24,25x. For a large dynamical range it
Fig. 1. Ža. Density dependence of the potential symmetry term for different Skyrme effective forces. Žb. Symmetry contribution to the mean field at I s 1r3 for neutrons Župper line. and protons Žlower line.: dashed lines asy–stiff, solid lines asy–soft.
L. Scalone et al.r Physics Letters B 461 (1999) 9–14
gives results quite similar to non-local forces of Gogny type w26x that well describe collective flow properties in symmetric systems with compressibilities around K s 220 MeV w25,27,28x. The transport equations are solved following a test particle evolution on a lattice w22,28x. In order to have a better overall control of this complex calculation and to optimize the parallel structure a new Object-Oriented code has been written in C q q language, the code DYN q q w29x. We use a general Skyrme-like form for the charge dependent part of the mean field potential: Uq s C
ri
ž / r0
tq q
1 E C r i2 2 Er s r 0
,
Ž 3.
where r s ' rn q r p and r i ' rn y r p are respectively isoscalar and isovector densities, and q s n, p, tq s q1 Ž q s n., y1 Ž q s p ..
11
In order to study the sensitivity of flow results to the density dependence of the symmetry term we will follow two rather extreme choices for C Ž r .: C s 32 MeV Ž Sk–Bonn of Fig. 1 . : SIII parameterization of Fig. 1:
asy–stiff choice asy–soft choice
We stress again that in this way we force the symmetric part of the EOS to be exactly the same in order to disentangle dynamical symmetry term effects. The corresponding symmetry contribution to the mean field, Eq. Ž3., has a density dependence shown in Fig. 1b for a N s 2 Z system Ž I s 1r3.: in a region just above normal density the field ‘‘seen’’ by neutrons and protons in the two cases is very different. As a consequence we expect to have two main direct effects on transverse flows: Ži. Proton-rich and neutron-rich nuclei will show different proton flows.
Fig. 2. Energy dependence of flows at bred s 0.45: full circles: Fe–Fe protons; open circles: Ni–Ni protons; squares: Fe–Fe neutrons. Ža. asy–stiff, Žb. asy–soft. Žc., Žd.: the same for s NN s 2 fm2 , without isospin dependence. The full diamonds in Ža. represent the balance energy data taken from Ref. w9x for the Fe–Fe and Ni–Ni systems.
12
L. Scalone et al.r Physics Letters B 461 (1999) 9–14
The difference will be much reduced for a symmetry term of asy–soft type. Žii. For neutron-rich nuclei proton and neutron flows will be also quite different in the case of a asy–stiff symmetry term. The effect will be particularly enhanced for semi-central collisions where we can get a very asymmetric piece of nuclear matter in the interacting zone. We have studied the p,n transverse flows for the collisions 58 Fe q58 Fe Žn-rich. and 58 Ni q58 Ni Žprich. in the energy range 55 to 105 A MeV, where nice data are existing for Z s 1,2,3 flows w9x and some calculations have been performed for proton balance energies w10,30,31x. For a given energy and reduced impact parameter, bred s brbmax , we evaluate the freeze-out momentum distributions of protons and neutrons and we construct the mean transverse component as a function of the parallel velocity Žrapidity.. In order to eliminate numerical fluctuations we average over twenty events for each fixed
initialization. The flow parameter is given by the slope at mid-rapidity. In Fig. 2a, 2b, we report the energy dependence of proton flows in Fe–Fe and Ni–Ni at fixed bred s 0.45 with the two choices of the symmetry term. In the asy–stiff case, Fig. 2a, we have a larger balance energy, of about 5 MeV, for the neutron rich system Fe–Fe. The values are in quite good agreement with the data of Ref. w9x Žthe full diamonds in Fig. 2a.. We show also the behaviour of neutron flows in the Fe–Fe case. As expected from the more repulsive mean field Žsee Fig. 1b. we have, with respect to protons, a smaller flow below the balance energy and a larger one above. We remark the increase of the difference at high energy: we are exploring higher densities in the interacting region and so we expect larger mean field differences between neutrons and protons. Everything will disappear in the asy–soft choice: in Fig. 2b we report the corresponding re-
Fig. 3. Centrality dependence of flows at beam energy 55 A MeV: stars: Fe–Fe protons; open circles: Ni–Ni protons; diamonds: Fe–Fe neutrons; squares: Ni–Ni neutrons. Experimental points: full circles ŽFe protons., full squares ŽNi protons.. Ža. asy–stiff, Žb. asy–soft. Žc., Žd.: the same for s NN s 2 fm2 , without isospin dependence.
