aa r X i v : . [ nu c l - t h ] F e b Phase diagram of dilute cosmic matter
Yoritaka IwataGSI Helmholtzzentrum f¨ur Schwerionenforschung, Darmstadt, Germany
Abstract.
Enhancement of nuclear pasta formation due to multi-nucleus simultaneous collision is presentedbased on time-dependent density functional calculations with periodic boundary condition. This calculation cor-responds to the situation with density lower than the known low-density existence limit of the nuclear pastaphase. In order to evaluate the contribution from three-nucleus simultaneous collisions inside the cosmic matter,the possibility of multi-nucleus simultaneous collisions is examined by a systematic Monte-Carlo calculation,and the mean free path of a nucleus is obtained. Consequently the low-density existence limit of the nuclear pastaphase is formed to be lower than believed up to now.
Nuclear pasta, whose existence and role were suggested inassociation with the collapse driven supernova explosions[1,2], is of great importance with respect to the astrophysi-cal synthesis of chemical elements. Indeed nuclear pasta isknown to change the transparency rate of neutrinos [3,4],which play a role of trigger in supernova explosions givingrise to a wide variety of chemical elements. Therefore theexistence of nuclear pasta at low densities is expected tohave a great impact on clarifying the historical evolution ofchemical elements and also on revising the low-density andlow-temperature part of the nuclear phase diagram [5,6]. Inparticular, here we are concerned with its existence in theoutermost part of protoneutron stars (for the review of su-pernova mechanisms, see [7]).In this paper, based on three-dimensional time-dependentdensity functional calculations, a dynamical formation ofnuclear pasta in the low-density and low-temperature sit-uations is demonstrated by 3-nucleus simultaneous heavy-ion collision. This calculation simulates the situation out-side of the known low-density existence limit of nuclearpasta phase. In order to examine the validity of this calcu-lation, the possibility of multi-nucleus simultaneous colli-sions is studied by a systematic three-dimensional Monte-Carlo calculation, and the mean free path of a nucleus ata given density and temperature is obtained. Note that forthis kind of nuclear pasta formation, the presence or ab-sence of the fast charge equilibration process [8] deter-mines the upper boundary of nuclear pasta phase with re-spect to the temperature [9].
Cosmic nuclear matter, which consists of both neutronsand protons, is studied. Here we are concerned with thedilute cosmic matter whose density is less than 10 % of thestandard nuclear density ρ . Nuclear matter of this kind isexpected to exist in the outermost part of the protoneutron star. In the following the dilute cosmic matter means thematter whose density is less than 10 % of the standard nu-clear density.Although the existence of nuclear pasta was suggestedin more dense situations, its existence on the outermostpart of protoneutron star has never been understood. Its ex-istence, however, is shown in this paper. We are concerned with heavy-ion collisions appearing inthe dilute cosmic matter. Here the mean free path of a nu-cleus is important, and multi-nucleus simultaneous colli-sions might play a role. Indeed, rather dense situations,which are not achieved on the earth, appear in the universe,so that the presence of multi-nucleus simultaneous colli-sions is expected. For reference, the density of targets inaccelerators is roughly estimated by ρ Lab = − ρ where the radii of nuclei and atoms are assumed to be 10 − m and 10 − m, respectively. In the following the density isdenoted by ρ , and the terminology of multi-nucleus simul-taneous collision exactly means simultaneous collisions be-tween more than two nuclei. A three-nucleus simultaneous collision is shown based onthree-dimensional time-dependent density functional cal-culations employing the Skyrme force parameter set SLy6[10]. The calculation is carried out in a spatial box 64 × ×
64 fm with a spatial grid spacing of 1.0 fm, whereperiodic boundary condition is applied. Three Zn are ini-tially located at (8,0,-8), (-8,0,-8) and (0,0,8), respectively(cf. the first frame in Fig. 1). The two nuclei initially lo-cated at (8,0,-8) and (-8,0,-8) are not given velocities in thecenter-of-mass frame, while the nucleus initially locatedPJ Web of Conferences
Fig. 1. (color online) Time evolution of a 3-nucleus simultaneous collision in a cubic space with periodic boundary condition. The boxsize is 64 fm ×
32 fm ×
64 fm. at (0,0,8) has the initial velocity along the z-axis. Accord-ingly, the total system consists of 90 protons and 144 neu-trons, and the total incident energy is set to 100 MeV in thecenter-of-mass frame.Figure 1 demonstrates a 3-nucleus simultaneous colli-sion resulting in nuclear pasta. The temperature and den-sity of this calculation are 0.5 MeV (see following discus-sion leading to Eq. (1)) and 0.039 ρ , respectively. Thissituation corresponds to the state with a density lower thanthe known low-density existence limit of nuclear pasta phase[5]. If we consider collisions with a higher energy ( T = ff erent settings of the ini-tial condition [9]. Although multi-nucleus collisions are suggested to enhancenuclear pasta formation, it is worthless if such a situationis quite rare in the cosmic matter.Let us confirm the validity of multi-nucleus collisionsin the cosmic matter. The numbers of simultaneous colli-sions are calculated utilizing the Monte Carlo method. Acubic space with side length L fm is given. Spheres (corre-sponding to nucleus) with the same radius r fm are intro-duced in the cubic space, where those positions are