a}: V . 5... . a .3! trawl—L F “x“. “a, . .5 4.3.2. ‘ fix .3? , #WH ‘xll 1,‘ .u. r fiflafidfl . . J.» at up I. r .h $5.1. . yam: v mafia I. :- . .11... . ‘3 ‘9'. ‘L‘OWHWFIH’ .31 fififié ‘5. "+583 9 1004 LIBRARY 50 é} :7 ’/ (c '1 Michigan State University This is to certify that the dissertation entitled TRANSPORT PHENOMENA IN HEAW-ION REACTIONS presented by LIJUN SHI has been accepted towards fulfillment of the requirements for the PhD. degree in Physics Pyi 00%- LQQJ (’4 Major Professor’s Signature (0/ 2 7/0 3 Date MSU is an Affirmative Action/Equal Opponunity Institution PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 6/01 c-JCIRC/DateDuepSSp. 15 TRANSPORT PHENOMENA IN HEA\-"Y-I()N REACTIONS By LIJUN SHI A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requii'vnwnts for the degree of DOCTOR OF PHILOSOPHY Physics 2003 ABSTRACT TRANSPORT PHENOMENA IN HEAVY-ION REACTIONS By LIJUN SHI This thesis is devoted to various aspects of transport in heavy-ion reactions. In the beginning, I give a brief introduction of heavy-ion reactions. transport theory for the reactions and transport simulations. The subsequent discussions are devoted to different issues. First, a plienomenological plraise-transition model for nuclear matter is introduced in order to understand the neutron enrichment in the midrapidity source in a heavy-ion reaction. The effect of cluster formation process on the neutron enrichment is discussed by considering droplet formation in the gas phase. The variational nature of the results in the 1)henomenoh)gical model is utilized to understand isospin transport process during a heavy-ion reaction. Moreover. microscopic transport theory. for uses in heavy—ion reaction transport simulations. is introduced relying on Landau theory. As one of the key ingredients of microscopic transport theory, the mean field interaction is introduced into the theory through the energy-density functional. The functional provides the nuclear equation of state (E08), and both the momentum independent and momentum dependent. mean fields are discussed. Given the recent interest. in systems with varying isospin content, I also discuss isospin dependence of the EOS within the functional method. The symmetry potential, which measures the difference in optical potentials between the proton and neutron. is parameterized in either momentum independent. or momentum dependent form. I also discuss some practical issues for transport simulations, like the initialization of a simulation and the numerical methods for integrating the Boltzmann equations. Besides the mean fields within transport. I discuss the in-medium cross section. Next. the. transport coefficients are derived from a systematic expansion of the Boltzmann equations. The isospin (:liffusiyity, shear viscosity and heat conductivity are calculated using free N—N cross section. Finally, transport theory is used to simulate heavy-ion reactions. Within transport simulations, I discuss the spe / < .40 > and T = 2/3(< E0 > / < .40 > —2MeV) are interpreted as the caloric curve of the liquid and the gas phase respectively, and the plateau region in between the two curves is believed to be related to the liquid- gas phase transition. (from [6]). The flow parameter F as a function of energy. The symbols represent the experimental result for sideward flow at different energies. The lines are theory prediction for the flow, and different equation of states are labelled by their corresponding compressibility value. (from [7]). An illustration of the momentum dependence of the. mean field. The optical potential U is not only a function of density p, but also a function of momentum k. The different lines represent different results from different microscopic theories. (from [8]). xiii 9 1.5 1.6 1.7 2.1 2.3 3.1 3.2 The concept of symmetry energy. The top line is the energy density for pure neutron matter and the lower line is that for symmetric nuclear matter. The difference of the two line is the symmetry energy. (from [9])- The concept of participant and spectators in a heavy-ion reaction. The regions initially labelled A and B collide with each other, go through a violent process and form a single hot emitting source, which is called the participant region. The regions labelled A’ and B’, on the other hand, are only moderately excited and do not actively participate the reacticm. They are called the. spectator regions. (from [10]). . . . The concept. of isospin diffusion in a heavy-ion reaction. Initial projec- tile and target have different isospin concentration, and are shown with different colors in (a). During the reaction, the projectile and target could exchange isospin content, as shown in (b). After the collision, both the projectile and the target isospin concentrations have changed. as shown in (c). The change of isospin concentration in the projectile (or the target) region measures isospin diffusion process in a. heavy-ion reaction. The asynnnetry coefficient C in the. phase transition model as a func- tion of density and temperature. The lines, from bottom to top, cor- respond to tmuperatures of 0. 2. 4, 6, 8. 10 and 12 MeV, respectively. The amplification factor R for the liquid-gas phase transition, as a function of temperature. The amplificatiim factor Ii’. as a function of cluster concentration a. The lines from top to bottom are for temperatures of 5, 6, 7, 8, 9, and 10 MeV. respectively. Density dependence of the net symmetry energy for four different IEOS: iso-SH, iso-NH. iso—NS, iso—SKM. . . . . Optical potentials for four different IEOS: iso—SH, iso—NH, iso—NS and iso-SKM, at asymmetries #3 = 0 and 0.4 for protons and neutrons. At zero asymmetry. the optical potential are the same for protons and neutrons because of assumed exact syn'imetry between proton and neutron. xiv 10 12 13 19 [0 [Q 3.3 3.4 3.5 3.6 3.8 3.10 Symmetry potentials for particle of zero momentum are plotted as a function of density. The four different isospin dependent nuclear EOS are iso—SH, iso—NH, iso-NS and iso—SKM respectively. Optical potential for protons and neutrons at normal density, for four of the MD IEOS: iso—SH, iso—NH, iso-NS and iso—SKM, in symmetric (t3 = 0) and asymmetric (3 = 0.4) nuclear matter, as a function of nucleon momentum 1). Optical potential for protons and neutrons at twice normal density. in symmetric (t3 = 0) and asymmetric (.3 = 0.4) nuclear matter, as a function of nucleon momentum p. Symmetry potential as a function of momentum p for different. MD IEOS. The three of the MD IEOS, iso—SH, NH and NS, give rise to the same symmetry potentials. for the same effective mass parameter rnfso. The different momentum depem’lencies for each the two types of parameterizations are labeled by respective rnfm/m values. Symn‘ietry potential at zero-momentum as a function of nuclear matter density p for different MD IEOS. Net. synnnetry energy as a. function of density for different MD IEOS. The iso—NS and iso—Slx' M cases almost overlap for the momentum de- pendent cases here. Symmetry potential as a function of momentum p for the iso—N H type of IEOS. The different lines correspond to the N H2 parameter set for the MD IEOS (see table 3.4), the iso—NH set for the MI IEOS (table 3.3) and the static form of the same MI IEOS in Eq. (355). The momentum-depcndence for the symmetry potential in MI IEOS stems from the use of scalar quantities in paraineterizing the effects of inter- act ions . The isospin momentum dependence parameter (1 controls the relative difference of proton and neutron velocities, as a function of changing momentum of the particles. A high value of the parameter a will sig- nificantly raise the velocity for protons relative to neutrons in neutron- rich matter by strengthening the momentum dependence of the optical potential of protons and weakening the dependence for neutrons. . 60 67 68 69 70 71 72 Kl 01 3.11 The isospin momentum dependence parameter a affects the effective 3.12 3.13 4.1 4.2 4.3 4.4 mass at non-zero asymmetries. Shown is the effective mass vs asymme- try 6 for the iso—NH type of MD IEOS. The split between the proton and neutron effective mass increases with the parameter a. Equilibrium proton fraction for the different MI IEOS: iso—SH, iso—N H, iso—NS, and iso—SKM, together with the critical proton fraction. Nucleon density profiles from solving the TF equations for MFs corre- sponding to K = 210 MeV, together with the empirical charge density profiles for 40Ca and 208Pb. The solid lines represent the empirical pro- files from Ref. [11]. The long- and short-dashed lines represent the pro- ton and neutron profiles, respectively, for the momenturn-independent field. The long- and short-dash-dotted lines represent the proton and neutron profiles, respectively, for the momentuni—dependent field pa- rameter set S3 in Table 3.2 that yields m‘ : 0.70 m. (From [12]). Isospin diffusion coefficient 1), in symmetric matter, for I ,7,- : 0. at different indicated densities, as a function of temperature T. In the high-temperature limit, the diffusion coefficient exhibits the behavior D, (X \/T/n. Correspondingly, at high temperatures in the figure, the largest coefficient values are obtained for the lowest densities and the lowest coefficient values are obtained for the highest densities. In the low-temperature limit, the diffusion coefficient exhibits the behavior D, (X 713/ 2 / T2 and the order of the results in density reverses. Isospin diffusion coefficient D, at normal density n = no = 0.16fm"3 and different indicated asymmetries (5, for U,- = 0, as a function of tem- perature T. An increase in the asymmetry generally causes a decrease in the coefficient, as discussed in the text. Thermal conductivity ['1‘- in symmetric nuclear matter, at different in- dicated densities in units of no, as a function of temperature T. The conductivity increases as density increases. Shear viscosity 7) in symmetric nuclear matter, at different indicated densities in units of no, as a function of temperature T. The viscosity increases as density increases. 79 83 I22 124 4.5 CI] ;_a Mean-field enhancement factor of the. diffusion coefficient in symmet- ric nuclear matter, R E D1(Ui)/D,(U,- = 0), at fixed density n, as a function of temperature T. The solid and dashed lines represent the factors for the assumed linear and quadratic dependence of the inter- action symmetry energy on density. The lines from top to bottom are for densities n = 2710, no, 0.5 no and 0.1710, respectively. At normal density the results for the two dependencies coincide. ......... Results from a BUU simulation of the the 197Au + 197An collision at 1 GeV/ nucleon and b = 8 fm, as a function of time: (a) the central densities of the participant pc and the spectator matter pwm (b—d) the midrapidity elliptic flow parameter v2. The results are from a simulation with the HM mean field, except for those in the panel (c) which are from a simulation with no mean field. The panels (b) and (c) show the elliptic flow parameter for all particles in the system while ((1) shows the elliptic flow for particles emitted in the vicinity of a given time. In the case of the HM calculations, also shown is u; when a high-momentum gate p, > 0.55 GeV/c is applied to the particles. Contour plots of the system-frame baryon density p (top row), local excitation energy E*/A (middle row), and of the density of bound baryons pbnd (bottom row), in the 124811 + mSn reaction at TM, = 800 MeV/nucleon and b = 5 fm, at times t. = 0, 5, 10, 15 and 20 fin/c (columns from left to right). The calculations have been carriered out employing the soft momenturn-dependent EOS. The contour lines for the densities correspond to values, relative to the normal density, of p from 0.1 to 2.1 with increment of 0.4. The contour lines for and are from 0.1 to 1.1 with increment of 0.2. The contour lines for the excitation energy correspond to the values of EVA at 5, 20, 40, 80, 120, 160 MeV. For statistical reasons, contour plots for the energy have been suppressed for the baryon densities p < 0.1 pg. Note, regarding the excitation energy, that the interior of the participant region is hot while the interior of the spectator matter is cold. ........... xvii 130 CI! 66 C51 ,3}. Qt C) CH x] Evolution of St‘lt‘CttKl quantities in the 124811 + 124Sn reaction at 800 MeV / nucleon and b=5 fm. from a calculation with a soft momentum- dependent EOS. The panel (a) shows the density at the center of a spectator p5,”.C (dashed line) together with the density at the system center pc (solid line). The panel (b) shows the average in-plane trans- verse momentum per nucleon of the spectator (PX / A) calculated using all spectator particles (solid line) and using only bound spectator par- ticles (dashed line). Two extra lines in the panel Show evolution of the momenta past the 40 fin/c of the abscissa. The panels (c) and (d) show, respectively, the spectator excitation energy per nucleon (E " / A) and the mass number (.4) from all spectator particles. ........ Spectator properties in the 800 MeV/ nucleon 12""811 + l“Sn collisions, as a function of the impact parameter, for four representative EOS: hard momentuni-dependent (HM), soft momentuni-dependent (SM), hard momentuni-independent (H) and soft momentumoindependent (S). Panel (a) shows the average iii-plane transverse momentum of the spectator per nucleon (PX/.4). Panel (1)) shows the change in the average net c.m. momentum per nucleon AMP/AH. Panel (c) shows the average excitation energy per nucleon (EV/1), and, finally, panel ((1) shows the average spectator mass (A). Open symbols represent re- sults obtained with reduced iii-medium nucleon-nucleon cross sections; filled symbols represent results obtained at b = 5 fm with free cross sections. ................................. Spectator properties in the 12"Sn + 12“Sn collisions at b = 5 fm, as a function of the beam energy, for four representative EOS: hard momenturn-dependent (HM), soft mornentum-dependent (SM), hard momentum-independent (H) and soft momentum-independent (S). Panel (a) shows the average in-plane transverse momentum of the spectator per nucleon (PX /4). Panel (b) shows the change in the average net c.m. momentum per nucleon A|(P/A)|. Panel (c) shows the average excitation energy per nucleon (E‘/4.) Finally, panel (d) shows the average spectator mass (.4) ....................... Baryon density as a. function of time at the center of the 124Sn + 124Sn system at TIM, = 800 MeV / nucleon and b = 5 fm, for different MFs. Average in-plane transverse momentum per nucleon of a spectator in b = 5 fm 124Sn + 1248n collisions at Tim) 2 800 l\-IeV/nucleon, as a function of time, for different EOS. ................... xviii 144 147 5.8 5.9 5.11 5.12 Landau effective mass m" : p/c, in units of free. nucleon mass, as a function of momentum at several densities in cold nuclear matter for S and SM MFs. The change in the net average c.m. momentum per nucleon A|(P/.4)| of spectators in the 197Au + 197An system at 'I‘lab = 1 GeV/ nucleon. Open symbols represents results obtained with reduced ill-medium nucleon—nucleon cross sections; filled symbols represent results obtained at b = 6 fm with free cross sections. A negative value of A|(P/A)| in- dicates a spectator deceleration. while a positive value indicates a net acceleration. The interaction part of the symmetry energy as a function of density for four different IEOS: iso—SH, iso-NH, iso—NS and iso—SKM. The in- teraction symmetry energy for the first three of the IEOS yields, by construction. the same symmetry energy at the normal density. while the iso—SKM yields a different value. In the left. panel. the isospin diffusion coefficients for nuclear matter for MFs with four different dependence on isospin and also without such dependence. plotted as a function of density; in the right panel, the isospin diffusion coefficients are normalized to that obtained with no isospin dependence in the MP. The temperature of nuclear matter is set at T r: 7 MeV. Isospin asymmetry of the projectile-like spectator region is plotted as a function of time, for four different reactions systems of 124Sii+mS1L 124SII+U2SIL “2811+l24Sn and “2811+“2811 at beam energy EN, = 50 MeV/ nucleon and impact parameter b = 6.5 fm. The top panel is from a simulation with a. stiff symmetry energy density dependence (iso—SH) and the lower panel is from a simulation with a. soft symmetry energy density dependence (iso—SKM). The isospin diffusion ratio as defined by Eq. (5.6) is plotted as a func- tion of time for two IEOS. The top panel is for the most stiff symmetry energy density dependence (iso—SH), and the lower panel is for the most soft symmetry energy density dependence (iso—SKM). Note the stabil- ity of the ratio after 100 fm/c. The shaded areas around the lines indicate the statistical error from averaging over multiple simulations. xix 154 101 162 164 167 5.14 The isospin diffusion ratios from the. simulations are compared to the experimental extracted isospin diffusion ratios. The symbols above the line R,- =-- 0 are for the projectile-like spectators in the 124Sn+”28n system, while those below are for the system 112811+124S1L The error bars reflect the uncertainties in the. experiment. or in the simulations. 170 LIST OF ABBREVIATIONS AMD ................................. Antisynmu—‘trized h‘lolecular Dynamics BUU ........................................... Boltzmann—Uheling—Ulenbeck EOS ........................................................ equation of state HG ............................................................. Hadron Gas HIC .................................................... Heavy-Ion Collisions IEOS ............... equation of state based on isospin dependent mean fields LGP ....................................................... Liquid-Gas Phase LHC .................................................. Large Hadron Collider MD ................................................. Momentum Dependence MD EOS ....... equation of state. based on momentum dependent mean fields MD IEOS .................. equation of state based on momentum dependent. mean fields sensitive to isospin asymmetry MF .............................................................. Mean Field MI ................................................ IVIomentum Independence MI EOS ...... equation of state based on momentum independent mean fields MI IEOS ................. equation of state based on momentum independent. mean fields sensitive to isospin asynnnetry NS ............................................................. Neutron Star QCD ............................................ Quantum Chromodynamics QMD ......................................... Quantum Molecular Dynamics QGP ................................................... Quark—Gluon-Plasma RHIC ........................................ Relativistic Heavy-Ion Collider RIA ................................................ Rare Isotope Accelerator SMM .................................. Statistical Multifragmentation Model xxi ’5: 6’0 ekin eint ‘1 grad ecoul éNM M IN MI LIST OF FREQUENTLY USED SYMBOLS Baryon number. Coefficient of the gradient term in the energy density in Eq. (3.21). Average mass number for a spectator. Impact parameter in a heavy-ion collision. Binding energy per nucleon. Isospin-asynnnetric thermodynamic driving force of Eq. (4.24). Binary diffusion coefficient. Isospin diffusion coefficient. Energy per nucleon. Net energy. Energy density per unit. volume. Energy density per unit. volume of Eq. (3.23), excluding is<)spin—dependent interaction energy. Isospin—dependent part of the energy density per unit volume in Eq. (3.23), associated with interactions. Kinetic energy density per unit volume in Eq. (3.1a). Potential energy density per unit volume in Eq. (3.1a). Energy density per unit volume from the gradient correction term defined in Eq. (3.21). Coulomb energy density per unit volume. Energy density for homogenmls nuclear matter in Eq. (3.19). Electric field in Chapter 4. Lattice energy within the. Hamiltonian Method Eq. (3.8;). Average kinetic energy per particle utilized in Chapter 4. xxii kin sym ,int 6 sym (EVA) Tn’iso Tl Kinetic contribution to the symmetry energy in Eq. (3.47). Interaction contribution to the symmetry energy in Eq. (3.47). Symmetry energy in Eq. (3.47). Excitation energy per nucleon for the spectatm. Free energy per particle. Force. Net free energy of a. system. Quasiparticle distribution function. Isospin-dependent part of the free energy per nucleon in Eq. (2.1). Pauli blocking factor f = 1 — f. Fermi-Dirac. distrilmtion in Eq. (4.17). Spin-antisymmetric Landau coefficient... Sideward-flow parameter. Spin-degeneracy factor. Linearized integrals for collisions lj)etw(rlen particle i and j. defined in Eq. (4.21). Collision integral for particle i in Eq. (4.13). Compressibility of nuclear matter at normal density. Coefficient ratio in Eq. (4.40). Loss term due to collisions in the Boltzmann equation (3.14). Gain term due to collisions in the Boltzmann ecpiation (3.14). Particle mass in vacuum. Medium—modified Lorentz mass. Quasiparticle effective mass at the Fermi momentum. Effective mass isospin dependence parameter defined in Eq. (3.65). N ucleon density in Chapter 4. Quasiparticle mmnentum. xxiii P P I)? PF p, I)? P? (PX/.4) AI (PM) I i Q , RV" 7121b T“" is t H X (It so (1,, (1,, l/grad Net momentum vector. Pressure. Electron Fermi momentum. Fermi momentum. C'ovariant momentum p" = (c, p). Neutron Fermi momentum. Proton Fermi momentum. Average in-plane transverse momentum per nucleon for the spectator. Change in the average c.m. momentum per nucleon for the spectator. Kinetic pressure tensor. Heat flux. Ratio of the real to imaginary parts of the neutron-neutrt)n scattering amplitude in Eq. (3.59). Ratio of the real to imaginary parts of the proton-neutron scattering amplitude in Eq. (3.59). Ratio of the real to imaginary parts of the prmon-proton scattering amplitude in Eq. (3.59). Form factor in the Lattice Hamiltonian Method. Beam energy. Energy-momentum tensor. Third component of particle isospin. Isospin equilibration time. Covariant velocity u." = (7, 7v). Isospin dependent part of the nucleon optical potential in Eq. (3.31). Neutron optical potential. Proton optical potential. Potential related to the gradient correction in Eq. (3.30). xxiv U opt (1,3 ym Uv «”91” Z Density-dc‘pendent factor in the isospin dependent part of the scalar potential. Optical I.)ote.ntia.l in Eq. (3.5). Symmetry potential in Eq. (3.39). Density-dependent factor in the optical-potential term stemming from the self—consistency requirement with regard to isospin dependence in Eq. (3.43). Potential resulting from the self-consistenev requirement in the case of a density-dependent velocity, given by Eq. (3.35). Elliptic-flow parameter. Neutron optical potential from the T-matrix approximation. Proton optical potential from the T-matrix approximation. Quasiparticle velocity. Critical proton fraction in Eq. (3.69). Charge number. Used to represent isospin asymmetry in Chapter 3. Lorentz contraction factor 7 = 1/ \/1 — v2. Dissipative particle flow. Isospin asymmetry. Chemical potential conjugate to isospin asymmetry in Chapter 2. Fermi energy. Proton Fermi energy. Neutron Fermi energy. Sl‘iear-viscosity. Quasiparticle energy. XXV r [‘12 a “12 1) H12 7' H12 PM. (1‘) [)0 {’0 Its.- firm 07'”) 0,", 0m.) 0N N (P6) e V Cull \ \ 4.me '— o \‘ Heat conductivity. Isospin asymmetric chemical potential in Sec. (4.3.3). Reduced density. Coefficient multiplying the asymmetry gradient in the. isospin asymmetric driving force in Eq. (4.29c). Coefficient multiplying the pressure gradient in the isospin asymmetric driving force in Eq. (4.2921). C(wfficient multiplying the temperature gradient in the isospin asymmetric driving force in Eq. (4.291)). Used to represent baryon density and, in Chapter 4, the net mass density. Charge density. Central density for participmnt matter. Normal density p” = 0.16 fin—3. Scalar I.)ary(i)n density. Bound particle density. Neutron-neutron cross section. Proton-neutron cross section. Proton-proton cross sectitm. Nucleon-nucleon cross section. Coulomb potential. Azimuthal angle. Spatial gradient. Tensor. Traceless tensor. Vector. Brace product. Square-bracket product. xxvi Chapter 1: Introduction Much research has been recently done on the properties of matter at extremely high density, which is many orders of magnitude higher than the ordinary matter we encounter everyday. The typical baryon density of nuclear matter is of the order of po = 0.16 fin-3, which. when converted into mass density. is about 10'27 g/cm3. For comparison, the density of water is p(water) = 1 g / cm", and even the density of the most dense metal is only p(Os) = 22.5 g/cm‘". When the high density is generated in reactions. the temperature of the matter is also often high. The typical temperatures of our interest here are in the range of a. few MeV to a few hundred MeV. If we ccmvert the scale of MeV to regular temperature scale. 1 MeV is equivalent. to about 10 billion degrees Kelvin. Room temperature is about 300 K and even the surface ten‘iperature of the Sun is only 6000 K. High temperature and density matter, or hot. dense matter, has some specific established and / or speculated properties. Fig. 1.1 is a schematic phase diagram of dense matter. The horizontal axis is density in units of normal density, and the. vertical axis is temperature. Normal nuclear matter at. (p = p0. T ~ 0) represents the liquid phase. The liquid-gas phase (LGP) transition region at the lower left corner of the figure is characterized by temperatures less than ~ 15 MeV and densities below normal density (p/Po < 1). The quark-gluon plasma (QGP) phase is likely characterized by temperatures of more than 170 MeV at low densities, and it might exist at lower temperatures if the densities are high. The hadron gas (HG) phase exists at intermediate temperature and density. The neutron star (NS) density region extends from low densities up to more than 10 times normal nuclear matter density. The typical temperatures are less than 10 MeV for newly born neutron stars and less than 0.001 MeV for cold neutron stars. The line that separates the QGP phase from the HG phase is the phase coexistence and / or transition region. One of the theories predicts a. cross-over Imam-transition at lower density, a first order phase transition at higher density and a critical point (the solid square symbol) that separates the two phase transitions, cf. Fig. 1.1. The chemical freeze-out points reached at SPS and RHIC experiments are plotted here as open symbols. The compression reached in heavy-ion collisions (HIC) is indicated by the solid line and the expansion of nuclear matter right after the maximum compression is indicated by the dashed lines. In the study of the hot. dense matter. heavy-ion reactions offer advantages over other methods. Though the dense matter exists in neutron stars. only indirect. information may be extracted from the astronomical observations. The Quark-Gluon Plasma phase and dense hadron gas phases. likely existed in the early stage of the Universe (about 15 billion years ago) but are quite inaccessible today. Nuclear structure studies have provided us much information about nuclear matter properties, but only at around normal density and at low temperature. Heavy-ion reactions, during which the matter goes through a compression and an expansion stage, are a true testing ground for the hot dense matter. The maximum compression in a heavy-ion reaction could reach a few times normal nuclear matter density in a head-on collision, and could possibly produce a QGP phase at the highest reaction energies. The expansion of nuclear matter after the compression stage usually leads to a freeze-out for interactions at subnormal densities (p < p0). The freeze-out temperature and density could be in the phase transition regions (QGP to HG phase transition , or LGP), or in the hadron gas phase. The specific regions of the phase diagram and the type of the strong—interaction physics that dominates the reactions depends on the incident energy. Thus, the highest energy reaction experiments at relativistic heavy-ion collider (RHIC) and future large hadron collider (LHC) explore the most dense matter, where quarks and 200 . , . , , ,, 150 temperature (MeV) u. S O o . ,- ~l a I ‘L "R. l 1 ,2 3 4 den81ty( 9/ Po) Figure 1.1: Schematic phase diagram for the. hot dense matter. The shaded region at the lower left corner is the liquid-gas phase (LGP) transition region for symmetric nuclear matter (adapted from [1]). The shaded region at low temperature but with densities ranging from subnormal density all the way up to several times normal nuclear matter density, refers to matter that occurs in neutron stars(N S). The hadron gas (HG) region lies at intermediate temperature and density region. The very high density and temperature region corresponds to the quark-gluon-plasma phase (QGP). The two lines that separate the hadron gas phase from the quark-gluon-plasma phase represent the phase transition region converted from the two lattice predictions [2] and [3]) with excluded volume corrections from [4]. The solid square is the critical point that separates a cross over phase transition from a first order phase transition [2]. The open symbols represent the chemical freeze-out points from SPS and RHIC experiments [4]. The head-on heavy-ion collision will likely follow the arrows on the solid line, starting at cold normal nuclear matter (p 2 p0, T = OMeV) ([5]). The maximum density and temperature achieved are. determined by the incident energy. After the compression to maximum density, the reaction system expands and cools as indicated by the dashed lines. Some low energy reaction systems will dive into the liquid-gas phase transition region. The high energy reaction systems could enter a QGP phase. gluons become the elementary degrees of freedom and quantum chrt)modynamics (QCD) is the basic theory. There are many interesting issues in the highest energy reactions, including the possible occurrence of the QGP phase, the degree of thermalization, the production of exotic particles and collective flow. Reactions at. lower energies, often termed as intermediate energies. are violent enough to excite the system to very high temperature, but not enough to break the internal structure of the hadrons. The description of the interactions with only hadrons and mesons is sufficient for the intermediate reactions. The. direct application of QCD theory at the low energies is somewhat difficult due to confinement; instead. effective nuclear interactions are often employed in the theoretical studies. My PhD. research has mostly concentrated on the heavy-ion reactions at intermediate energies. The research at int ern'iediate energies involves many topics of interest. including the nuclear equation of state (EOS). collective flow. the possible liquid-gas phase transition, and recently also isospin physics. The nuclear equation of state describe the energy-density relations in nuclear matter. The flow phenomena are results of the compression of nuclear matter during heavy-ion reactions, and is a primary tool for the studies of EOS. The liquid-gas phase transition has been predicted long ago. but its actual impact on a heavy-ion reaction is still under debate. Isospin physics is relatively new. and is mostly spurred by the development in rare isotope facilities and by the success in the. nuclear structure research. Here, I will give a brief introduction to these topics at. intermediate energies. Compared to the phase transitions common in everyday life. the liquid-gas phase transition in heavy-ion reactions occur in an enviromnent complicated by small Size of the reacting system. the strong nature of the nuclear forces and the nonequilibrium aspects of the reaction. The multifragmentation process in heavy-ion reactions, as a critical phenomena. is often thought to be closely related to the LGP transition. The. multifragn'lentation process refers to the. process where large number of particles were produced in the energetic heavy-ion reactions; and the produced particles include free nucleons. deuterons. tritons. alpha particles. and all the way to very heavy clusters. Studies of inultifraginentation processes reveals its universal behavior, and critical exponents are determined from experiment. [13, 14, 15]. Some recent work has concentrated on the caloric curve extracted from heavy-ion reactions, even though the use of isotopic temperature. is still questioned. One of the experimental caloric curves is shown in Fig. 1.2, and the appearance of a plateau in the isotope temperature as a. function of excitation energy is attributed to the latent heat in the phase transition. The liquid-gas phase transition and phase coexistence, if present. in heavy—ion reactions, could bring in new phenomena. which are often explored on a phenomenolt)gical basic. One example of such exploration for the liquid—gas coexistence region will be shown in Chapter 2. The research 011 the nuclear equation of state combines both experimental and theoretical effort. The development of collective flow is closely related to the pressure build-up during the compression stage of reactions, and gives us information about the pressure and particle density relation (that. is, the EOS). But such information is hard to extract directly because of the complex evolution of the reaction system. The particles measured in the experiments are produced through out the emitting process from the beginning of compression stage till the end of the expansion stage, and it is difficult to know when the particles are prmluced. However, microscopic transport theory could simulate the dynamic reaction. reproduce the particle emission process, and link the observed flow to the nuclear EOS. Transport simulation has become. an important tool for understanding the dynamics of the heavy-ion reactions. As an example of the research on nuclear equation of state, Fig. 1.3 shows one of the flow signals F measured in experiments as well as simulation results for the flow signal. The flow F measures the magnitude of the sideward flow, which 12 I I l I I 1 1 l I I I l 1 ‘l 1 F Ifi’ o '97Au+'97Au, 600 AMeV n”C,"’O +"°'Ag,'97Au, 30—84 AMeV 10 — AzzNe+'°'To, 8 AMeV . — «10 / I ' THeLi (M 6V) O 14 n n I n . L4 J 1 1 i 1 LL L4 1% 0 5 10 15 2o / (MeV) Figure 1.2: Experimental caloric curve of nuclear matter. The symbols represent the isotope temperature T “em as a function of the excitation energy per nucleon. The dashed lines for T = t/lO < E0 > / < A0 > and T = 2/3(< E0 > / < .40 > —2MeV) are interpreted as the caloric curve of the liquid and the gas phase respectively, and the plateau region in between the two curves is believed to be related to the liquid-gas phase transition. (from [6]). represents the collective deflection of particles away from the beam direction in the reaction plane. At different beam energy, the compression and pressure build up in the reaction region will be different, thus causing different. deflection of particles. The different lines in the plot represent the simulation results with different EOS and are labeled by the corresponding compressibility value for that particular EOS. Systematic studies of the sideward flow from experiment and theory were able to greatly constraint the uncertainties of the nuclear equation of state, eliminating some of the very stiff or very soft type of EOS (see [16] for a more comprehensive review on this subject). pmax/903 ~2 ~3 ~5 ~7 _' U l I I I l I I I II I U I I I I I II I r - - 0 FOPI a 0'4 .— 0 EOS '2 : 0 E895 : b 9 E877 ’—'?‘\-~ K=380MeV - 0-3 T I E917 /,{:-_.-\:‘i:\. t g 2 DATA ‘4‘, 380 w/PT i > '_ K .2 8 0 2 .. \\ 300 . LL. : 210 I 0.1 - _,.I - : """" ‘ \ 167 j u- 0 X‘ 'O, .l - cascade - 0.0 f j bLl l l 1 I l l I ll 1 L 1 I L [J IL I l d 0.1 0.5 1.0 5.0 10.0 Elab (GeV/A) Figure 1.3: The flow parameter F as a function of energy. The symbols represent the experimental result for sideward flow at different energies. The lines are theory pre- diction for the flow, and different equation of states are labelled by their corresponding compressibility value. (from [7]). In microscopic transport simulations, the nuclear equation of state often enters through a parameterized form for practical reasons. The nuclear EOS obtained from various microscopic theories are often quite complicated, and the results based on different assumption for the nuclear force often differ considerably. On the other hand, as transport simulations are already quite involved. a simplified form of the EOS can simplify transport simulations. The validity of the adopted EOS form is checked when comparing results from simulations and experiments. As the guidelines for the 1,)arameterization of the EOS, the parameterized form should be. simple enough to be easily incorporated into transport simulations and also be versatile enough to accommodate the different features of the microscopic predictions. 