L. Scalone et al.r Physics Letters B 461 (1999) 9–14
13
Fig. 4. Mean transverse momentum in the reaction plane vs. reduced rapidity for light clusters in the collision Fe–Fe at beam energy 55 A MeV for a semiperipheral impact parameter bred s 0.62.
sults. In particular we get, within the uncertainties of small flow evaluation, the same balance energy for proton flows in the two systems, in disagreement with experiments. A detailed comparison of the flow dependence on the centrality of the collision is presented in Fig. 3a, 3b, at a beam energy 55 A MeV, i.e. below the balance energy. In the asy–stiff case the Fe–p flows are well above the Ni–p ones, in good agreement with the experimental points also reported in the figure 1. To remark the asymmetry of the effect with respect to bred s 0.5, more pronounced for more peripheral collisions: this is a nice evidence of the larger asymmetries we can reach in a ‘‘skin-skin’’ interaction of neutron rich ions. We finally stress the difference between proton and neutron flows in the Fe–Fe case, completely absent for Ni–Ni collisions. Again everything is disappearing for asy–soft calculations, including the agreement with data. In order to evaluate the relative importance of isospin effects on mean field and NN cross sections
1
We have deduced the experimental proton data just averaging over the results for Zs1,2,3 ions of Ref. w9x. We expect our ‘‘single’’ proton results to be systematically above these extracted data just due to the predicted reduced flow of the neutron component.
we have repeated the calculations with a constant Žno-isospin-dependent, no-density-dependent, isotropic. total NN cross section of 20 mb Žsee Figs. 2c, 2d, and 3c, 3d.. From Fig. 2c we see that the isospin dependence of the mean field is able to keep the balance energy difference between proton flows in Fe and Ni, although in a reduced way. Moreover the impact parameter variation is not anymore in good agreement with data Žsee Fig. 3c.. As expected both isospin dependences are important, on cross sections and on mean fields, but our simulations, and also the data, seem to show a quite noticeable sensitivity to the symmetry term of the EOS. From this analysis we can draw some quite clear conclusions: Ži. Collective transverse proton flows for charge asymmetric systems are very sensitive to the density dependence of the symmetry term of the nuclear EOS; Žii. All present data seem to indicate the need of a quite stiff symmetry term for densities up to 1.5 r 0 ; Žiii. We expect to have a clear difference in neutron and proton flows, in presence of an interacting asymmetric piece of NM. This should be seen in a detailed study of the flows of different isotopes of light ions. The latter point is particularly stimulating since it could allow to disentangle among various symmetry terms just from the flow analysis at one beam energy, below or above the balance value.