chosen randomly (The left panel of Fig. 2). Accounting for the ef-fect from the Pauli principle (Fermionic feature) and theCoulomb force, a sphere overlapping with the other sphereis discarded from the beginning. Together with periodicboundary condition, thermal equilibrium of nuclear mat-ter is simulated. The density of a sphere is fixed to ρ , sothat the density of matter is given by adjusting the valuesof L ( L is fixed in actual calculations), r and the number ofspheres. Another sphere (nucleus) with exactly the samefeature (radius: r fm, density: ρ ), which corresponds tothe projectile, is coming from the outside of the cube. Thetemperature of the system is determined by the relative ve-locities between spheres. Here it is given by the relativevelocity of the projectile to the cubic space. The projectileexperiences collisions with spheres, when it goes thoughthis periodic space.Let the typical time interval of low-energy heavy-ioncollisions be 1000 fm / c = . × − s. Collisions aredefined to be simultaneous, if they occur within this timeinterval. The statistical values for the number of simulta-neous collisions is obtained by 10 trial calculations. Let A denote the nuclear mass number. Three cases with di ff erentradii of the sphere ( A =
50, 100, 200): r = . × A / fm , are calculated for several densities and temperatures. Inparticular, with respect to low-energy heavy-ion collisions,we consider three di ff erent temperatures T = , ,
10 MeV.According to the Stefan-Boltzmann relation, the tempera-ture of matter in heavy-ion reactions is estimated by therelative kinetic energy, σ T = E ∗ , (1)where T and E ∗ mean the temperature and the incident en-ergy, respectively.USION11 L [fm]L [fm] L [fm] ρ ρ ρ Probability [%] Collision timesT =1 MeVA =100
Fig. 2. (color online) In the left panel the setting of the Monte Carlo calculation is shown, where periodic boundary condition is applied.Spheres, whose spatial distribution is random, stand for nuclei. The passing of the projectile going through nuclear matter is simulated. Inthe right panel the result of Monte Carlo calculation is shown. The number of collisions during a typical interval of low-energy heavy-ioncollisions is presented.
Table 1.
Mean free path [fm] of a nucleus in nuclear matter,which consists of nuclei with mass number A .(a) A = T \ ρ ρ ρ ρ
10 MeV 19.5 4.6 2.75 MeV 30.9 6.9 4.11 MeV 54.4 14.2 8.2(b) A = T \ ρ ρ ρ ρ
10 MeV 23.6 5.9 3.45 MeV 40.2 10.4 5.61 MeV 70.1 20.1 10.8(c) A = T \ ρ ρ ρ ρ
10 MeV 31.2 7.4 4.35 MeV 42.5 11.1 6.41 MeV 89.7 34.9 16.1
Table 1 shows the mean free path of a nucleus at given tem-perature and density. The mean free path tends to be largerfor matter including heavier nuclei. It seems to be reason-able; for a fixed density and volume of the cubic space,there are more nuclei in lighter case. The mean free pathbecomes smaller for dense and high temperature situations.Comparing two cases in ρ = . ρ and T = ff erence of mean free path is noticed depend-ing only on the size of the nuclei. Note that the cases with T = ρ > . ρ . Table 2.
Transition density [10 − ρ ] from dominance of 2-nucleus collisions to that of multi-nucleus collisions. The dom-inance of 2-nucleus collisions is defined to be true, if 2-nucleuscollisions are the most probable among other finite-nucleus col-lisions (not taking into account collisionless cases). The densitydominated by the 2-nucleus collisions (the left side of arrow) andthat dominated by the multi-nucleus collisions (the right side ofarrow) are shown in each case. T \ A
50 100 20010 MeV 1.5 → → → → → → → → → The right panel of Fig. 2 shows the statistics of simultane-ous collisions for T = = ρ = . ρ , the situationis dominated by 2-nucleus collisions. However, depend-ing on the increase of the density, 2-nucleus collisions losetheir dominance below ρ = . ρ , and multi-nucleus col-lisions becomes dominant instead. It is also seen that three-nucleus simultaneous collisions are not expected (their ex-pectation values are negligibly small), if we are only inter-ested in collisions in the laboratory ( ρ = ρ Lab ).Table 2 summarizes the transition from the situationdominated by 2-nucleus collisions to that dominated bymulti-nucleus collisions. The transition density is found tobe of the order of 10 − ρ , where its temperature depen-dence is small. In addition simultaneous collisions betweenmany nuclei are expected for the nuclear matter with 1 %of the standard density. Indeed, for ρ = ρ and T = A =
50, 100, 200, respectively. Note again that this cal-culation is carried out by giving a typical time interval(1000 fm / c), so that the transition density becomes larger ifwe assume the smaller time interval. Consequently, multi-PJ Web of Conferencesnucleus simultaneous collisions become important and areexpected to play a role in the dilute cosmic matter withdensity higher than 0 . ρ , and the validity of the calcula-tion shown in Fig. 1 is confirmed. Enhancement of nuclear pasta formation due to 3-nucleussimultaneous collisions has been presented based on time-dependent density functional calculations (using SLy6 andSKI3 force parameter sets). Multi-nucleus simultaneouscollisions between more than two nuclei have been con-firmed to occur in cosmic matter with density higher than0 . ρ . Therefore, nuclear pasta formation is possible inthe outermost part of protoneutron stars, and neutrino radi-ation is not as easy as expected. Eventually, the low-densityexistence limit of nuclear pasta phase is suggested to belower than believed up to now.The author thanks to Profs. K. Iida, N. Itagaki, J. A.Maruhn and T. Otsuka. This work was supported by theHelmholtz Alliance HA216 / EMMI.
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