1 will introduce the parameterization method in Chapter 3, and the parameterized mean field will then be used in the simulations in Chapter 5. In microscopic transport theory. the mean field is often used in either a momentum independent (Ml) or a nnnnentum dependent (MD) form. The MI form, with only density dependence. was used to explain the systematics of the sideward flow and was quite successful. But pure density dependence is not enough to explain the more detailed experimental results, especially the momentum dependence of the elliptic flow signals. Elliptic flow is a measure of the squeeze-out of nuclear matter and has been successfully used for determining the momentum dependence of the. mean field. The momentum dependence of the mean field can be seen in Fig. 1.4. where the optical potential is plotted as a function of both density p and momentum k. The implementation of the momentum dependence is more difficult, and will be also shown in Chapter 3. The recent progress in rare isotope facilitics has also raised interest in the isospin degree of freedom at high densities. The term isospin refers to the pair of similar particles. the proton and the neutrml. which are almost identical in nuclear matter when the electric charge difference is ignored. In many transport simulations, the nuclear interaction differences between protons and neutrons are simply ignored, or in other words. these simulations only explore the reactions in the symmetric nuclear matter limit. Such a simplification is used because of the limited beam and target. combinations, but needs to be refined in view of the rare isotope beam reactions offered by the future facilities. The isospin dependence of the nuclear equation of state is often expressed in terms of symmetry energy. An U(MeV) :‘r, ‘cr l I I‘T' V bi— 0'1 O I 2 3 k um") Figure 1.4: An illustration of the momentum dependence of the mean field. The op— tical potential U is not only a function of density p. but also a function of momentum k. The different lines represent different results from different microscopic theories. (from [8]). elementary illustration of the concept of synnnetry energy is shown in Fig. 1.5. Symmetric matter is represented by the lower line, while pure neutron matter is represented by the upper line. The difference between the two lines is the symmetry energy, which expresses the effect of the isospin on nuclear matter energy density. As with the case of the symmetric matter, the nuclear equation of state based on isospin dependent mean fields (IEOS) could also be put into either a MI form or a MD form. Specific parameterizations of IEOS are shown in Chapter 3. ' r Y I I l U I V U r ' V V I 1 r I f' I I U — O E/A (MeV) o -10 I I I | L 4 1 A A 0.00 0.05 0.10 0.15 0.20 p (fm-3) Figure 1.5: The concept of symmetry energy. The top line is the energy density for pure neutron matter and the lower line is that for symmetric nuclear matter. The difference of the two line is the symmetry energy. (from [9]). In the simulation of l'reavy-ion reactions. microscopic transport tluém‘y has a close relation to the hydrodynamic theory. In fact, microscopic transport theory could reproduce the hydrodynamic equations for high density matter. Such a close relation makes it possible to use certain concepts in both theories. If the hydrodynamic limit is valid for heavy-ion reactions, then the nuclear EOS would exclusively determine the dynamics of the reaction. l\~'Iicroscopic transport theory, which does not make such an approximation, still shows the importance of the nuclear EOS on the evolution of the reaction system. In hydrodynamic theory, the elementary transport properties of matter are often characterized by transport coefficients. Even though a direct application of the transport coefficients is not easy because of the complex nature of the reactions, the transport coefficients still 10 could be used in the understanding of the basic transport process in the reactions and serve as bridge to the transport process in the infinite nuclear matter limit. The, transport coefficients could be calculated analytically from transport theory in the small perturbation limit. As an example, the isospin diffusion coefficient. is calculated in Chapter 4. The calculated isospin diffusion coefficient is then used in Chapter 5 to gain an understanding of the isospin transport process in heavy-ion reactions. The analytic solutions. such as that for the transport coefficients, are limited to a few cases where transport. equations could be solved exactly. For heavy-ion reactions. the transport equations with complicated initial conditions are difficult to solve, and we often resort to simulations. Transport simulations not only reproduce the particle emission patterns. but also provide a model space for understanding the dynamics of the reactions. There are many factors that affect the simulation results. and we can vary each individual factor separately in the model to find the most relevant factors. We can find not only what happens after the reaction. but also what happens during the reaction through a simulation. Such information about the intermediate steps gives us additional help in the understanding of the heavy—ion reactions. I will present two examples of transport simulations in Chapter 5. In the first example of transport simulations. I try to understand the. interaction of the spectator with the participant region during a heavy—ion reaction. The concept of the spectator and the participant regions is illustrated in Fig. 1.6‘. The participant region refers to the strongly interacting hot zone in the middle, where the opposite moving nuclear matter collide with each other; while the spectator regions refer to the forward and backward moving zone that do not directly interact with the opposite moving matter. The interaction between the participant and the spectators has two important effects. First, the interaction strongly affects the (.levelopment of collective motion in 11 BEFORE AFTER ._,. I l 00.80%) ‘— go. 0. . -—-> 000 ° . Figure 1.6: The concept of participant and spectators in a heavy-km reaction. The regions initially labelled A and B collide with each other, go through a. violent process and form a single hot emitting source, which is called the participant region. The regions labelled A’ and B’, on the other hand, are only moderately excited and do not actively participate the reaction. They are called the spectator regions. (from [10]). the participant region and thus links collective flow to the nuclear EOS. Second, the interaction also affects spectator motion. The link between the participant matter motion and the nuclear EOS has been explored extensively. but less is known about the spectator motion after a heavy-ion reaction. To gain a more complete understanding of the reaction dynamics, I will explain the spect.ator-participant interaction and the impact on the spectators in Chapter 5. In the second example of transport simulations. I will try to understand isospin diffusion process during a heavy-ion reaction. The basic idea of isospin diffusion is illustrated in Fig. 1.7. In the initial system, the projectile and the target have 12 different isospin concentration, and are shown with different colors. During the reaction, the projectile-target could interchange isospin content. After the reaction, the projectile (or the target) picks up or loses isospin. The change of isospin concentration in the projectile (or the target) region could be measured experimentally and thus provides a measure of isospin diffusion process in a heavy—ion reaction. It is interesting to know that isospin diffusion process is governed by a single isospin diffusion coefficient in the quasi equilibrium limit and is thus connected to the isospin diffusion coefficient I calculated in Chapter 4. Figure 1.7: The concept of isospin diffusion in a heavy—ion reaction. Initial projectile and target have different isospin concentration, and are shown with different colors in (a). During the reaction, the projectile and target could exchange isospin content, as shown in (b). After the collision, both the projectile and the target isospin con- centrations have changed, as shown in (c). The change of isospin concentration in the projectile (or the target) region measures isospin diffusion process in a. heavy—ion reaction. 13 Chapter 2: Isospin Asymmetry and Cluster Formation in the Liquid-Gas Phase Transition Region As mentioned before, the nuclear Liquid-Gas phase (LGP) transition has attracted much attention in the field of heavy-ion reactions. Here I will introduce a simple model for the LGP transition in isospin asymmetric nuclear matter. This phase transition model is then used to explore phase equilibrium conditions in isospin asymmetric matter and the effect of cluster formation on those conditions. The imbalance of isospin asymmetry in the phase transition region is described in terms of an isospin asynunetry amplification ratio R. The ratio is used to explain the neutron em‘iclnnent in the low density source as compared to the high density source in reactions. A brief introduction of the physical background, and a simple statistical phase space argument for the features of isospin asymmetry in the phase transition region, are given in section 2.1. A formula. for the free energy of isospin asymmetric nuclear matter is given in section 2.2. The isospin equilibrium condition in the liquid-gas phase transition region, and the effects of the cluster formation in the gas phase. are discussed in section 2.3. The isospin asymmetry amplification factor R is also defined in section 2.3. Results from the current phenomenological model analysis are compared in section 2.4 to the experimental results from analyzing the midrapidity source. A discussion about the complexity of the isospin flow in realistic heavy—ion reactions is given in section 2.5. The effects of isospin asymmetry and cluster 14 formation in the LGP transition region in this model are summarized in section 2.6. 2.1 Introduction to the Liquid-Gas Phase Transition Physical Background The liquid-gas phase transition in nuclear matter is an analog of that in ordinary matter. Such a phase transition is often explained by the Van der VVaals type of interaction between molecules in ordinary matter, like water. The interacticm between nuclecms exhibit. similar characteristics as the Van der \V-(‘rals force: repulsion at short distance and attraction at long distance. Based on this similarity. the liquid-gas phase for nuclear matter was proposed more than two decades ago (see. the review on this subject in [17] and [18]). Many theoretical models were proposed for the. LGP transition in nuclear matter [19. 20. 21, 17. 22, 23]. Search for such a phase transition has always been the focus of interest, and some evidence has been proposed for the critical behavior within the phase transition region [13. 24]. Moreover. the recent experimental studies on the caloric curve for nuclear matter [6] have been an indication of a transition and have inspired many related studies [25. 26. 27, 1, 28, 29, 18]. The nuclear liquid-gas phase transition in isospin asymmetric matter is generally different from that in symmetric matter. The theoretical models were often based on symmetric nuclear matter, but experimental studies mostly concerned the asynnnetric matter. As pointed by Miiller and Serot, the additional isospin degree of freedom relaxes the system and changes the phase transition from first-order to second-order [1]. Additionally, the recent experimental temperature measurements in [6] are intimately related to the isospin asymmetry of the system. 15 So it is important to learn more about the isospin effects on the nuclear liquid—gas phase transition. Some recent data analysis also tried to explore isospin observables and to relate them to a possible occurrence of the phase transition [30]. One possible occurrence of the liquid-gas phase transition is in the midrapidity region formed in heavy-ion collisions [30, 31, 32]. In simulations of semiperipheral collisions, formation of a low-density neck region between the nuclei is observed that likely contributes to the midrapidity emission [153]. The low-density region in contact with high-density regions (the projectile and target) opens up the possibility for a liquid-gas phase coexistence and phase conversion. In a dynamical simulation with the Boltzmann—Uehling-Uhlenbeck equation, Sobotka et (il. [.33] observed neutron enrichment in the lmv-density neck region. However. a high n/ p ratio (much higher than in the composite system) was found when counting only free nucleons in the neck region. 216. excluding nucleons in clusters. The paper argued that the symmetric clusters (deuterons and alphas) contributed much to the enrichment. of neutron in the neck region. Specific results were, however, purely numerical. In this chapter, I shall discuss the isospin asymmetry in the phase transition region in a heavy—ion collision and the effect of clusters on that asymmetry. I will first follow crude statistical phase space arguments, to gain a general understandings of the physics involved. Thereafter, 1 will construct a model for the asymmetric nuclear matter, and used the phase equilibrium condition to reach more concrete results illustrating the general ideas. Qualitative Discussion In the general discussirm, I will consider the isospin effect on the liquid-gas phase transition region without complications from cluster formation in the gas phase. For a given temperature and density, a large isospin asymmetry will increase the total energy, which is unfavorable. The extra energy needed for maintaining the 16 same asymmetry will be much larger in a dense phase than in a dilute phase. Thus. if a dilute phase is in isospin equilibrium with a dense phase, the asymmetry in the dilute phase will be larger. This argument, when applied to the heavy-ion reactions, could explain the neutron enrichment in the neck region. The initial nuclei are in the liquid phase, while the low density neck formed during the reaction could be viewed as in gas phase. For the scenario of a neck region (gas phase) neighbored by a. dense region (liquid phase), the isospin asymmetry in the liquid phase is close to that of the whole system, while asymmetry in the neck region could be much larger than in the composite system. In a neutron-rich reaction system, as often the case in lieavx-r-ion reactions, the result is neutron enrichment of the neck region. Next, I will consider letting the clusters be formed in the gas phase. Then the available phase space for liquid does not change while the phase space for the gas phase increases. The added phase space, which corresponds to clusters, has an overall n/p mean value lower than the old phase space for gas. From a. statistical equal-partition point. of view. partition into the new liquid and gas phase space will make the whole gas phase more symmetric. If the percentage of clusters is small, however, then there is essentially not much change in the phase space distribution, and asymmetry in the gas phase excluding clusters should not change much. This idea. also applies to the neck region and points to a. reduced neutron enrichment as a result of cluster formation. 2.2 Flee-Energy Formula The free energy for the isospin maynnnetric nuclear matter can be expanded in powers of the isospin asynnnetry parameter 6 = (N — Z ) /(N + Z): f 2 f/A :f()+fy Z fo‘thz, (2.1) 17 where [0 and C are functions of both temperature and density. The second term on the r.h.s. of Eq. (2.1) is due to isospin asynnnetry and may be called the asymmetric free energy. Since nuclear interaction is symmetric with respect to proton-neutron interchange, the expansion of the free energy contains no odd powers of 6. Only the lowest two terms in the power expansion are retained in Eq. (2.1), and numerical analysis indicates that a quadratic form in 6 is adequate up to almost 6 = 1 (similar conclusions have been reached in [34, 35]). Isospin asynnnetric nuclear matter can be viewed as a two component interacting Fermi gas of neutrons and protons. and the mean field interaction can be represented by an energy density consistent with the enmirical nuclear equation of state (EOS). For simplicity I assume no temperature dependence for the interaction energy. and the Coulomb interaction is not considered here. The total free energy of the system is then a sum of the free energies of two non—interacting Fermi gases and of a density-depeiulent, nuclear potential energy. For a single phase at. ten‘iperature T = 0 and density p. the free energy per nucleon may be written out explicitly: f I f/-‘l = ”I (/’//)())2/3 + (12 (P/I’U) + (13 (IV/MTV] + (a, (/)//)U)2/3 + (15 (mm) 5'2. (2.2) where p0 = 0.16f1n—3 is the normal density. The (/)/p0)2/3 terms come from the non-interacting Ferm' gas. The terms (12 (/)//)0) + (13 (p/pu)"_l are. associated with a. simple parameterization of the nuclear EOS [10, 36‘, 37]. As I am only concerned with the isospin asymmetry in the liquid-gas phase transition, details of the parameterization do not affect the later discussion (though the exact numerical results may change). Given that the interaction generally contributes to the asymmetry energy [38] . I adopt a simple parameterization in Eq. (2.2) for that ('(‘mtril'nitimL 0f the form (15 (p/po) (52. At '1‘ > 0, the free energy could not be 18 written in a simple analytic form, but one can still expand the free energy per nucleon about 6 = 0. The forms for f0 and (1' will be more complicated at non-zero temperatures and will not. be shown here. In the numerical calculation, I use [)0 = 0.16 fin—3, (7 : 2.1612, (12 = -—183.05 MeV, (13 = 144.95 MeV, (1.5 = 11.72 MeV. and at T = 0, a, 22.10 MeV, and a4 = 12.28 MeV The total symmetry energy a4 + (15 2 25 MeV could be obtained from optical potential analysis [39] or from the mass formula [40]. This value is a little low in view of a more recent analysis of the symmetry energy [41]. C (MeV) 00 0.05 0.10 0.15 p (fin—3) Figure 2.1: The asymmetry coefficient C in the phase transition model as a. function of density and temperature. The lines, from bottom to top. correspond to temperatures of 0, 2,4, 6, 8, 10 and 12 MeV, respectively. Figure 2.1 shows the calculated the asymmetry coefficient C in the current model as a function of density and temperature in the Fermi gas. The general trend is that C increases with increasing density and tenmerature. Therefore, at a given temperature, a dense phase will need more extra energy for maintaining a given 19 asymmetry than a dilute phase. 