14
L. Scalone et al.r Physics Letters B 461 (1999) 9–14
In order to show a quantitative evaluation of the effect we have performed a light ion flow study just using a phase-space coalescence approach to form clusters at the freeze-out time. We have considered a mean Boltzmann–Vlasov trajectory as an average over NTP ensembles, where NTP is the number of test particles per nucleon. In each ensemble, randomly chosen, we have introduced a clustering procedure in r- and p-space with parameters D R s 2.4 fm and D P s 200 MeVrc. We get an exponential mass Žcharge. distribution which reproduces a Poisson-like behaviour of purely combinatorial nature. This is not surprising since few body correlations are not explicitly included in our mean field approach. Such estimation of light ion yields can be reliable in a fast expansion scenario of the participant zone. From 200 events of Fe–Fe, 55 A MeV at bred s 0.62 Žasy–stiff case. we can accumulate enough statistics to perform a flow analysis of light clusters, 3 H,3 He, presented in Fig. 4. We can estimate a 20% larger flow Žnegative. for 3 He ions with respect to 3 H, just opposite to what expected from Coulomb effects. This is a clear indication of the contribution of a much reduced neutron flow in the case of a repulsive symmetry term Žsee Fig. 1b.. The effect is absent within the asy–soft choice. Finally, we would like to stress that the effects discussed here can be clearly seen also with ‘‘exotic’’ but not radioactive beams. Indeed all the flow data analysed here are from stable beams. The results presented here are in a sense complementary to other predictions of a quite large sensitivity of the main dissipative reaction mechanisms to the symmetry term w32x. A possibility is emerging of obtaining in terrestrial accelerator laboratories important information on the symmetry term of large astrophysical interest. It appears essential to have good charge asymmetric beams available at intermediate energies and to perform more exclusive experiments.
References w1x H. Mueller, B.D. Serot, Phys. Rev. C 52 Ž1995. 2072. w2x Bao-An Li, C.M. Ko, Nucl. Phys. A 618 Ž1997. 498.
w3x M. Colonna, M. Di Toro, A. Larionov, Phys. Lett. B 428 Ž1998. 1. w4x V. Baran, M. Colonna, A. Larionov, M. Di Toro, Nucl. Phys. A 632 Ž1998. 287. w5x I. Bombaci, T.T.S. Kuo, U. Lombardo, Phys. Rep. 242 Ž1994. 165 I. Bombaci, Phys. Rev. C 55 Ž1997. 1 M. Prakash et al., Phys. Rep. 280 Ž1997. 1. w6x I.M. Irvine, Neutron Stars Oxford Univ. Press 1978. w7x C.J. Pethick, D.G. Ravenhall in The Lives of Neutron Stars Ed.s M.A. Alpar et al., NATO ASI Series C Vol.450 Ž1995., pp.59-70. w8x V.A. Urpin, same Proceedings, pp.163-166. w9x G. Westfall, Nucl. Phys. A 630 Ž1998. 27c. w10x Bao-An li, C.M. Ko, W. Bauer, Int. J. Mod. Phys. E7 Ž1998. 147. w11x M. Di Toro et al., Proc. Int. Winter Meeting on Nuclear Physics, Ed. I. Iori, Univ. Milano 1998 p.319. w12x M. Di Toro et al., Isospin Dynamics at Medium Energies, Progr. Part. Nucl. Phys. Vol. 42 Ž1999. ŽPergamon., in press. w13x H. Krivine, J. Treiner, O. Bohigas, Nucl. Phys. A 336 Ž1980. 155. w14x H. Stoecker, W. Greiner, Phys. Rep. 137 Ž1986. 277. w15x H.H. Gutbrod, A.M. Poskanzer, H.G. Ritter, Rep. Progr. Phys. 52 Ž1989. 1267. w16x S. Das Gupta, G.W. Westfall, Physics Today 46 Ž1993. 34. w17x G.F. Bertsch, S. Das Gupta, Phys. Rep. 160 Ž1988. 189. w18x W. Bauer, C.K. Gelbke, S. Pratt, Ann. Rev. Nucl. Part. Sci. 42 Ž1992. 77. w19x Ch. Gregoire et al., Nucl. Phys. A 465 Ž1987. 315. w20x A. Bonasera et al., Phys. Rep. 244 Ž1994. 1. w21x A. Bonasera, G.F. Burgio, M. DiToro, Phys. Lett. B 221 Ž1989. 233, A. Bonasera, F. Gulminelli, Phys. Lett. B 259 Ž1991. 399. w22x TWINGO code: A. Guarnera Ph.D. Thesis, Univ. Caen, July 1996. w23x G.Q. Li, R. Machleidt, Phys. Rev. C 48 Ž1993. 1702 and Phys. Rev. C 49 Ž1994. 566. w24x C. Gale, G.F. Bertsch, S. Das Gupta, Phys. Rev. C 35 Ž1987. 1666. w25x M. Prakash, T.T.S. Kuo, S. Das Gupta, Phys. Rev. C 37 Ž1988. 2253. w26x J. Decharge, D. Gogny, Phys. Rev. C 21 Ž1980. 1568, and references therein. w27x V.de La Mota, F. Sebille, M. Farine, B. Remaud, P. Schuck, Phys. Rev. C 46 Ž1992. 677. w28x V. Greco, A. Guarnera, M. Colonna, M. DiToro, Dynamical effects of momentum dependence of the nuclear mean field in medium energy heavy ion collisions, Phys. Rev. C, in press; V. Greco, Master Thesis, Univ. of Catania 1997. w29x DYNqq Code: L. Scalone, Master Thesis, Univ. of Catania Dec. 1998. w30x R. Pak et al. Phys. Rev. Lett. 78 Ž1997. 1026. w31x E. Daffin, W. Bauer, Effects of isospin asymmetry and in medium corrections on balance energy Preprint MSUCL 111, Aug.98. w32x M. Colonna, M. DiToro, G. Fabbri, S. Maccarone, Phys. Rev. C 57 Ž1998. 1410.
Physics Letters B 461 Ž1999. 9–14
Transverse flows in charge asymmetric collisions L. Scalone, M. Colonna, M. Di Toro Laboratorio Nazionale del Sud, and UniÕersity of Catania, Via S. Sofia 44, I-95123 Catania, Italy Received 2 April 1999 Editor: J.-P. Blaizot
Abstract Collective flows in heavy ion collisions are shown to be largely dependent on the symmetry term of the nuclear Equation of State ŽEOS.. A comparison with data leads to a clear evidence for a strong increase of the repulsion of the symmetry term above normal density. This will also imply noticeable differences in the neutron and proton components to be seen in the transverse flows of light ions. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 21.30.q Fe; 25.70.-z; 25.75.Ld; 26.60.q c Keywords: Asymmetric nuclear matter; Collective flows; Isospin
The availability of exotic Žradioactive. beams has driven a large interest on nuclear structure studies of b-unstable nuclei. However, it is clear that essential complementary information will come from charge asymmetry effects on non-equilibrium nuclear dynamics. In particular we can show that it would be possible to extract some unique information on the symmetry term of the nuclear EOS in regions away from normal density. Quite stimulating predictions exist on new phases of asymmetric nuclear matter ŽNM. that eventually could be reached during heavy ion reaction dynamics with radioactive beams. The onset of coupled mechanical and chemical instabilities is envisaged w1–4x, that should lead to clear experimental signatures. All these effects are expected to be very sensitive to the behaviour of the symmetry potential in low density regions. In this letter we shall focus the
attention on collective flows in medium energy heavy ion collisions showing the dependence on the symmetry term of the EOS, this time in high density regions. In particular we will show how contributions from mean field and nucleon-nucleon ŽNN. cross sections can be separated allowing a direct insight into the EOS of asymmetric NM. We can put then some quite general experimental constraints on the theoretical predictions used in astrophysical contexts w5x, where such information is essential in understanding the lives of neutron stars w6–8x. In the last years some first data have appeared on isospin effects in reaction dynamics, with a few theoretical analyses. See the recent reviews w9–12x. Although the data are mostly of inclusive type and the theoretical studies are not always focussing the effect of different charge symmetry terms, it emerges quite clearly a noticeable dependence of the reaction mechanisms on charge asymmetry.