20 2.3 Phase Equilibrium and Clusterization in the Gas Phase Phase Equilibrium Condition Now I will consider a system that has two phases of liquid and gas, respectively, in contact with each other. The general conditions for phase equilibrium require the pressure, temperature and chemical potentials to be the same in the two phases. If the isospin equilibrium is the only concern, it is more convenient to use the isospin chemical potential defined by All. 2 (‘9 f / 06 , and the equilibrium condition is: Au, 2 Aug . (2.3) Here Am and Aug denote the isospin chemical potential in the liquid phase and in the gas phase respectively. Specifically for the phase transition model presented in the last section, the isospin chemical potential is just An = 2C6, and the equilibrium condition is now: C, a, : Cg (59, (2.4) At a. given temperature, the liquid phase is denser than the gas phase, and the coefficient C is a monotonically increasing function of density, C1 > C9. Thus, the asymmetry in the gas phase 69 is always larger than that in the liquid phase 6,. To characterize the relative asymmetry of the two phases, one may define the isospin asymmetry amplification ratio: R = C'z/Cq = 69/6, (25) Figure 2.2 displays H vs. temperature for the phases in equilibrium. For my 21 20 15 10 lILJllllllllJJlllllll IIIIITITIIIIFIIIIII N H N T (MeV) Figure 2.2: The amplification factor B for the liquid-gas phase transition, as a function of temperature. model calculation, the ratio R stays always larger than 1, which means that the gas phase will always have a higher neutron content than the liquid phase. Notably the amplification ratio is independent of the net isospin asymmetry of the whole system. The ratio R, decreases as temperature increases, so that a large overall 71. / p ratio in the gas phase is more easily reached at low temperature. In the case of a. nonequilibrium process, Eq. (2.3) is still of a use due to a. variational origin of the equation. If a local equilibrium assumption is met, i.€., if statistical variables are still valid locally, then Eq. (2.3) tells us the direction of development for the system. The gradient of chemical potential could result in a net flow of isospin asymmetry, which tries to restore the isospin equilibrium condition Eq. (2.3). The flow direction is to the steepest decrease of isospin chemical potential Au. If there is a gradient of AM in a nonequilibrium system, then there could be a flow of isospin asymmetry in the system, with the direction to the lower isospin chemical potential region. Isospin diffusion process will be discussed in some detail 22 in Chapter 4. Effect of Cluster Formation in the Gas Phase One knows that, if the nucleon density is not too low, the mean field description is quite good. But when the density is low, particle-particle correlations become important, and the validity of a. mean field description worsens. One way to incorporate particle correlations is to allow for the formation of clusters in the system (as is done in the BUU calculations [42]). Since clusters are in practice only important for the. gas phase, I will only allow clusters there and no clusters in the liquid phase at all. To further simplify the discussion, I shall adopt a droplet model for the clusters (as used by Goodman ['27] and many others). I will assume that droplets have the same properties as the liquid phase. that is the same density and asymmetric coefficient; for the. present discussion I shall ignore the surface energy term. Suppose the average size of droplets is A. and asymmetry in terms of average proton and neutron numbers in droplets is (5.1. The density of nucleons in clusters may be represented as [2,; : up and of free nucleons as p f = (1 — 0);), where p is the density of the gas phase. The asymmetric free energy of the new (free nucleons + droplets) gas phase is: fy = (1 — (1) Cf 6; + 0 Cd 63. (2.6) Here, the subscripts f and (I refer to free nucleons and droplets, respectively. To get the isospin equilibrium condition, I can carry out a similar variation of asymmetry parameters in the liquid, free—nucleon gas, and in droplets. as before, obtaining: (5d 2 ($1, and (If/(51 = (ll/(If . (2.7) As the density of the gas phase is low, one may use the ideal gas EOS p 2 [2T for 23 clusters in a. calculation. And adding clusters will necessarily decrease p f in order to satisfy the mechanical equilibrium condition. Hmvever, in Fig. 2.1 one can see that C decreases only slightly as density decreases. To first order, one can take szC , so that 6; is nearly the same as the in old gas phase. Overall, the asynnnetry of the new gas phase is: 6 = a (5(1+(1— (r) (if. (2.8) This may be compared to the asymmetry for the old gas phase, (59 z 6, , which is much larger than (5.1 = 6,. It is clear that the more droplets are added to the gas phase, the more it looks like the liquid phase. The amplification ratio now is: R = (5/6, = a + (1 — o)("[/Cf z (i + (l — (1)5)“. (2.9) where R0 : (ll/Cg >> 1. The case of (r = 0 («in'responds to no cluster formation in the reaction, and the isospin amplificaticm ratio reaches then the maximum R0. The gas phase acquires then the largest possible net. asymmetry at a given temperature. On the other hand, a = 1.0 corresponds to the gas phase with only clusters and the same net asymmetry as for the liquid phase. Figure 2.3 shows the decrease of the amplification factor R as a function of the cluster concentration a. As one adds more clusters, the low-density gas phase will need more energy for the same isospin asymmetry, comparable with that of the liquid phase. As a result, the density and asymmetry in clustered gas will both approach those in the liquid phase. 2.4 Data Analysis Short of simple tools to estimate typical relathe numbers of free neutrons, free protons and clusters in the gas phase in a reaction, one may seek help from 24 O l l I I l I l 1 ll IIPJLLJLJIIIIJ O. O O N O .p .0 03 C CD H C Figure 2.3: The amplification factor R as a function of cluster concentraticm a. The lines from top to bottom are for temperatures of 5, 6'. 7. 8, 0. and 10 MeV, resjwctively. experiments. Different. regions of velocity space are generally believed to reveal characteristics of different. sources. with the midrapidity particle region revealing the. low-density neck region. Several intermediate-energy experiments pointed out to a neutron-rich midrapidity source in peripheral heavy-ion collisions [30, 32, 43]. Sobotka et al. [44] measured neutron and 4He emission from a midrapidity source formed in mid-central l"29Xe +120 Sn collisions at 40 MeV / nucleon. They compared their results with results of the IN DRA collaboration for the same system [43, 45] and gave a quantitative description of the midrapidity source. About half of the charged particles from this source are 4H e and only 10% are free protons. Similiar results have been obtained in other papers [32, 43, 45, 30]. The number of neutrons is approximately the same as the number of charged particles, or 10 times the number of protons in this source [44]. If one takes the average cluster size in the midrapidity as about 5 [46], then the percentage of nucleons inside clusters will be o~80%. The N /Z ratio for the midrapidity source is found to be higher than for 25 the full system [44]. Thus the midrapidity source has (N /Z )mid ~ 1.65 or 61nid ~ 0.25 while the system has (N /Z )_,y,, ~ 1.39 or 6,”, ~ 0.16. The asymmetry amplification ratio is then H. ~ 1.5. For a. mid-rapidity source formed in peripheral heavy-ion collision at similiar energy, a fully consistent comparision of different experiments is not easy. Neverthless. comparison of the peripheral data from [30, 32, 47] also suggests a high cluster com-entration and a high n/p ratio for free neutrons and protons. In the current model calculation. Fig. 2.3 shows that for the. cluster concentration a as high as 80%. the asymmetry amplification ratio R. will decrease by more than a. half when compared with the nonclustered gas phase. This large decrease of R will largely limit the isospin asymmetry in the gas phase when the asymmetry in the liquid phase is fixed. Sobotka cf 01. [44] (.rxtracted temperature for the midrapidity source as 6 ~ 7 MeV". For this tr—anperz‘iture and the cluster concentration n w 80‘} . one can read off from Fig. 2.3 the corresponding equilibrium value as R w (1.9 ~ 2.1). This value is higher than the extracted R ~ 1.5 in the experiment, which means that. the system only achieved a partial isospin equilibrium and the asymmetry an'iplification in the gas phase did not reach its full value. As another interesting test case, I will consider the isospin equilibrium condition between the free nucleons (as gas phase) and the clusters (as liquid phase) produced in central collisions. The isospin asymmetry of the free nucleon gas could be extracted from the isototm yield ratios, and a value of 0.429 is obtained from a fitting in the “2811 +112 S n system at beam energy of 50MeV / nucleon [48]. The total liquid phase should have about the same asymmetry as the original system, which is 0.107 for the 1128/) +1” Sn system. And thus the amplificaticm ratio is R 2 4.0. Similar argument for the ”“311 +12" Sn system gives 1? 2 3.6. Tracing back these values in Fig. 2.2. I find the temperature is in the range of 9 ~ 9.6 MeV. This temperature is in agreement with the thermal teumerature obtained from the central collisions of similar reaction system at this energy [49]. 2.5 Isospin Flow in Heavy-Ion Collisions While equilibrium consideration provide general indications for the role of asymmetry in liquid-gas phase-transition region, the asymmetry evolves in a reaction in a process that is principally a nonequilibrium one. As such, the asymmetry may deserve more thorough investigations than can be comprised in the simple analysis, possibly requiring simulations. For now, I will only give a general discussion of the possible isospin asymmetry dmr'elopment in the system. Because of the transient nature of heavy—ion collision. the development of isospin equilibrium depends on two time scales. One. time scale is for the separation of the midrapidity source from the remaining sources in the reaction. and the other is for isospin equilibration. At high enough energy. the. three sources separate quickly before isospin equilibration could set in between sources. The isospin asymmetry is then determined by the reaction geometry and the isospin content of the target and projectile. Isospin equilibration and cluster formation operate only as post-reorganization processes, changing only isospin asymmetry for free nucleons and clusters within individual sources. The large isospin asynnnetry for free nucleons could be the result of Clusterization in the low-density phase, with clusters taking over the role of the liquid phase, consistently with the arguments by Sobotka et al. [33]. From the previous discussion, the R ratio in Fig. 2.2 sets an upper limit to the asymmetry of free nucleons in the midrapidity source. On the other hand, if the energy is low enough, partial isospin equilibrium will set in before different sources separate from each other. and the reaction scenario becomes more complex. As the two heavy ions collide against each other, initial compression of the participants produces a dense phase in the center. while the two 27 spectators remain less dense. As the asymmetry coefficient for the dense phase is larger than for the less dense phase (cf. Fig. 2.1) at the interfaces between the two spectator regions and participant region, there could be a local density gradient from the center out to the two spectators. The isospin asymmetries of the participant and the spectator regions are almost the same in the early stage of reaction because of the short time scale of compression, and gradient of isospin asymmetry could be ignored until later in the expansion stage. From the arguments following Eq. (2.3) and (2.4), the net effect of compression is to produce a gradient of isospin chemical potential. There then could appear an isospin asymmetry flow, and it would be out to the two spectators. As the compression stage ends. the center region begins to expand, and the density drops. the asymmetry coefficient also drops as a result. When the gradient of the asymmetry coefficient. changes direction, so does the isospin chemical potential, and so should the flow of isospin asymmetry. Cluster formation in the center region counteracts the decrease of the. asymmetry coefficient, and thus delays the change of flow direction. Further development of the system separates the three sources, and net isospin asymmetries for different sources do not change after the separation. But Clusterization still plays a role changing the isospin asymmetry of free nucleons within individual sources. Since dynamical simulations suggest. a much longer expansion time than the compression time, one could expect that the isospin asymmetry flow to the midrapidity region dominates. This could give rise to an enhanced asymmetry in the midrapidity region. The experiments also suggest a. neutron-rich midrapidity source, which is consistent with the present picture. 28 2.6 Summary In ccmclusion, I have. investigated the isospin asymmetry in the nuclear liquid-gas phase-transition region. In the framework of the two-component. Fermi-gas with a parameterized interaction, under the assumption of isospin equilibrium, I found that. a neutron enrichment in the gas phase is due to the density-dependent part of the asymmetry energy. Meeting the isospin equilibrium condition, Eq. (2.3) and (2.4), drives extra neutrons out to the low—density phase. The formation of clusters, which have average asymmetry smaller than the gas phase, will make the gas phase more liquid-like, and counteract neutron enrichment in the gas phase. The ‘1 He clusters will be the most important due to their predominance in the neck region [32, 30]. Considering the gradient of local isospin chemical potential. the flow of isospin asynnnetry was suggested to establish the global isospin-equilibrium requirement. Since the midrapidity undergoes compression and expansion, I also suggested a possible change of the direction of the isospin asymmetry flow during the evolution of the system. The flow of isospin asymmetry will be discussed in some detail in Chapter 4. 29 , Chapter 3: Transport Theory In this chapter, I will formally introduce microscopic transport theory for heavy—ion reaction simulations. A brief introduction of the physical background for transport theory is given in section 3.1. The Landau-Fermi Liquid theory for nuclear matter. and the concept of quasiparticle excitations for heavy-ion reactions are presented in section 3.2. The Boltzmann equation set for heavy-ion reactions is introduced in section 33. The nuclear equation of state is discussed in section 3.4, where both the momentum independent and momentum dependent interactions are considered. The recent advances in rare isotope facilities have stimulated much interest in the miclear reaction community to investigate isospin effects, and I will introduce the isospin related physics. and especially the isospin dependence of the nuclear equation of state in section 3.5. For transport simulations of reactions, the reaction system needs to be properly initiated. The set of Boltzmann equations needs to be properly integrated. The quasiparticle collision cross sections will determine the pace of the thermalization process and the production of most energetic particles. All these details contribute to the interpretation of the results from the simulations. and will be discussed in section 3.6. 3.1 Introduction The applicability of the semiclassical transport theory to the heavy-ion collisions generally improves as the bombarding energy increases. Thus, in the ground-state, the characteristic de Broglie wave length is A = h/p Z h/PF ~ 1 fm. However, at an incident energy of 1 GeV / nucleon, the incident momentum is p = 1.7 GeV/c. This yields A = h/ p = 0.12 frn, which is much smaller than either the characteristic mean-free path Am”, 2: 1/(0NN no) 2 2.1 flu, or the characteristic 30 size of nuclei: If 2 1.2/11/3 2 5.6 fin for A = 100. Early on in the research of heavy-ion reactions, the cascade model and the classical hydrodynamic model had been used for reaction simulations. The cascade model, which includes collisions between elementary classical particles, is able to explain the inclusive proton energy spectrum in heavy-ion reactions at intermediate and high energies. The classical fluid hydrodynamical model, which includes the mean field interactions, had success in predicting qualitative features of the collective motions for nuclear matter. Semi-classical transport models. which combine the inter-particle collisions with a semiclassical movement in the mean field, have been quite successful in describing a variety of data. The Boltzmann type of t.rans1.)ort models [10]. often labelled as BUU (for Boltzmann-Uheling-Ulenl)eck). have become an important tool for the. studies of heavy-ion reaction dynamics [16. 50. 7]. The Boltzmann equation set is appealing because it naturally reproduces the rare gas dynamics in the low density limit and the hydrodynamic equations in the high density limit. The Boltzmann type transport simulations are able to reproduce the inclusive proton spectrum from inclusive experiments [10] at intermediate to high energies. and they allow the linking of collective flows observed in exclusive experiments to the empirical nuclear equation of state (EOS) [51, 36', 12. 52, 53]. The efforts carried both from the sides of simulations and experiments allowed extraction of the nuclear EOS at densities from normal density up to neutron star densities (see [16] and reference therein). Nuclear stopping power, the development of a shock wave in nuclear collision, the spectator-participant interactions could be explained within the transport models (see [7] for a review of the success of transport models in heavy-ion reactions). Such transport models were also combined with the statistical rnultifragmentation model (SMM) or coalescence models to calculated the production of heavier fragment produced in reactions. The Boltzmann equation of utility for nuclear reactions may be derived from the Kadanoff-Baym equation [54] in non-equilibrium field theory by following different approximations, including the weak gradient expansion [55], and / or small scattering amplitude approximation [55, 56, 57]. Numerical solution of the Kadanoff-Baym equation turns out to be very difficult [58], and only limited success has been achieved in extremely simplified situation [59, 60, 61, 62]. While the application of transport equations has been very successful, a number of improvements are continuously being added to the understanding of the basic approach. The Boltzmann equation for quasi—particle interactions was. first used by Bertsch et al. [10], the inclusion of three particle interaction for cluster formation was introduced by Danielewicz and Bertsch [:12] and relativistic covariant transport equations have been introduced by Blattel et til. [63. 64]. The relativistic structure for the single particle potential has been introduced by Danielewicz and Fan [51] and Weber et a1. [65]. Finally, the fully self-consistent energy functional method for the parameterization of the mean field was introduced by Danielewicz [12]. Moreover, momentum dependence of the mean field was investigated [66, 67, 68], and the relation to elliptic flow was established [12, 53]. The effects of off-shell transport [69, 70, 71, 72], and non-local collisions [55] have also been investigated. Originally, the Boltzmann equation was derived for a classical dilute gas with predominantly binary collisions. Its extension to quantum gas was postulated by Uehling and Uhlenbeck [73, 74]. A systematic expansion of the Boltzmann equation in density gradients give rise to hydrodynamic equations [75]. In the zeroth order, one obtains the Euler equations [75, 76]. In the first order. one finds the equations of dissipative hydrodynamics with coefficients given through the linearized Boltzmann equations [75. 