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 8 3 5 - 7
L. Scalone et al.r Physics Letters B 461 (1999) 9–14
10
In the nuclear equation of state we have a symmetry term in the form: E A
Ž r,I. s
E A
Ž r. q
Esym A
Ž r. I2 ,
Ž 1.
with I s Ž N y Z .rA asymmetry parameter, given by kinetic and potential contributions: Esym A
Ž r. s
eF Ž r . 3
q
CŽ r . 2 r0
r.
Ž 2.
In Fig. 1a we report the density dependence of the potential symmetry contribution Žsecond term of Eq. Ž1.. for three different Skyrme effective interactions, SIII, SKM ) ŽRef. w13x. and Skyrme–Bonn, obtained to reproduce the EOS of microscopic nuclear matter calculations using the Bonn A potential w3x. While all curves obviously cross at normal density, quite large differences are present in low density and particularly in high density regions. Collective flows in heavy ion collisions give important information on the dynamical response of excited nuclear matter w14–16x, and references therein. Particularly interesting for isospin effects is the directed transverse flow, measured on the reaction plane, which provides a direct signature of the expected transition from mean field to hydro-dynamics in the evolution of dissipative heavy ion collisions at intermediate energies w9,16x. Since the transition takes place from a delicate balance between EOS properties Žmean field and compressibil-
ity.. Coulomb repulsion and nucleon-nucleon cross sections in the medium, we will see a noticeable isospin dependence. In particular we expect a large sensitivity to the form of the symmetry term in regions above normal density Žup to 1.5 r 0 . which are explored in this energy range Žfrom 40 to 120 A MeV.. We solve microscopic transport equations w14,17– 22x where asymmetry effects are suitably accounted for. A density dependent symmetry term is used also in the ground state construction of the initial conditions, isospin effects on nucleon cross section and Pauli blocking are consistently evaluated. Medium effects on NN cross sections are included from the work of Refs. w23x, based on a Dirac–Brueckner approach with the Bonn NN interaction. In general we have a reduction with respect to the free case, partially due to the effective mass partially to a weakening of the force. The latter is particularly clear for the np channel due to the disappearance of the deuteron correlation at high density. This will be important for the isospin dependence of two-body collisions and the related influence on flows. In the energy range of interest here it is essential an appropriate treatment of the momentum dependence of the mean field: with local fields the transition energy can be reproduced only with a non realistic high stiffness w16x. In our calculation we use the momentum dependence suggested by Gale– Bertsch–DasGupta ŽGBD. in a Skyrme-like effective mean field w24,25x. For a large dynamical range it
Fig. 1. Ža. Density dependence of the potential symmetry term for different Skyrme effective forces. Žb. Symmetry contribution to the mean field at I s 1r3 for neutrons Župper line. and protons Žlower line.: dashed lines asy–stiff, solid lines asy–soft.
L. Scalone et al.r Physics Letters B 461 (1999) 9–14
gives results quite similar to non-local forces of Gogny type w26x that well describe collective flow properties in symmetric systems with compressibilities around K s 220 MeV w25,27,28x. The transport equations are solved following a test particle evolution on a lattice w22,28x. In order to have a better overall control of this complex calculation and to optimize the parallel structure a new Object-Oriented code has been written in C q q language, the code DYN q q w29x. We use a general Skyrme-like form for the charge dependent part of the mean field potential: Uq s C
ri
ž / r0
tq q
1 E C r i2 2 Er s r 0
,
Ž 3.
where r s ' rn q r p and r i ' rn y r p are respectively isoscalar and isovector densities, and q s n, p, tq s q1 Ž q s n., y1 Ž q s p ..
11
In order to study the sensitivity of flow results to the density dependence of the symmetry term we will follow two rather extreme choices for C Ž r .: C s 32 MeV Ž Sk–Bonn of Fig. 1 . : SIII parameterization of Fig. 1:
asy–stiff choice asy–soft choice
We stress again that in this way we force the symmetric part of the EOS to be exactly the same in order to disentangle dynamical symmetry term effects. The corresponding symmetry contribution to the mean field, Eq. Ž3., has a density dependence shown in Fig. 1b for a N s 2 Z system Ž I s 1r3.: in a region just above normal density the field ‘‘seen’’ by neutrons and protons in the two cases is very different. As a consequence we expect to have two main direct effects on transverse flows: Ži. Proton-rich and neutron-rich nuclei will show different proton flows.
Fig. 2. Energy dependence of flows at bred s 0.45: full circles: Fe–Fe protons; open circles: Ni–Ni protons; squares: Fe–Fe neutrons. Ža. asy–stiff, Žb. asy–soft. Žc., Žd.: the same for s NN s 2 fm2 , without isospin dependence. The full diamonds in Ža. represent the balance energy data taken from Ref. w9x for the Fe–Fe and Ni–Ni systems.
12
L. Scalone et al.r Physics Letters B 461 (1999) 9–14
The difference will be much reduced for a symmetry term of asy–soft type. Žii. For neutron-rich nuclei proton and neutron flows will be also quite different in the case of a asy–stiff symmetry term. The effect will be particularly enhanced for semi-central collisions where we can get a very asymmetric piece of nuclear matter in the interacting zone. We have studied the p,n transverse flows for the collisions 58 Fe q58 Fe Žn-rich. and 58 Ni q58 Ni Žprich. in the energy range 55 to 105 A MeV, where nice data are existing for Z s 1,2,3 flows w9x and some calculations have been performed for proton balance energies w10,30,31x. For a given energy and reduced impact parameter, bred s brbmax , we evaluate the freeze-out momentum distributions of protons and neutrons and we construct the mean transverse component as a function of the parallel velocity Žrapidity.. In order to eliminate numerical fluctuations we average over twenty events for each fixed
initialization. The flow parameter is given by the slope at mid-rapidity. In Fig. 2a, 2b, we report the energy dependence of proton flows in Fe–Fe and Ni–Ni at fixed bred s 0.45 with the two choices of the symmetry term. In the asy–stiff case, Fig. 2a, we have a larger balance energy, of about 5 MeV, for the neutron rich system Fe–Fe. The values are in quite good agreement with the data of Ref. w9x Žthe full diamonds in Fig. 2a.. We show also the behaviour of neutron flows in the Fe–Fe case. As expected from the more repulsive mean field Žsee Fig. 1b. we have, with respect to protons, a smaller flow below the balance energy and a larger one above. We remark the increase of the difference at high energy: we are exploring higher densities in the interacting region and so we expect larger mean field differences between neutrons and protons. Everything will disappear in the asy–soft choice: in Fig. 2b we report the corresponding re-
Fig. 3. Centrality dependence of flows at beam energy 55 A MeV: stars: Fe–Fe protons; open circles: Ni–Ni protons; diamonds: Fe–Fe neutrons; squares: Ni–Ni neutrons. Experimental points: full circles ŽFe protons., full squares ŽNi protons.. Ža. asy–stiff, Žb. asy–soft. Žc., Žd.: the same for s NN s 2 fm2 , without isospin dependence.
L. Scalone et al.r Physics Letters B 461 (1999) 9–14
13
Fig. 4. Mean transverse momentum in the reaction plane vs. reduced rapidity for light clusters in the collision Fe–Fe at beam energy 55 A MeV for a semiperipheral impact parameter bred s 0.62.