76, 74]. Successive expansion of the Boltzmann equation in density and velocity gradients will give higher order corrections for the hydrodynamical equations [76, 75]. 32 While the semi-classical method for QED plasmas has long been established [77, 78], the most recent progress has been in the development of the semiclz-issical transport equations for the hot QCD plasmas produced in ultra-relativistic heavy-ion collisions. In the non-equilibrium dense quark matter, the soft quantum fields are well approximated by soft classical field and the weakly coupled hard excitations behave like quasiparticles [70], the transport. equations were derived on resuming the hard thermal loops [86, 81, 79, 82, 83]. The resulting non-Abelian transport equations, which contain nonlinear color field coupling, have been used to simulate the production of the Quark—Glutiii-Plasmas in ultra-relativistic heavy-ion reactions [84. 85]. Another initmrtzmt type of transport models in the simulz-itions of the relativistic l'leavy-ion reactitms are the Quantum I\'l(‘)lecular Dynamics (QMD) [86] and closely related antisymmetrized molecular dynamics (AMD) [87. 88]. Though nominally theoretically less rigmous, the QMD equations are similar to the Boltzmann transport equations. Rather than sampling over the quasiparticle distribution and follow the time evolution of the ensemble, the QMD method follows each initial random condition on an event-by-event basis. and the ensemble average is taken at the end [86, 87, 88]. The QMD approach naturally incorporates particle correlations into (.lyuamical simulations and provides insight. into the cluster formation process in an energetic heavy—ion reaction. AMD uses antisymmetrized wave function for the reaction system, and can nicely reproduce the ground state properties of composite particles [87, 88]. 3.2 Landau Theory Standard descriptions of heavy-ion reactions rely on the Wigner distributions for the particles in the reactions. The natural framework for the description of .33 systems exclusively in terms of Wigner functions is the Landau quasiparticle theory [10]. In this section, I will give a brief introduction to the Landau theory, to the nuclear equation of state within that theory, and the relativistic transformations for quasiparticle motions that are relevant to relativistic reaction simulations. Energy Density and Quasiparticle Excitation Quasiparticles in Landau theory are excitations of the strongly interacting system (in this case, nuclear matter), and, for a. normal Fermi system, they have a one-to-one correspondence to the particles in the system. In other word. each quasiparticle has the same quantum number as the cmresponding particle, including baryon number, charge, spin, isospin and possibly other quantum numbers [89]. The energy ex and momentum px of the quasiparticle are. however, related differently than for a free particle. The occupation of these (piasiparticle states are described by the quasiparticle distribution function fx(p, r, t.) [89]. The total energy of the system in the theory is the sum of two terms: the quasiparticle energy from summing over the quasiparicle distribution function and the additional term from quasiparticle interactions; the total momentum of the system is a sum of momenta for the quasiparticles. E : fdré Z /(1r (6km + (tint) (I - . : /dI‘/‘——‘(27:))3 €X(p)fx(p.r)+/dr€int~ (3'18) (1 P = [dr/fipfydpm). (3.1b) In Eq. (3.1a), the energy density (7 is divided into a. kinetic energy density em and a potential energy density an, to be further discussed. If the system contains multiple components, then there is an additional summation over all particle species in the. above expressions. The degeneracy factor for spin and isospin, which could also be 34 viewed as a special type of summation over particle species, is also omitted for simplicity. The particle Immber, total charge and other additive quantum numbers could be easily obtained following the correspondence between the. quasiparticles and the particles: (1 A = / dr / (7"), AX Law), (32) where AX is the baryon number (or other quantum numbers) of the (plasiparticles, and A is the total baryon number (or total number of other sununationa] quantum number) in the system. The scalar baryon density is defined by: (I where the factor 7 is the lorentz contraction factor. Once the total energy of the interacting system is known. the quasiparticle energy and momentmn (ex .px) are easily derived: 6E _ = , 3.4; (X (sfx ( I) (5P px (ifx ( ) The relation between the quasiparticle energy and momentum reflects the dynamical behavior of the excitation, and is called the dispersion relation. In free space, the dispersion relation is just. (X(p) 2 «mi, + [)2. In the strongly interacting system, the relation generally gets modified clue. to interaction. and the optical potential U0,” is often used to characterize the change: (1”,, : (Mp) — 4mm). (3.5) From the dispersion relation, one can also get the Euler equations for quasiparticle motions under the mean field interactions: v _ (1r _ 06 (3 6) _ (It _ (9p, ' d- dp 0c 2 — = —— . ' . 'l, (1! 8r (3 6 )) The dynamics of the quasiparticle motions should also include the collisions between quasiparticles. Collisions are described by Boltzmann equations and are discussed in section 3.3. Nuclear Equation of State The energy-density relation for nuclear matter. often simply referred to as the nuclear equation of state (EOS), determines the motion of quasiparticles under the influence of the mean field through quasiparticle energy 6 in Eq. (3.6). Since quasiparticle energy could be derived from either Eq. (3.4a) or (3.5), so the nuclear EOS could be uniquely specified by providing the form of the energy density (3, or equivalently in the form of the optical potential Um. The nuclear EOS is often discussed in terms of energy per particle e(p), defined by e = (7 / p. The nuclear EOS has been the subject of intense investigation in l'lea\-'y-ion reactions in the context of transport simulations. As mentioned in Chapter 1, the nuclear EOS in simulations could be classified into two types, the momentum independent (MI) or mmnentum dependent (MD) EOS, depending on whether the momentum dependence of the optical potential is ignored or not. Since the quasiparticle energy 6 in Eq. (3.4a) generally depends on the distribution function, so does the optical potential Um". and the most simple implementation of this dependence is realized through the density dependence of the optical potential (this is the case. of momentum independent EOS). To characterize the density dependence of the nuclear EOS, the compressibility is often defined: 0P K = 9— 3.7 where the pressure P is defined by: 8E . 8e P = ——.—— = 2— . The cmnpressibility of nuclear matter at normal density is often used to label different density dependencies of the EOS. In the more general momentum dependent case, the optical potential in Eq. (3.5) is also a function of momentum, Uop, = (.70,,,(p, p). The effective mass for the quasiparticle is often used to signify the momentum de1:)en('lence of the nuclear EOS: ) m’ = L, (39) v where p = [p], and u = [v]. The velocity vector v for the quasiparticle is defined as: ()6 Op The effective mass at the Fermi surface is often used to label the different MD EOS. As mentioned in Chapter 1, the nuclear equation of state is also isospin dependent. The isospin dependence is related to the additional summation over particle species in Eq. (3.1) (here, over protons and neutrons respectively). The isospin dependence results in different dispersion relations and different optical potentials for protons and neutrons, and thus different Euler equations for the corresponding quasiparticles in nuclear matter. Isospin dependence has recently raised some interest. in nuclear reaction studies, and the relevant. physical ideas and 37 parameterization of the isospin dependence will be discussed in detail in section 3.5. Relativistic Transformations As u'iicroscopic transport theory will be frequently applied to heavy-kin collisions at relativistic energies, the relativistic covariance transformations of the quasiparticle motions becomes important. The relativistic transformations have been derived by Baym and Chin [96]. and I will only list the covariant formula that are relevant to transport theory. The total energy and momentum of the interacting system forms a relativistic 4-vector 1’“ = (E, P). From Eq. (3.4) and the scalar structure of the distribution functirm fx(p. r) . we find that the quasiparticle. energy and momentum also form a covariant vector p” = ((.p). The covariant velocity is u“ = (7,7v), where e, = 1/\/1 — ('2. In a moving frame. the particle density is multiplied by the boost factor a, for the moving frame. but the scalar density p_,, remains invariant. In general. the energy-1nomentuu1 tensor is: I 721111 2 /(,‘—P])llll.uf(p, r) + gin/Ci,” . (3.11) I And the the energy-nuunentum conservation law is just. (9,,T’w = 0. The energy density 6 is a component of the energy-nu)mentum tensor: (.3 = T00 : /(1p€f(p.r)+ an, (3.12) In view of Eq. (3.11), one could see that the first term and second term in Eq. (3.12) have different Lorentz structure and transformation laws. The interaction term 67,-n, is Lorentz invariant, while the kinetic term will be different in different reference frame. In terms of the energy-momentum tensor, the total energy and momentrnn 38 could be. written as: E = /dr7m, P,- = /drT°". (3.13) The tensor structure of the energy—momentum tensor in the Eq.(3.11) is important for accessing of the excitations of nuclear matter, and can be used to test energy and momentum conservation in transport simulations. 3.3 Boltzmann Equation Set As the basic equations in transport. theory, the Boltzmann equations describe the quasiparticle motions in the nuclear medium under the influence of mean field interaction and inter—particle collisions. In this section, I will introduce the Boltzmann equation set, and the covariant form of the collision integral. In the quasiparticle approximation, the state of a system is completely specified when the phase—space distributions fx E fx(p, r, t) are given for all particle species. The distribution function satisfies the Boltzmann equation: 7 *‘_——‘—=’C<’1 — > . 3.14 g); + 0p (2)1. Dr 0P .\ f T IX) ’Cx fX ( ) The l.h.s. accounts for the motion of particles in the MF, while the r.h.s. accounts for collisions. If the collision terms are ignored, one arrives at the Vlasov equations, that is, the mean field dynan’iics for the quasiparticles. Factors IC< and K) on the r.h.s. of (3.14) are the feeding and removal rates. The factor (1 2}: fx) is the reduction or enhancement factor in the Fermi-Dirac or Bose-Einstein system. The degrees of freedom in the Boltzmann equations are usually nucleons, pious, A and N " resonances. While most BUU models do not. describe cluster formation, a 39 description for the production of light (:1 g 3) clusters. an option in the current. model, was developed by Danielewicz and Bertsch [~12]. The combination of relativity and momentum dependence brings in some peculiarities into the collision rates and cross sections. beyond what is encountered in the non-relativistic dynamics. Thus, to be consistent with the Fermi Golden Rule for transition rates and the requirements of covariance. the contribution of binary collisions of particles X to the removal rate in (3.14) is written as: - a. (ng (1p; / (1p; 1— K’> : - —. ———..— — M __.. r 2 x(2w)36(p1+ p2 — p3 - p72) XLTT(S((1+€2 _(,l _(,2)f'2(l_f[)(1—f2) ' ' ' .. 2 (Ix ((1)2 1 , p/ —f :-- —— 1:_____ -_.~.2 / (2am? ‘2,/‘S2 47rz~,*’a*’ln'[M1X ml 1 I2 ’12 >. (3.15) (152' In the above. W represents a squared invariant. matrix element for the scattering. averaged over initial and summed over final spin (lira-tions. The factors 7' are associated with the respective velocities and dp/e, is the invariant measure in the momentum space. The starred quantities in Eq. (3.15) refer to the two-particle. c.m. quantities defined by the vanishing of the three-niomentum in the entrance channel, P = 0. where P" = [if +11]; is initial system 4-momentum. The cross section in (3.15) is given by: 2 (10 p" WU: Zfitgr ,1 filfitlvlt, |A/12,,._.2,.,|2 . (3.16) The relativistic reli-rtive velocity in Eq. (3.15) and (3.16) is defined through the 40 invariant form: . /2 (P- (1.2 ul — (P - ul ((2 2 l _ 7172012 = —[ ) P2 ) J . (3.17) The above definitions ensure the standard form of the detailed balance relation, i.e.: [,2 do _ “2 (10 (19" ’ 1" (1S2: ‘ (3.18) In the em. frame, the relative velocity reduces to the velocity difference. The factor of 1/2 in front of the angular integrations in Eq. (3.15) accounts. in the standard manner, for the double-counting of the final states in scattering for like particles. 3.4 . Nuclear Equation of State As mentioned in Chapter 1. the nuclear equation of state plays an essential role in the dynamics of the heavy-ion reaction. and transport. simulatitms often employ parameterized forms of the EOS. In this section and the next section, I will show the parameterizatit)ns for the nuclear EOS. The parameterized EOS will be used in transport sinnilations, and some examples of the simulations will be. given in Chapter 5. While many of the mean field parameterizations start directly from the 01,)tical potential U . I will introduce the energy density functional method. As seen in Section 3.2, the optical potential U completely determines the mean field dynamics for the quasiparticles through the Euler equations(3.6). So it is sufficient to parameterize the optical [grotential for all transport simulations, the only pitfall is that the formula for the total energy and the excitation energy may become quite complicated. As will be apparent, the energy density functional method is superior: the single particle energies are determined self—consistentIy thrcmgh Eq.(3.4). the 41 total energy and excitation are easily accessible, the relativistic transformaticms are already derived in Section 3.2. The nuclear equation of state based on momentum-independent mean fields (MI EOS) will be used in exploring the compression effects in nuclear reactions; the nuclear equation of state based on momentum—dependent mean fields (MD EOS) will be used for exploring the additional effects of changed particle velocities. The energy density functional method was first used as the starting point for the parameterization of the EOS in [12]. 3.4.1 Energy Density Functional As already mentioned in section 3.2. the energy density functional in Eq.(3.1) uniquely determines the quasiparticle energy and the optical potential, and thus completely specifies the mean field dynamics. In this section, I will start from a general functional form for the nuclear equation of state, and the next two subsections will be devoted to the details of the momentum independent. and the momentum dependent mean field 1)arameterizations. With finiterange effects in the system energy. the energy density of the reaction system could be written as: 6 : éNAl + C:grud + (icoul - (3-19) The last term in Eq. (3.19) is the Coulomb energy. When beam energy is not so high, the radiation retardation effect can be ignored and the Coulomb interaction is given in a static form: 1 (teen! 2 73/)ch(r)q)(r) ' (320) where (I)(r) is the. Coulomb field produced by all other particles and p,.;,(r) is the charge density at position r. 42 The second term in Eq. (3.19) involves spatial gradients of the distribution function. It allows us, primarily, to account for the effect of the finite range of nuclear forces, which is similar to the lowest—order quantal effect of the curvature in the wave functions. In a finite nuclear system. the energy density due to density gradient corresponds to the additional energy required to form a surface (other than the change of the volume energy at the surface region). ~ (1.3 . egrad : _ [)(V)2/)._ (321) 200 After partial integration, I find that the surface energy is positive definite if (1,, is positive. Egret! I \/(]I‘(~’gmd Z 905 /(1r(V/))2. (3.22) ~00 . This gradient term is import ant in the Thou1a..‘—Fer1ni (TF) initialization of the nuclei [91, 92] for the reaction sinmlations. I take the coefficient. in (3.21) equal to as = 21.4 MeV fm'2 for the density dependent MFs. For the momenturn-dependent fields, I take a bit lower (1,, :2 18.2 MeV f1112 from adjustments to ground-state data on nuclear shapes. The initialization of the nuclei for the simulation is described in the subsection 3.6.1. The energy density for nuclear matter is often written as the sum of the energy density term (.70 with no isospin interaction considered and an additional contribution 51 when isospin interaction is considered: BN1” 2 50 + (7] . (3.23) The term 60 will also contain isospin dependence if the protons and neutrons are treated differently. but is not sufficient to explain fully the total isospin dependence 43 of the nuclear EOS. The additional isospin dependent part «2”,, which is usually smaller in magnitude than the term a), will be discussed in detail in section 3.5. The isospin independent strong-interaction field is chosen to act only on baryons in the current calculations. Pions, which should also contribute to the mean field interaction, are infrequent in the energy range I am interested in and are ignored in the consideration of mean field, except in the case of isospin dependence, in order to simplify the energy conservation. I should note that, when the vector and scalar type MFs may be momentum dependent with no exclusive dependence on the vector and scalar densities, there is neither a benefit nor a phenomenological basis, in the absence of spin dynamics, for a. separate consideration of these fields. The parameterizations of the nuclear equation of state are constrained by known physical properties of nuclear matter. Nuclear matter saturates at normal density p0 = 0.16 fm—3 and the binding energy per particle at normal density is about 16 MeV. Also at zero density. the strongly interacting system became a. free. system and the quasiparticles became free. particles. In a mathematical form, I require the EOS to satisfy: 9.) I((/):l)0) = 0., (3.2%) ()p e(p : p0) :— —16 I\. (3.40) 0 I 2 2])() 2 The Lorentz mass above depends both on the scalar density and isospin asynnnetry. through the scalar optical potentials: mx(p,,, i3) = mx + AX U ([I,,) + 2I;;If,7'[:(/)S)i3 . (3.41) The energy functional in Eq. (3.40) does not incorporate momentum dependence in the isospin part of the scalar potential. The. resulting EOS will be referred to as MI 53 IEOS, 218., EOS based on n'ion‘ientum—independent MFs with isospin sensitivity. The single particle energy derived from Eq. (3.40) with inclusion of finite-range and coulmb terms, is then: . , I . 