sults. In particular we get, within the uncertainties of small flow evaluation, the same balance energy for proton flows in the two systems, in disagreement with experiments. A detailed comparison of the flow dependence on the centrality of the collision is presented in Fig. 3a, 3b, at a beam energy 55 A MeV, i.e. below the balance energy. In the asy–stiff case the Fe–p flows are well above the Ni–p ones, in good agreement with the experimental points also reported in the figure 1. To remark the asymmetry of the effect with respect to bred s 0.5, more pronounced for more peripheral collisions: this is a nice evidence of the larger asymmetries we can reach in a ‘‘skin-skin’’ interaction of neutron rich ions. We finally stress the difference between proton and neutron flows in the Fe–Fe case, completely absent for Ni–Ni collisions. Again everything is disappearing for asy–soft calculations, including the agreement with data. In order to evaluate the relative importance of isospin effects on mean field and NN cross sections
1
We have deduced the experimental proton data just averaging over the results for Zs1,2,3 ions of Ref. w9x. We expect our ‘‘single’’ proton results to be systematically above these extracted data just due to the predicted reduced flow of the neutron component.
we have repeated the calculations with a constant Žno-isospin-dependent, no-density-dependent, isotropic. total NN cross section of 20 mb Žsee Figs. 2c, 2d, and 3c, 3d.. From Fig. 2c we see that the isospin dependence of the mean field is able to keep the balance energy difference between proton flows in Fe and Ni, although in a reduced way. Moreover the impact parameter variation is not anymore in good agreement with data Žsee Fig. 3c.. As expected both isospin dependences are important, on cross sections and on mean fields, but our simulations, and also the data, seem to show a quite noticeable sensitivity to the symmetry term of the EOS. From this analysis we can draw some quite clear conclusions: Ži. Collective transverse proton flows for charge asymmetric systems are very sensitive to the density dependence of the symmetry term of the nuclear EOS; Žii. All present data seem to indicate the need of a quite stiff symmetry term for densities up to 1.5 r 0 ; Žiii. We expect to have a clear difference in neutron and proton flows, in presence of an interacting asymmetric piece of NM. This should be seen in a detailed study of the flows of different isotopes of light ions. The latter point is particularly stimulating since it could allow to disentangle among various symmetry terms just from the flow analysis at one beam energy, below or above the balance value.
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L. Scalone et al.r Physics Letters B 461 (1999) 9–14
In order to show a quantitative evaluation of the effect we have performed a light ion flow study just using a phase-space coalescence approach to form clusters at the freeze-out time. We have considered a mean Boltzmann–Vlasov trajectory as an average over NTP ensembles, where NTP is the number of test particles per nucleon. In each ensemble, randomly chosen, we have introduced a clustering procedure in r- and p-space with parameters D R s 2.4 fm and D P s 200 MeVrc. We get an exponential mass Žcharge. distribution which reproduces a Poisson-like behaviour of purely combinatorial nature. This is not surprising since few body correlations are not explicitly included in our mean field approach. Such estimation of light ion yields can be reliable in a fast expansion scenario of the participant zone. From 200 events of Fe–Fe, 55 A MeV at bred s 0.62 Žasy–stiff case. we can accumulate enough statistics to perform a flow analysis of light clusters, 3 H,3 He, presented in Fig. 4. We can estimate a 20% larger flow Žnegative. for 3 He ions with respect to 3 H, just opposite to what expected from Coulomb effects. This is a clear indication of the contribution of a much reduced neutron flow in the case of a repulsive symmetry term Žsee Fig. 1b.. The effect is absent within the asy–soft choice. Finally, we would like to stress that the effects discussed here can be clearly seen also with ‘‘exotic’’ but not radioactive beams. Indeed all the flow data analysed here are from stable beams. The results presented here are in a sense complementary to other predictions of a quite large sensitivity of the main dissipative reaction mechanisms to the symmetry term w32x. A possibility is emerging of obtaining in terrestrial accelerator laboratories important information on the symmetry term of large astrophysical interest. It appears essential to have good charge asymmetric beams available at intermediate energies and to perform more exclusive experiments.
References w1x H. Mueller, B.D. Serot, Phys. Rev. C 52 Ž1995. 2072. w2x Bao-An Li, C.M. Ko, Nucl. Phys. A 618 Ž1997. 498.
w3x M. Colonna, M. Di Toro, A. Larionov, Phys. Lett. B 428 Ž1998. 1. w4x V. Baran, M. Colonna, A. Larionov, M. Di Toro, Nucl. Phys. A 632 Ž1998. 287. w5x I. Bombaci, T.T.S. Kuo, U. Lombardo, Phys. Rep. 242 Ž1994. 165 I. Bombaci, Phys. Rev. C 55 Ž1997. 1 M. Prakash et al., Phys. Rep. 280 Ž1997. 1. w6x I.M. Irvine, Neutron Stars Oxford Univ. Press 1978. w7x C.J. Pethick, D.G. Ravenhall in The Lives of Neutron Stars Ed.s M.A. Alpar et al., NATO ASI Series C Vol.450 Ž1995., pp.59-70. w8x V.A. Urpin, same Proceedings, pp.163-166. w9x G. Westfall, Nucl. Phys. A 630 Ž1998. 27c. w10x Bao-An li, C.M. Ko, W. Bauer, Int. J. Mod. Phys. E7 Ž1998. 147. w11x M. Di Toro et al., Proc. Int. Winter Meeting on Nuclear Physics, Ed. I. Iori, Univ. Milano 1998 p.319. w12x M. Di Toro et al., Isospin Dynamics at Medium Energies, Progr. Part. Nucl. Phys. Vol. 42 Ž1999. ŽPergamon., in press. w13x H. Krivine, J. Treiner, O. Bohigas, Nucl. Phys. A 336 Ž1980. 155. w14x H. Stoecker, W. Greiner, Phys. Rep. 137 Ž1986. 277. w15x H.H. Gutbrod, A.M. Poskanzer, H.G. Ritter, Rep. Progr. Phys. 52 Ž1989. 1267. w16x S. Das Gupta, G.W. Westfall, Physics Today 46 Ž1993. 34. w17x G.F. Bertsch, S. Das Gupta, Phys. Rep. 160 Ž1988. 189. w18x W. Bauer, C.K. Gelbke, S. Pratt, Ann. Rev. Nucl. Part. Sci. 42 Ž1992. 77. w19x Ch. Gregoire et al., Nucl. Phys. A 465 Ž1987. 315. w20x A. Bonasera et al., Phys. Rep. 244 Ž1994. 1. w21x A. Bonasera, G.F. Burgio, M. DiToro, Phys. Lett. B 221 Ž1989. 233, A. Bonasera, F. Gulminelli, Phys. Lett. B 259 Ž1991. 399. w22x TWINGO code: A. Guarnera Ph.D. Thesis, Univ. Caen, July 1996. w23x G.Q. Li, R. Machleidt, Phys. Rev. C 48 Ž1993. 1702 and Phys. Rev. C 49 Ž1994. 566. w24x C. Gale, G.F. Bertsch, S. Das Gupta, Phys. Rev. C 35 Ž1987. 1666. w25x M. Prakash, T.T.S. Kuo, S. Das Gupta, Phys. Rev. C 37 Ž1988. 2253. w26x J. Decharge, D. Gogny, Phys. Rev. C 21 Ž1980. 1568, and references therein. w27x V.de La Mota, F. Sebille, M. Farine, B. Remaud, P. Schuck, Phys. Rev. C 46 Ž1992. 677. w28x V. Greco, A. Guarnera, M. Colonna, M. DiToro, Dynamical effects of momentum dependence of the nuclear mean field in medium energy heavy ion collisions, Phys. Rev. C, in press; V. Greco, Master Thesis, Univ. of Catania 1997. w29x DYNqq Code: L. Scalone, Master Thesis, Univ. of Catania Dec. 1998. w30x R. Pak et al. Phys. Rev. Lett. 78 Ž1997. 1026. w31x E. Daffin, W. Bauer, Effects of isospin asymmetry and in medium corrections on balance energy Preprint MSUCL 111, Aug.98. w32x M. Colonna, M. DiToro, G. Fabbri, S. Maccarone, Phys. Rev. C 57 Ž1998. 1410.