6X00 : \/PZ + 772.3(0)... i3) + §UTCI32 + I’lx Ugmd + Z.v‘1’~ (3.42) The first order term in ,3 enters into the energy through the Lorentz mass m x(Ps, [3), while the self—consistent isospin dependent MF always intrcxluces additional potential of the order i132, through: 20(UT/p5) 0p, (3.4.3) Wczm In principle, there is an at'lditioiml correction factor for UK: which comes from the derivative of the scalar density with respect to distribution function. The correction term is only on the order of 10‘3UTC and will be neglected here. Note that the .32 term in eq. (3.42) is a second-order correction that arises from the self-consistency requirement, and is very small for small asymmetries. The optical potential calculated from Eq. (3.42) is approximately linear in isospin asynnnetry: , 1 ,. (10])! N “(#3) ”l" 2I3II,JT(I)3)II + él’frlY'I-jz ‘ N U(/),.,.) + 2t.31.I./T(ps);3, (3.44) and so is the difference between the proton and neutron potentials. With this, UT may be viewed as the symmetry potential defined before, and can be compared to the symmetry potential in [121]. 54 Symmetry Energy For the isospin density dependent MF paraineterization in Eq. (3.40), the synnnetry energy is of the form: 1]);‘2 + 41sz + 3UT 1 3s m = — - - U 6 ' y 3 26].: 4(F 2 T (3'7”.2YUIZ' 10g mx + Bi (3 45) 4 p}. 7 6p + pp (p i I where 1);: is Fermi momentum for the corresponding symmetric nuclear matter at a given density in the local frame, and ('1: : \/(p1.~)2 + (mX(/),, 3 = 0))2, (3.46) is the Fermi energy. The last term in Eq. (3.45) contributes to less than 5% of the total symmetry energy at normal density. but became significant at higher densities for the iso-stiff type of IEOS. The first term in Eq. (3.45) is associated with the Fermi motion of the qt1asi-particles, and will be called the kinetic symmetry energy c'”" The last two terms in Eq. (3.45), with powers of the synnnetry potential Ur. syrn ’ could be. called the interaction symmetry energy e123". , _ fun + #11! (.3 47) (5.1/"I _ (sym (syn) ° ‘ ' At around normal densities in the rum-relativistic reduction, if one uses (p ~ mx ~ m and ignore higher powers of the synnnetry potential UT than first order, the symmetry energy in Eq. (3.45) reduces to: 1 7);} 1 ,,,,, 2 -— —U~. 3.48 P'J 3 2m + 2 I ( ) At normal density [)0 = 0.16 fin—‘5, I find the kinetic contrilmtion to the symmetry energy is about 12 MeV for a free Fermi gas. The experimental data on binding energy suggest a net. symmetry energy value of 27 — 36 MeV [107]. Thus, the interaction part of the symmetry energy is about the same magnitude as the kinetic part. The recent. analysis of the binding energy of known isotopes, with separate fit parameters for volume and surface asymmetry terms, suggested a value of around 27 — 31 MeV [41]. Many theoretical models have been tuned to yield the synnnetry energy within the range of the previous values of 27 — 36 MeV. The Isospin Dependent Potential The asynnnetry part of the synnnetry potent-ial UT is often represented as a. simple. power of density: Uflfls) = 2.0.4167. (3.49) where £_ = /)/P(,) is the reduced density, the normalizing constant A,- is the interaction contribution to the synnnetry energy at normal density and the exponent T defines the stiffness of the density dependence. Because the kinetic symmetry energy scales as 6p ~ p2/3 in the non-relativistic limit, I will define the iso—stiff type as T > 2/ 3 and iso—soft type otherwise. The isospin super stiff (iso—SH) case, T = 2. gives a very stiff density dependence similar to the dependence in some of the Skyrme—Hartree-Fock models [108] or in the non-linear coupling model in relativistic mean field theory; the isospin normal stiff (iso-NH) case, 1' = 1, which has the most naive form for density dependence, corresponds to the linear coupling model in relativistic mean field theory and is close to some of the Skyrme model dependencies. The normal isospin soft (iso—NS) case, 7' 2 1/ 3, mimics the results from the Brueckner-Hartree-Fock calculations [35] and the density dependence from the relativistic mean field calculations with density or momentum dependent couplings 56 Table 3.3: Parameters for the nuclear equation of state based on momentum inde— pendent mean fields sensitive to isospin asymmetry (MI IEOS), all MI IEOS have the same. synnnetry energy at normal density egg", 2 30ilIeV. Ml IEOS A,- B,‘ Ci T lVleV MeV iso-SH 19.17 2.0 iso—NH 10.17 1.0 iso—NS 10.17 1/3 iso-SKM 38.34 -15.34 0.2 [120, 110]. Another type of utilized isospin soft EOS (iso-SKM) is of the form: m£+flfi U, :20 , , r 1+(KZ (3.50) where the constants .i-l,. I}, are adjusted to reproduce the synnnetry energy at normal density. and the ('1‘ term serves to limit the symmetry potential at high densities. This type of symmetry energy density dependence has been suggested by a special type of Skyrme interaction [101], and is similar to the dependence from chiral perturbation theory [114] and from variational many-body theory [8]. The synnnetry potential from iso—SKM in [101] will generally fall below zero at high densities if the constant B, is negative. The negative total symmetry energy wrmld make the energy of pure neutron matter lower than nuclear matter with a finite proton fraction. Thus. for this density (‘lependence, a neutron star would contain no protons at high densities. The net. synnnetry energies for different IEOS are plotted as a function of density in Fig. 3.1. The IEOS are named after the behavior of the symmetry energy at densities higher than normal: the IEOS with the most stiff symmetry energy density dependence is the iso—SH, followed by iso—NH, while the iso—NS and iso—SKM are subsequently softer. The features of the synnnetry energy are directly related to the features of the optical potential experienced by the particles. as shown in 57 I /fi/ T W SH / - / NH / / ”« ///”’Ns ‘ / 4 “~\SKM 2 3 5 P/Po Figure 3.1: Density dependence of the net symmetry energy for four (.lifferent IEOS: iso—SH, iso—NH, iso—NS. iso—SKM. Fig. 3.2. Notice the optical potential lines for proton and neutron at .3 = 0.4 in Fig. 3.2 are almost equal spaced on the two side of the optical potential at zero asymmetry. this justifies the last approxiination in Eq. (3.44). A Non-Relativistic Reduction In the past, a non-relativistic form of the isospin interaction was used for simulations. To facilitate comparisons with the past results, I will give here the optical potential formula in the non-relativistic limit. If the isospin potential or the isospin asynnnetry is small, the energy density Eq. (3.40) will reduce to Eq. (3.25) plus an extra isospin dependent energy density given by: 1 . (7] = 5(1'115‘52 . (3.51) The isospin dependent part. of the optical pt.)tent.ial derived from Eq. (3.51) will be: UY " i so 1 _. = 21:3.v(«7'1‘2’3 + 51"l.vU7‘('2-32- (Ii-52> Uopt (MCV) Figure 3.2: Optical potentials for four different IEOS: iso—SH, iso—NH, iso-NS and iso— SKM, at asymmetries ,3 = 0 and 0.4 for protons and neutrons. At zero asymmetry, the optical potential are. the same for protons and neutrons because of assumed exact synnnetry between proton and neutron. The explicit expression for the power-law types in particular, is: (IX 2 4I;,X.4,£_T,d + (r — 1 ).:l,€T,L.32. (3.53) ISO The isospin dependent part. of the optical potential for the iso—SKM type is slightly more. complicated: (IX 4! 11,5 + 8’8 .3 Bi — 2:1,(35 — [3,0152 .‘ : :;‘Y . ————.. , 2.132 . 25.34 1+(‘,£2 ‘ . (1+(‘,£2)2 E . ( )) 100 . , . r . , * MI IEOS SH / . P/Po Figure 3.3: Symmetry potentials for particle of zero momentum are. plotted as a function of density. The four different isospin dependent nuclear EOS are iso—SH, iso—NH. iso—NS and iso—SKM respectively. The symmetry potential will then be: .‘Ipr, for power-law type; - V ‘ rr (szm : U] _ (3'00) (Ag + 8,52)/(1 + (LE2). for iso—SKM type. The non—relativistic formulation of isospin depei‘idence is generally sufficient at low energies, where the non-relativistic reduction works well. As many of the experiments on isospin related heavy-ion reaction studies have concentrated on the low energy side. the non—relativistic reductions introduced here will be useful due to the siinl'flicity of the formulas. ($0 3.5.3 Momentum Dependence of Symmetry Potential The synnnetry potential, which represents the optical 1.)ote1'itial ('lifference for protons and neutrons in isospin asynnnetric nuclear matter. is not only density dependent, but. also momentum dependent. Such momentum dependence for IEOS stems from isospin dependence of the non-local nuclear force as well as from exchange interaction [118]. An analysis of the forward nucleon scattering data by Grein [122] demonstrates a definite isospin momentum dependence of the scattering amplitude that is proportional to the optical potential S in the impulse approximation. Both the Brliecker-Hartree—Ft)ck [123. 35] and the Dirac-Bruecker—Hartree-Fock [113. 124] calculations. which use the G-matrix generated from the various nuclear force. show a strong isospin momentum dependence for the nuclear mean fields. The relativistic mean field theories with explicit derivative coupling [110] or with exchange interaction [118. 117] and the chiral perturbation theory [114] also result. in a. strong isospin momentum dependence. In this section. I will present a general functional form for the IEOS, which could incorporate the isospin momentum ('lependence seen in various i'nicroscopic theories. The common features of results from the microscopic calculations suggest the general functional form for the energy density. Given that, under the assumptimi of charge synnnetry. the interaction between p—p and n-n should be the same. the proton and neutron should have the same properties in the symmetric nuclear matter. In neutron rich nuclear matter, the proton and neutron potentials and single particle energies will in general be different and the momentum dependence is seen to be different in the theoretical calculations. For low momentum particles. the n-p interaction is more attractive than the corresponding p—p or n-n interaction, resulting in a more attractive potential and a lower effective mass for the particles at a lower ccnicentration (that is, typically the protons). But at higher momei'itum, 61 at around p ~ 2 GeV / c, the interactions between nucleons are repulsive and the optical potential becomes just. the reverse of that at low momentum. Within a T-matrix analysis for high momentum particle. to be presented, I find that the proton optical potential is more repulsive in neutron rich matter than the neutron potential. The different behavior of the optical potential at high and low momentum suggests a cross-over that should occur at some intermediate momentum for the asymmetric nuclear matter. Such a cross—over is also directly supported by some theoretical results [123, 35, 121]. Optical Potential Within the T-matrix Approximation As I have mentioned earlier. the optical potential for particles with a high momentum could be (‘il‘itained from an analysis of the forward elastic scattering data within the T-matrix approximation (also called impulse ap]:;n'oxiination). In what follows, I will present the T-matrix approximation and employ it to obtain the optical potential for high-momentum particles in asymmetric matter, using nucleon-nucleon scattering data. The resulting high momentum characteristics will be used in the C(mstruction of the energy functional form in the present section. Within the T-matrix approximation. valid for high-mon‘ientum particles, the optical potential is related to the real part of the elementary forward scattering amplitude. 47f ((0,, = [)T = ——F:R(f). (3.53) Here, p is the density of the. scatterers, T is T—matrix and .7: is the scattering amplitude. The forward elastic scattering data have been analyzed by Grein [122]. This analysis, utilizing Coulomb-nuclear interference, produced ratio of the real-to—imaginary forward scattering amplitude It 2 :‘R(}7)/K‘s(f). As the imaginary part of the scattering amplitude in the forward direction is related to the total cross section through the optical theorem, 475 UNIV 2 —(\S\‘(.7'-) . (3.57) p One can express the optical potential in terms of cross section and the amplitude ratio R: (10,, = —p%ft0~~. (3.58) In the two component system, the optical potential for a. given particle is the sum of optical potential contritmtions from each species: /' I) )D " )7) It}, : —E(pp 0],],11’.” + p,, OWN ). . ) ., _ . \in : —%(/)p (7,," RP" + p,, 0,,.,,H"") . (3.59) For the momenta. p S 2 GeV/ c. the ratios R are hm” 2 —0.59 and Rm’ 2 —0.28. and the total cross sections are (Jpn 2 am, 2 4 fm2. At. the normal density of p = 0.16 fut—3 and )3 = 0.2, the optical potentials at p S, 2 GeV/c, assuming charge symmetry, are V 2 43.2 MeV and V" 2 37.5 MeV. This kind of considerations give guidance concerning the behavior of optical potentials at. high momentum: the difference Vp — V" is small and positive and increases only linearly with increasing density. This will help in constructing the energy functional for mean fields with combined momentum and isospin dependencies. The specific values of the optical potential should not be treated quite literally, because of possible limitations of the impulse approximation and because of uncertainties in measuring proton-neutron interactions. The Dirac phenomenological analysis of the nucleon-nucleus scattering data yield (3.9. 1"}, = 25.8 MeV for a 2 GeV/c momentum particle ([125] fit 1). Similar values for the optical potential were also obtained in the relativistic mean field model [126]. Both results are not 63 very far away from the proton optical potentials estimated in this section. Energy Functional Extending the development of energy functional for isospin independent but momentum dependent mean field in section 3.4.3, I parameterize the energy density in the local rest frame as: ~ (II) p I a: I , 6 = 2:9.Y/(27r)3fX(P)(mX+/O dl)'vx(1)./).fl)) ” 1 .. .3 1 . + / (Ip' U(/)') + —UT p52 — ‘a—pVQp + —p;. (3.60) 0 2 200 2 The essential difference here is the isospin dependence of the single particle velocity: , , Til/V AX €(1.0 — 2(3109’) 2 , , 3* , ‘ d : ' '2 ” 2 1+ ,'-—— ‘ , . 3.01 1X“) g I ) p/ p + 7er /( ( my (1 + Avg/mi)2 ( ) where the extra. term (1.0 - 2t3£ug3) is different for protons (13$ 2 -—1 / 2) and neutrons (t31. = 1/2), and gives rise to a different momentum dependence for those particles. In the case of zero asymmetry, the dispersion relations are exactly the same for protons and neutrons. The parameter a is within the range 0.0 _<_ a. _<_ 1.0, and it controls the isospin n‘ion'ientum dependence of the IEOS. Since the velocity is just the slope of the quasiparticle energy as a function of momentum, the larger values of a produce a bigger difference in the slope between protons and neutrons. I will later introduce a model independent parameter that quantifies the sensitivity of momentum dependence of the synnnetry potential. The energy density in Eq. (3.60) gives rise to the single. particle energy in the 64 local rest frame of the form: 1) 6X(p,/)) : mx + / (l)/U}(p,{,,.3) + AxU(p) + (1320)) 0 I . + 2f3xUTfi + §(]T( 332 + Angmd + Zx‘f) . (3.62) As compared to Eq. (3.42), the additional isospin dependent potential U5} stems from the variation of the density dependent velocity and is given by: 1' (IP I) I 07"”: . Ux(/)) = ZQY / 63—7173 foP) (/o (11’ c')p:) - (3-63) v . The energy functionals in Eqs. (3.61) and (3.62) l1'1(."()1‘}f)()1"dt9. additional nunnentum dependence in the isospin part of the scalar potential. The resulting EOS will be referred to as MD IEOS. z'.c., EOS based on momentuni-dependent MFs with isospin sensitivity. The isospin dependent potential term UT has the same form as for the momenturn-independent mean fields given by Eq. (3.49) and (3.50). Since they are independent of momentum, these potentials do not change the velocity as a function of momentum. N otably, the density dependence of the current IEOS parameterizations is mostly determined by the potential term UT. The effects of asymmetry on the optical potential at normal density and twice normal density are displayed in Fig. 3.4 and Fig. 3.5. As may be apparent in Fig. 3.4 in asymmetric nuclear matter at normal density, the optical potentials for protons and neutrons cross at some intermediate momentum. As mentioned before, this cross-over is consistent with the T-matrix analysis. The parameterizations generate the expected behavior for the symmetry potential at low and high momenta for three of the IEOS. However, different density dependencies of the isospin dependent optical potential show up clearly at twice normal densities in Fig. 3.5. The synnnetry energy in 65 zero—tenmerature matter is both impacted by the synnnetry potential UT(/)) and by the parameter a. describing the sensitivity of particle velocities to asymmetry. The synnnetry potential as a function of momentum is plotted in Fig. 3.6. The density dependence of the synnnetry energv for a few MD IEOS is next plotted in Fig. 3.8. Three of the MD IEOS have identical symmetry potential at normal nuclear matter density, because the three power-law forms for Ur in Eq. 3.49 yield identical values at normal density. The synnnetry potential for zero-momentum particles is also plotted as a function of density in Fig. 3.7. The density dependencies of the synnnetry potential for both MI IEOS and MD IEOS stem from the potential of Uflp) , which is exemplified by the similarities between Fig. 3.3 and Fig. 3.7. For those parameters that already appeared in the isospin-independent parameterization of velocity in section 3.4.3, I will take the parameters from set 83 in Table 3.2, which were tested in the previous BUU simulations [12] and were found to agree well with the data on elliptic flow. The parameters for the MD IEOS are listed in Table 3.4. Because of the sensitivity of momentum dependence to isospin, the parameters .4,- and B,- in the potential UT need to be adjusted so as to produce the. synnnetry energy of nuclear matter at normal density of around 30 MeV. For the iso—SKM type of density dependence, the above implementation of the sensitivity to isospin results in some pathological behavior at higher densities. Specifically, as the synnnetry potential becomes negative at high densities, the proton optical potential at zero momentum gets higher than the neutron optical potential in neutron rich nuclear matter. If the momentum dependence is still governed by Eq. 3.61, the proton and neutron optical potential difference is going to increase with increasing momentum. Such behavior is not expected at high momentum given the results from the T-matrix approximation and the forward N N scattering data. To allow the optical potential difference between protons and neutrons decrease as momentum increases at high density, I have to allow for a sign 66 Uopt (MeV) -100 Figure 3.4: Optical potential for protons and neutrons at normal density, for four of the MD IEOS: iso-SH. iso—NH, iso—NS and iso—SKM, in symmetric (B = 0) and asymmetric (,3 = ().4) nuclear matter, as a. function of nucleon momentum p. 67 Uopt (MeV) .'_. o o dz. 0 s . . -100 0.0 0.5 1.0 1.5 2.0 Figure 3.5: Optical potential for protons and neutrons at twice normal density, in symmetric (d 2 0) and asymmetric (,3 = 0.4) nuclear matter. as a. function of nucleon momentum p. 68 \ - l ~ 0.22.” -------- l l 0 L MD iso-SKM 9 g -40- MD iso-SH, NH, NS E. I 0.0 0.5 1.0 p (GeV/c) Figure 3.6: Symmetry potential as a function of momentum p for different MD IEOS. The three of the MD IEOS, iso-SH, NH and NS, give rise to the same synnnetry potentials, for the same effective mass parameter mi“). The different momentum dependencies for each the two types of parameterizations are labeled by respective m ‘ 'iso / m. values. 100 . , . f MD IEOS 3H2 A P/Po Figure 3.7: Symmetry potential at zero-momentum as a function of nuclear matter density p for different MD IEOS. 70 100 I l r I v I ' I 7 r MD IEOS / ‘ ’ 3 P/Po Figure 3.8: Net symmetry energy as a functim'i of density for different MD IEOS. The iso—NS and iso—SKh’I cases almost overlap for the momentum dependent cases here. 3OF—*:\:———_—st—ati—c__: 9 \\\ O) \\ E20 . \\\ _ MIlSO-NH \\\ E a 5910 D -I MD iso-NH . -10- - l n l 1 J_ 0.0 0.5 1.0 1.5 2.0 p (GeV/c) Figure 3.9: Symmetry potential as a function of momentum p for the iso—NH type of IEOS. The different. lines correspcmd to the NH2 parameter set for the MD IEOS (see table 3.4), the iso—NH set for the MI IEOS (table 3.3) and the static form of the same MI IEOS in Eq. (3.55). The moment.mn—dependence for the symmetry potential in MI IEOS stems from the use of scalar quantities in parameterizing the effects of interactions. Table 3.4: Parameters for the MD IEOS, where the parameters are adjusted so that they have the same synnnetry energy at normal density esym —- 30 MeV. MD IEOS 11,- B. C,- T a b 7111'“) / 111. MeV MeV SHI 16.3417 20 0.2 0.10 SH2 21.2522 2.0 0.45 0.15 8H3 22.2405 2.0 0.5 0.16 8H4 28.2159 2.0 0.8 0.22 NHI 16.3417 1.0 0.2 0.10 NH2 21.2522 1.0 0.45 0.15 NH3 22.2405 1.0 0.5 0.16 NH4 28.2150 1.0 0.8 0.22 N81 10.3417 1/3 0.2 0.10 N82 21.2522 1/3 0.45 0.15 N83 22.2405 1/3 0.5 0.10 N84 28.2159 1/3 0.8 0.22 SKh‘ll 28.4207 41.3707 0.2 0.2 0.4 0.075 SKhI2 33.28 -I3.312 0.2 0.5 0.4 0.10 SKRB 38.3007 45.3227 0.2 0.8 0.4 0.13 change for the isospin momentum (.leljiendence parameter at. some. interim—*diate density. For the case of iso—SKM type of IEOS. this can be accomplished with: _ly Ax {11.0 — 21....mo—bo/<1wot)? (3.04) v‘ = ) )2 + r112 1 + c . . . X l/ I X/( "IX (1+ 0112/7”?le The additional parameter 1) controls the sign change of the 18081011 momentum dependence. However. it. should be cautioned that. such a sign change is neither based on any physical argument. nor supported by any microscopic theory. The Isospin Momentum Dependence The sensitivity of momentum dependence to isospin asymmetry is characterized by the parameter a. (or a and b in the case of iso—SKM). while the sensitivity of density dependence to asymmetry is characterized by the potential parameters :1, and T (or A. 13,-, and (2', in the case of iso—SKM). The impact of the isospin momentum dependence parameter a is demonstrated in Figs. 3.10 and 3.11. Note that the difference in effective masses has both contributions from the difference in proton and neutron Fermi momenta and from the sensitivity of the optical potential to isospin momentum dependence. To characterize the sensitivity of the momentum dependence to isospin, in a model independent way, one can introduce the following parameter to characterize the isospin dependence of the effective mass: 74 J 1 - -100 1.5 2.0 I A l p (GeV/c) Figure 3.10: The isospin momentum dependence parameter (1 controls the relative difference of proton and neutron velocities, as a function of changing momentum of the particles. A high value of the parameter a will significantly raise the velocity for protons relative to neutrons in neutron-rich matter by strengthening the momentum dependence of the optical potential of protons and weakening the dependence for neutrons. CI"! LO . , . TfifT . .. 09" 3:0.2 ——-n: 0.8. —"P. 0.7 ’ ______________ . 0.6 \ .‘ 0.5:%:1,:}.Lirnr 09- 0.8» """""" ‘ ___—_,.. 0.7 . 0.6 \ : m*lm ALL4414.1 00 (M2 04 (M5 08 L0 Figure 3.11: The isospin momentum dependence parameter a affects the effective mass at non-zero asymmetries. Shown is the effective mass vs asymmetry 13 for the iso—NH type of MD IEOS. The split. between the proton and neutron effective mass increases with the parameter a. 76 3.5.4 Direct Urca Process in Neutron Stars The isospin dependence of the nuclear eqln-ition of state has direct astrophysical implications. I will discuss here the effect of IEOS on the direct Urca cooling process in this section. The direct Urea. process is an efficient cooling process in hot neutron stars. and at. high proton fraction, the direct. Urca process will dominate over the modified Urca process and lead to a fast, cooling of neutron stars. The direct Urea process in neutron stars is believed to be an important cooling process in hot neutron stars [105]. Recent data from Chandra. observatory demonstrates a fast cooling of neutron stars [1‘27]. in support of the direct Urca process. W'ithin the direct L'rca process. neutrinos are prmluced in a two step cycle: neutrons decay into protons and emit antineutrinos, subsequently the protons convert back into neutrons via electron capture and emit neutrinos. Thus. the cycle is: n —+ 12+ 6" + 17... F +1) —’ 11 +14. (3.66) For the sequence to take place. momentum conservation requires: 1"} +11} 2 I)? 2! 1”} -1'} | - (3-67) Because of the. ultra-relativistic nature of the electrcm, the energy constraints are less stringent than the mmnentum constraints and could be ignored for most densities and forms of IEOS. The charge neutral condition requires equal proton and electron density pp 2 p( and consequently equal Fermi momentum, p”, = p}. (3.68) 77 One also knows that Fermi mmnentum scales as 1), ~ [)1/3, so that the above condition gives the critical proton fraction: yer — pl) — l -p n I will show how the different IEOS will affect the. onset of the direct Urca process. Here I will only consider a. cold neutron star, and examine the equilibrium proton concentration in a neutron star. If the equilibrium proton concentration yeq is larger than the critical proton fraction y”. for momentum conservation, then the direct Urca. process becomes a. favoral.)le cooling process for the corresponding hot neutron star. A more thorough investigation would require calculation of a reaction rate for the modified and direct U rca processes at non-zero ten1peratures in order to determine the dcnninant prm-ess. The electron gas is treated here as an ultra-relativistic ideal gas. where single particle energies are r. : \/ [)2 + 171?. ~ 7). and the energy density corresponds to the sum of single particle energy up to the Fermi momentum. The variation of the total energy with respect. to the proton fraction gives the equilibrium condition: y. = )1" —/1.p. (3.70) where the chemical potentials are the respective single particle energies at the Fermi surface. From this condition. I find the equilibrium proton fraction y needs to satisfy the relation: 4833117701 _ 2”) : pr/I/3 ° (3'71) where pp is the Fermi momentum for the corresponding symmetric nuclear matter. The equilibrium proton fraction in a neutron star depends sensitively on the ratio of the nuclear-matter Fermi-momentum to the net synnnetry energy for the given density. If the synnnetry energy turns negative, the equilibrium proton fraction vanishes, i.e., pure neutron matter becomes energetically favorable; if the symmetry energy stays positive, then equilibrium proton fraction will be always non-zero. On the other hand, if the symmetry energy is positive and large, the symmetric nuclear matter may become favorable. Such a conclusion is independent of the density dependence of the terms in the EOS or momentum dependence of the optical potential. . . . . . T . , - . . 0 4 _ —— iso-SH - — — — iso-NH * — — —1so-NS 0.3 _ —-—-—iso-SKM _ 5 — — cr. frac. ’5‘ 3 ,3: 0.2 s: o ‘5 a 0.1 0.0 1 l n 1 1 L L I 1 l . Figure 3.12: Equilibrium proton fraction for the different MI IEOS: iso—SH, iso—NH, iso—N S, and iso—SKM, together with the critical proton fraction. The proton equilibrium fraction in a neutron star and the critical proton fraction for direct Urca process are plotted in Fig. 3.12 for four of the IEOS with no momentum dependence in optical potentials. The iso—SH and iso-N H interactions will give rise to high proton concentrations at high densities, and thus will enable the direct Urca. process and fast cooling in hot neutron st ar; while the equilibrium 79 proton fractions for the other two types of EOS (iso—NS and iso-SKM) are. always below the critical proton fraction for the direct Urca process, and not allow for a fast cooling in a neutron star. Since the momenturn-independent iso—SKM case will lead to a negative symmetry energy at high densities, the equilibrium proton fraction vanishes at high densities, leading to pure neutron matter in the center of neutron stars. The critical proton fraction of yr, 2 1/ 9 is also plotted for reference. 80 3.6 Aspects of Transport Simulations of Heavy-ion Collisions This section deals with details of transport. theory implementations in heavy-ion collisions. The self-consistent initialization of the nuclear ground state is important for the description of the excitations during the reaction, and is given in section 3.6.1. The quasi-particle ensemble method is used for integrating the transport equations, and it is discussed in section 3.6.2. The lattice hamiltonian method is used for achieving a high accuracy in the numerical integration, and it is described in section 3.6.3. Both the iii-medium cross section and free space N N cross section are used in transport simulations. and a. few of the connnon used parameterizations of the iii—medium cross section are discussed in section 3.6.4. 3.6.1 Initialization of a Reaction System At the start of the reaction simulation the nuclei need to be initialized in their ground states. The minimization of the energy functional for the ground state nuclei, leads to the 'I'homas-Fermi ('I‘F) equatitms for nuclear densities. Specifically, to determine the ground state properties of a. nucleus, I try to minimize the total energy of a system under the condition of fixed total neutron and proton numbers in the nuclear frame. The condition of fixed total N and Z are imposed through the introduction of Lagrange multipliers. The condition of the, minimal energy subject. to the constraints. under the variatimi of the distribution function for protons and neutrons then yields: 0 : (1(1):) — (1,, V2 (is) + ("7:0 + (I) — [1],, (3.72) 81 0 = 5,, (pf) — (1,, V2 (E) + ((11.0 — [1", (3.73) [)0 These are just the T hmnas-Fermi equations in the local frame. In the above equations, 11,, and an are the Lagrange n'iultipliers for the proton and neutron numbers, respectively. The consistency for the nuclear density requires that Vp = 0 at the edge of the density distribution. The role of the derivative correction in (3.72), (3.73), and (3.29) is to reduce the effect of the negative MF when the density distribution in the vicinity is primarily concave and to enhance the effect of the field when the density is convex. Such a result. would be obtained for a. finite-range effective two-lmdy interaction convoluted with density expanded in position to second order. Not surprisingly, the derivative correction to single particle energy is small but it. becmnes important when the energies balance, such as in Eqs. 3.72 and 3.73. permitting an at'lequate description of the density in the ground state. In finding the ("lensity profile. it is coi'ivenient. to transform the TF equations into: —,—I‘— :—F'+ 170). Each test particle represents a possible phase space location for the quasi-particle. An average over all these possible phase space configurations within some phase space volume gives the quasi—particle distribution f(p,r), (2:)3 ffp‘“) :NZM P PM )Wr— rr(f)). (3.76) where it is understood that both sides are to be averaged over the phase space volumes. Without the collision in the Boltzmann equations, the quasiparticle distribution follow Vlasov equations. For the distributitm in Eq. (3.70), a. Vlasov mutations yields: — Zmérp—paawe—rke» — Zrk6(p- pd t))5 (Fri-(1)) + 3%ZMP— Pk())<5(1‘—I‘k(’)) _ {33 zap — pk6(r..)._ (3.84) where . _ (z. 1 , ("[‘mr 0 :; ———. 2_ o —— o AliAi _— o—AliAl ~ ._ ;(r) WEN). ( Mr) p 0. Since 1111 F1 + 1112 F2 2 0, One has ml [)1 = mg Dz. For the differential flow, one has 01/ 1") 1 — 1/ 86 II; = —11.Dl,+— + 111.72 ——(—.—-)- = —n [)6 ,—. (4.5) ()r 0r dr Here, the differential coefficient is Dg : (1)1 + 0;) / 2. So far, I assumed a system at a. uniform pressure and temperature, with only the concentration changing with position. If the variations in a system are more complex, other nonequilibrium forces than the concentration gradient can drive the diffusion. This will be explored later in this chapter. General guidance regarding the forces which can contribute is provided by the Curie principle. This principle exploits symmetry and states that the driving forces must. have the same tensor rank and parity as the flux they generate. For the system of neutrons and protons, the. differential concentration 6 becomes a concentration of the isospin and the differential flow becomes the isospin flow, F5 E F]. 1\=Iore(,)ver. the differential diffusion coefficient. becomes an isospin diffusion coefficient. 1),,- E 1),. and for equal masses one expects D, = 1),, = D". It is popular to relate the com-ept of a diffusion coefficient to a diffusion equation. Indeed. if one considers a uniform system of protons and neutrons at rest, but with the nucleon cm‘icentration changing in space. then, from the contimiity equation for the differential density '9 p.15 —‘ 1,” 1 : —v-I‘,. (4.6) ()1 I get the familiar equation '16 _ . - ‘7, : p,v2(), (4.7) ()1. Here, for D], I have assumed a. weak dependence on the concentration 6. Before turning to a derivation of rigorous results for the diffusion and other transport coefficients, it may be instructive to produce simple mean-free—path estimates for those coefficients. Let us consider components of equal mass (the mass then becomes a simple normalization coefficient in density that may be factored out) and consider the gradient of concentration along the .1: axis, in the medium at rest. If one takes the three coordinate axes, then 1 / 6 of all particles will be 96 primarily moving along one of those axes in the positive. or negative (:lirection, with an average thermal velocity K = (MT/m, for the distance of the order of one mean free path A, without a collision. Considering the particles 1 moving through the plane at :1: = 0, they will be reflecting density at a distance /\ away. Including the particles moving up and down through the plane, I find for the flux 1 . 1 ') F1%6(111(;1:— A) —1'11(.17+ 1)) y s 711%. With (4.4), I then get for the (.liffusion coefficient 1 1 T 3 '11. a 3m. with A ~ 1/(110). A more thorough investigation shows that it is the cross section 171-2 for interaction between the two species that enters the diffusion coefficient. Let us now evaluate the magnitude of the. isospin diffusion coefficient. At temperature T ~ 60 MeV and normal density 710 = 0.16 fm’3, with 0",, ~ 40 mb, one finds 1), ~ 0.2 fm c. As will be seen, this is in a rough agreement with thorough calculations. Similarly to the above, one could employ the mean-free path arguments to determine the better investigated coefficients: shear—vist*osity 17 and heat conduction K. One finds 7) ~ %n mKA and K. ~ %n K /\ Cy, where cv is the specific heat per particle. For T ~ 60 MeV and 0 ~ 40 mb, I find 1) ~ 30 MeV/(fin2 c) and K. ~ 0.06 c/f1112. Up to factors, the shear viscosity and heat conduction coefficients play the role of diffusion coefficients in the diffusion equation for velocity vorticity and in the heat conduction Fourier equation identical in form to the diffusion equation. In the estimates above, I just considered the free motion of particles in-l.)etween collisions. If self-consistent mean fields produced by the particles depend on 97 concentration, then this dependence, on its own, contributes to the diffusion. In the case of nuclear matter, the interaction energy per nucleon may be well approximated in a form that. is quadratic in isospin asynnnetry, e”” (52, where syn: irit (S : ("p — 11,,)/n and 65y", is the interaction contribution to the symmetry energy esym. At normal density, the interaction symmetry energy is 6"” 5:1 14 MeV. The syni naive expectation for tw<")—body interactions is that 623’", is linear in density. At constant net density, the quadratic dependence of the interaction energy on 6 leads to the force FM : 42(4 W” /n,) (011,, / Or), of opposite sign on protons and neutrons. ”syrn The direction of the force for positive eff", is to reduce nonuniformity in isospin. Under the influence of this force. a proton accelerates for a typical time between collisions A1 = A/L and then, in a collision. resets its velocity. The described polarization effect. augments then the proton flow by . 7'1 )1 011, AF], 2 11,, AK}, : —4c;','l'm J ————’—’. (4.9) - 1'1. 2 m L ()r In comparing with (48) after correcting for the local center of mass motion, I find that the polarization increases the diffusion coefficient by -. . I DI] N (1 _ ()2)()lnf 0 53/772 71?: 1 ‘ (4.10) where 1)? represents the previous estimate in Eq. (4.8). It is apparent that the contribution of the polarization effect is negligible for temperatures T >> em .93)!!!“ However, at temperatures comparable to 623%,13118 contribution could be significant; notably, at those temperatures Fermi effects also need to play a role. The isospin diffusion induced by mechanical forces has analogy in an electric current induced by the electric fields. Indeed, for large enough systems, the Coulomb interactions can contribute currents altering the concentration and, for completeness, I evaluate the conductivity (7E for nuclear matter, relating the. isospin 98 flux to the electric field, P120158, (4.11) where 8 is the local electric field. 4.3 Fluxes from the Boltzmann Equation Set 4.3.1 Coupled Boltzmann Equations The two components of the binary system will be described in terms of the quasiparticle distribution functions f,(p,r,1). The local 1nacroscopic (‘jiiantities 11(1‘, 1.) are expresst d as momentum integrals of f . I r. ( I.’ I, r‘ s 10 hr (‘2.7r1’1)3 . .\ A where g is the intrinsic degenm'acy factor. Different standard expressions for macroscopic quantities in terms of f , such as pressure and heat flow, are listed in Appendix A. The components are assumed to follow the nonrelativistic set of coupled fermion Boltzmann equatirms, without 1111)mentuni—dependence in the MFs. f). ‘1, 9', (f'+p 1+1? .(._‘/_. — ' ' = ' '- 4. 3 (’11 In,- 111‘ 1 (9p 1' ( 1 1 The terms on the 1.11s. account for the changes in .1,- due to the movement of quasiparticles and their acceleration under the influence of mean-field and external forces, included in F,, while the r.h.s. accounts for the changes in f, due to collisions. In the following. I shall often denote the l.h.s. of a. Boltzmann equation as 19,-. With (fa/(IQ and 11‘ representing the differential cross section and relative 99 velocity, respectively, the collision integral for particle 1 is «11 = Jii+J12 g :1 1,: ([011 7 7 I I *1”! W 111-119 ( d9) (1. 11.1. f..—1. 11.1.1...) -/ i. (10 ” 7 I I ~I ~I 4.5—7,9,5, / 1131121112 v ((19122) (that—1.1.1.1.). (4.14) Here, f = 1 — f is the Pauli principle factor. The factor of 1/2 in front of the first r.h.s. J11 term, compared to the J12 term, compensates for the double-counting of final states when integration is done over the full spherical angle in scattering of identical particles. The subscript. (1 and the primes in combination with the particle subscripts 1 and 2 are used to keep track of incoming and outgoing particles for a collision. Other than in the context of particle components, such as here, the 1 and 2 subscripts will not be utilized in this chapter. The collision integral .12 for particles 2 follows from (4.14) upon interchange of the indices 1 and 2. As it stands, the set of the Boltzmann equations (4.13), with (4.14). preserves the number of each species. In the macroscopic quantities (4.12). the distribution function f gets multiplied by the degeneracy factor 9. When considering changes of macroscopic quantities (4.12) dictated by the Boltzmann equation (4.13), the changing distribution function f continues to be multiplied by g. In these equation, the factor of f for the other particle in the collision integral J is accompanied by its own factor of 9. As a consequence, in the variety of physical quantities I derived, the factor of f is always accompanied by the factor of g. while, however, f is not. To simplify the notation, in the derivations that follow, I will suppress the factors of g. onLv to restore those factors towards the end of the. derivations. When the Boltzmann equation set is used to study the temporal changes of densities of the quantities conserved in collisions. i.e. number of species. energy and momentum, local conservation laws follow. Those ccmservation laws are discussed in 100 Appendix B. 4.3.2 Strategy for Solving the Boltzmann Equation Set Irreversible transport takes place when the system is brought out of equilibrium such as under the influence of external perturbation. Aiming at the transport coefficients, I shall assume that the deviations from the equilibrium are small, of the order of some parameter c that sets the scale for temporal and spatial changes in the system. Then the distribution functions may be expanded in the power series in c [76, 75] fzflm+fln+flm+nn (in) where f “‘1 represent. the consecutive terms of expansion and [(0) is the strict local equilibrium solution. The terms of expansion in f may be nominally found by expanding the collision integrals in 6, following (4.15), expanding, simultaneously, the derivative terms in the equations and by demanding a consistency, 1 2 . n 1 2 Dfl+Df’+...:.1,”+Jj’+.1jl+.... (4.16) Here, one recognizes that the derivatives, themselves, bring in a power of ( into the equations and, thus, the derivative series starts with a first order term in 6. While I nominally included the zeroth-order term in the expansion of the collision integral J,, the integral vanishes for the equilibrium functions ff” 2 1/{exp [(W — [1]) /T] +1} , (4.17) where 11.], v and T are the local kinetic chemical potential. \v'elocity and temperature which are functions of r and 1., consistent with the Euler equations (B.7). Notably, the vanishing of the collision integrals is frequently exploited in deriving the form of 101 the equilibrium functions, leading to the requirement that f,- /f~J is given by the exponential of a linear combination of the conserved quantities. In the context of specific transport coefficients, the boundary conditions for the Euler equations (8.7) may be chosen to generate just those irreversible fluxes, and forces driving those fluxes, that are of interest. The equation set (4.16) can be solved bv iteration, order by order in c, requiring 19‘.“ = .11”. (4.18) l , (0) .' .7 _ . _. (1). ..°- _ (1 Thus, f ~ may be introduced mto Di, producing ”D. and allowing to find f . J ., z - J . . 1 . . 2 . 2 Next, Inserting f j 1 into D. yields ”DE 1 that. allows to find f] 1 and so on. For finding the coefficients of linear transport, only one iteration above is necessary, Since f: 1, as linear in gradlents, yield d1ss1pative fluxes that are lmear in those gradients. The local equilibrium functions on its own produce no dissipative fluxes, as the species local velocities V j and heat flux Q vanish. while the kinetic pressure tensor P is diagonal, _ 13' 111V?) 2 [(11) p f‘°’(p.r~t)=0. (4.193) 2wh)3 nu Qw) : Z/(d'gp .1.—’2 —f(m(prt)—() (4.191)) 27111)3 )2n5 n5 :(0) pig-_E (0) 2n : . = z/e’e— —:r .1..— (4.19) in the frame where the local velocity vanishes v(r, t) = 0, with E representing the local kinetic energy per particle. The above fluxes reduce the local continuity equations to the ideal-fluid Euler equations. 102 4.3.3 Boltzmann Set in the Linear Approximation I now consider the terms linear in derivatives around a given point, i.e. the case of k = 1 in (4.18), for the Boltzmann equation set. On representing the distribution functions as f j = fjm) + f j“), I expand the collision integrals Ji, to get terms Jim linear in f)”. Upon representing fil) as f)” = fjo‘) leO) 961-, I get for the k = 1 2' = 1 version of (4.18): 01:”) +3.81?” +11 01“” (”H ml (9r 1 . (9p = ‘Ill((rb) _ 112015) 1 (420) where 1 {131)- (1'0“ [1‘7 7" = 1 4219f” -—‘2 1(9)) 11.0., / (271).“ ‘ (do) (0) (0) ~(())’ .~(())’ fi fja fi fja (Q15 + (16.111 _ d): — (163(1) 9 (4.21) and where I have utilized the property of the equilibrium functions (0) (0) ~(Ol’ ~(0)’ ~(0) ~(0) . (0)' (“l’ . f. fja f; fja : f.” fja ft fja - (422) The result for 1' : 2. analogous to (4.20), is obtained through an i1‘1terchange of the indices 1 and 2. The l.h.s. of Eq. (4.20) contains the derivatives of equilibrium distribution functions with respect to t, r and p. These derivatives can be expressed in terms of the parameters describing the functions (4.17), i.e. 11., T and v. Through the use of the Euler equations (Appendix B) and equilibrium identities (Appendix C), moreover, the temporal derivatives may be eliminated to yield for the rescaled l.h.s. of (4.20) It?! T 9012(0) _9 ‘3 p 0T pop. 0 p f (0) ~(U) 9 _ ' — + — - 7‘! + — -d12. (4.23) [1 f1 4711 01‘ ()r m 1 T ml [)1 103 Here, a. synnnetrized traceless tensor is defined as xy = % (xy + yx) — g (x - y) T, Elll(l {)1 {)2 F1 F2 3 #1 #2 5 E1 E2 8T (1 . : __._ __ T _ _ _ _ _ 12 P l( 7711 + n1.2)+ (91‘ (m1 T mgT) + 3T (m1 mg 8r ’ . F F. ('9 . s 3 8T 2 P1P2[(__1_+_2_)+l__(_lil__l_l2_)+(_l___2_) —] , (4.24) {2 ml 711;; dr ml 1712 m1 7712 Or where s,- is the entropy per particle for species 1', 3,: = (5_E_, / 3 — 11,-) / T. The result for species 2 in the Boltzmann equation is obtained by interchanging the indices 1 and 2 in Eqs. (4.23) and (4.24). Note that d2, = —d12. The I‘e}.)resentati(111 (4.23) for the l.h.s. of the linearized Boltzmann equation (4.20) exhibits the thermodynamic forces driving the dissipative transport in a medium. Thus. one has the tensor of velocity gradients fiv contracted in (4.23) with the tensor from particle momentum. The distOItion of the momentum distribution associated with the velocity gradients gives rise to the tensorial dissipative momentum flux in a medium. As to the vectorial driving forces, they all couple to the momentum in (4.23) and they all can contribute to the vector fluxes in the medium, i.e. the particle and heat fluxes, as permitted by the Curie law. The criterion that I, however, employed in separating the driving vectors forces in (4.23) was that of synnnetry under the particle i1‘1terchange. When considering the diffusion in a binary system, with the two components flowing in opposite directions in a local frame, one expects the driving force to be of opposite sign for the two species. On the other hand, in the case of heat conduction, one expects the driving force to distort the distributions of the two species in the same direction. Regarding the antisynnnetric driving force in (4.24), one may note that for conservative forces we have '9 R=—#m. Hfi) One can combine then the first with the second term on the r.h.s. of (4.24) by 104 introducing the net. cl11~2111ical }_)ote11tials 11: = u,- + U,- and obtaining ,) )« t) 1.’ 1.’ s 81 (9T d12=“” .— iL—il + —‘— 2 7—. (4.26) p (9r m 1 771.2 m 1 7712 ()r For a constant. temperature. T, the driving force behind diffusion is the gradient of the difference between the chemical potentials per unit mass, [1‘12 = 113 /m1 - 113/mg, as expected from phenolnenological considerations [148]. However, the temperature gradient can contribute to the diffusion as well, which is known as the thermal diffusion or Soret effect. Note that the vector driving forces in (4.23) vanish when the temperature and the difference of net che1'1'1ical potentials per mass are uniform throughout. a system. Given the. ty}.)ical constraints on a system, it. can be more. convenient to obtain the driving forces in terms of the net pressure 1” , temperature T and concentration 6, rather than “’12 and T. Thus, on expressing the potential difference as 11'” : [132(1”, '1’. 6). I get ")p'. 0/1'. 011‘ I, '. : ‘ '2 m —_'—2 dT . £2 15, 4.27 (”12 ((1)1): 726 ( + 5” 1".6 + 00 PT U ( ) and (1,, = (“p/’2 (nfive + 1133, VT + 113‘2 V6) , (4.28) that I will utilize further on. The coefficient functions are, 0/1' “F: 2 ( 12) 1 (4.298.) 12 apt T96 {ill} 81 82 HT‘ : ( 12 ——— . 4.2% 12 ( (97" )P‘,6 + ("11 7712 ( ) ‘ 8,1! 11". Z (12) ~ (4.291,) ‘2 ()6 m. and specific expressions for those functions in the. nuclear-matter case are given in Appendix D. Notably, however, the concentration 6 111ay not be a convenient variable in the phase transition region where the transformation between the chemical potential difference and 5 is generally not invertible. With the l.h.s. of the linearized Boltzmann set (4.20) linear in the driving forces exhibited on the r.h.s. of (4.23), and with the collision integrals linear in the deviation form-factors 1,1"). the form factors need to be linear in the driving forces, (5)1 : —A1 ' VT —‘ E]: - C1 ‘dm, :1): 31 —A2 -VT—§2: ll (1'52 — C2 - d... (4.30) where A, B and C do not depend on the forces. ()11 inserting (4.30) into (4.20), one gets the following equations. when keeping alternz-itively a selected exclusive driving force finite: /)p1[‘fim.~i0) : 111(C) + 112(C)1 (4313‘) 1 _l)p1[~ . 2“) .320.) : [22(C) + 121(0) s (4311)) 2. when keeping d1), _ P215 («1) ~<0) _, fi I fi 432 .,111'[‘ 1 f1 _ 11( )+ 12( ) (I ) and another one, with indices 1 and 2 interchanged. when keeping Vv, and. finally, i — 9F. p ‘0’ “”1 = I (A) + 1 . (A) (4 33) 2m. 341 ml T2 l l H 12 ’ ' and another one, with 1 and 2 i1’1tercha1’1ged, when keeping VT (while d12 = ). The linearized collision integrals 12'1‘ cannot change the tensorial character of objects upon which they (‘11.)erate. Moreover, the only vector that can be locally utilized in the object construction is the momentum p. This i111plies. then, the 106 following representation within the set (4.30): C, : (fl-(1)2)B, (4.3421) 01 2 r' p a p . A1: .2 ———.—1. —. 4.341) a (p ) 2m, 3_') 7H,:T2 ' ( ) E = 11,012);qu . (4.3%) Here, the tensorial factors are enforced by construction. The factorization of the scalar factors is either suggested by the respective linearized Boltzmann equation or will be convenience later on. The unknown functions a, b and c can be principally found by inserting (4.34) into Eqs. (4.31)-(4.33). The resulting equations are. however, generally quite complicated and analytic solutions are only known in some special cases. I11 practical calculations, I shall content. myself with a. power expansion for the unknown functions. It has been shown that any termination of the expansion will produce lower bounds for the transport coefficients and that the lowest terms yield a. predominant contribution to the coefficients [76']. 4.3.4 Formal Results for Transport Coefficients Before solving Eqs. (4.31)-(4.33), I shall obtain formal results for the transport. coefficients. assuming that solutions to (4.31)~(4.33) exist. I shall start with the diffusion. The velocity for species 1 is 1 (13]) p 1 (13]) p ~~ V ___ __ ____(5 :_ _' 401(0) *1 71.1 (27rh)3 ml f1 r11 / (27rh)3 ml 951f1 ' l . d3!) = T. Weh [111(C)+112(C)l T (13]) : _ 'f_ _._.__ . , , V 3 / (27Tfi.)3 A1 1111(0) + 112(C)l T (l3 1 —d. 3 / (3:11? 0. ~ (111(0) + 112(0)] . (1.35) 107 where I have utilized (4.30) and (4.34). The contribution of a tensorial driving force to the vector flow drops out under the integration over mtnnentum, as required by the Curie principle. W'ith a result. for X2 analogous to (4.35), I get for the difference. of average velocities (utilized for the sake of particular symmetry between the components) T (13]) L—x. : 4713 / WA.~[I.+I.1 (13]) +/WA2 . [122(0) + 121(0)] T 13) —d12 — [ET—’11? C1 '1111(C)+ 112(cll (13]) + / W C2-[122(C)+121(C)l : {I}: ((A.C)vr+ {C.C}d,2). (4.30) where the brace ]_)roduct {-. } is an abbreviation for the integral ccnnlnnations of vectors A and C. multiplying the driving forces. The brace product. has been first introduced for a classical gas [76']. The fermion generalization of the prmluct and its properties are discussed in the Appendix E; see also [147]. The diffusion coefficient is best. defined with regard to the most. common conditions under which the diffusion might occur, i.e. at uniform pressure and temperature. but varying concentration. I have then, cf. (4.3), y, — y, = ” I‘.S : —ME 1).-V6. (4.37) (ml + H12)H1712 p, [)2 where 71112 is the reduced mass, 1/11112 = 1/ml + 1/7712. Respectively, when P' and 108 T vary, with dlg given by (4.28), I write the r.h.s. of (4.36) as - HP ‘ D. (V0 + i,— VP’ + 1.. VT) , (4.33) p 71 17112 /)1 P2 Xi’X'z: where, shnplifying the notation, I dropped the subscripts 12 on coefficients TI. The diffusion coefficient in the above. is given by _ T 1" . . 2 1).— H— (mp2) {0.0}, (4.39) 377112 11 p and _ IT: 1 /) {A,C} — IT‘S H15 [)1 [)2 {(3.0} . One can note that the expressions above contain 11‘5 in the denon1inators. Normally, the positive nature of the derivative (4.20c) is ensured by the demand of the system stability. However. across the region of a phase transition the concentration generally changes while the chemical potentials generally do not, so that II‘5 = 0. While the coefficient D5 above is the one I am after as the standard one in describing diffusion, in the phase transition region it can be beneficial to resort to the description of diffusion as responding to the gradient of the potential difference in (4.28). Noti-rbly, as explained in the Appendix E, the brace product. { C, C} in (4.39) is positive definite. This ensures the positive nature of D6 away from the phase transition and, in general, ensures that, at a constant temperature, the irreversible asymmetry flux flmvs in the direction from a. higher potential difference #32 to lower. As to the Soret effect. i.e. diffusion driven by the temperature gradient, described in (438)-(440). it has its counterpart in the heat flow driven by a concentration gradient. termed Dufour effect. Transport coefficients for counterpart effects are related through Onsager relations [149] that are also borne out by my 109 results. Diffusion driven by pressure is rarely of interest, because of the usually short times for reaching mechanical equilibrium in nuclear systems, compared to the equilibrium with respect to temperature or concentration. However, an irreversible particle flux may be further driven by external forces, such as due to an electric field 5. With the flux induced by the field given by 1“,; 2 (IE 5, where (IE is conductivity, with the first equality in (4.37), and with (4.36) and (4.24), I find for the conductivity T [)1 P2 2 (12 (11 ([2 q] n g 2 ——— CC = ———— —D~, 4.41 where q, is charge of species '1'. One could see, that conductivity is closely tied to diffusivity. While my primary aim is to obtain coefficients characterizing the dissipative particle transport. due to the generality of the results I can also obtain the coefficients for the transport of energy and 1110111entum. Thus. starting with the expression (A.1e) in a local frame. and proceeding as in the case of (4.35) and (4.36), I get, with (4.33), T 5 Q1 ‘1' Q2 : —§ ({A,A} VI +{C,A}d12)+ 3(E17'11V1—E2712V2) T = —§({A,A} VT+{C,A}d1-2) (4.42) 1 E E +§(———'—:2) Mon—v.2). ml 7712 p where in the second step I make use of the condition on local velocities ,01V1 + [)2 V2 = 0. The standard procedure [148] in coping with the heat flux is to break it into a contribution that can be associated with the net. movement of particles and into a remnant, driven by the temperature gradient, representing the heat conduction. With this, the driving force d12 needs to be eliminated from the 110 heat flux in favor of the species velocities. Using (4.36), I find Q1+ Q2 ——— —V'1‘:§({A.A}— {354) {QC} _ , é _Ei_§g 91,02 {C,A} , , +(V1 V2) 13(1111 m) p +{C,C} . (4.43) The coefficient - _ T {CA}2 , h. _ -3— ({A,A} — m) , (4.44) relating the heat flow to the temperature gradient, is the heat conduction coefficient. me (4.44) and considerations in Appendix E, it follows that A“ given by Eq. (4.44) is positive definite. The final important coefficient that I will obtain, for completeness. is the viscosity. The modification of the momentum flux tensor (AM). on a('-('-1;1unt of the distortion of momentum distributions described by (4.30), is =‘” (13]? “13:5 (131) :2- P = —6 _o . [(271703 mi fl+,/(27rh)3 171.2 f2 (13]) :1) = i (0)10) (13p p—rpt = '=' (0)10) Z ‘ — B I — —— B :v . /(27rh)3 "’1 ( l Vv) 1 fl [(271103 m2 1 v) f2 f2 D: (137) = fi (0) ~(0) (13]) = :1) (0) ”(0) Z __ V . B 2 —— ' _____I B I __ ‘_ ‘ 5 V (/ (2775-13 1 ml 1 1 + / (27th.)3 2 1112 f2 .2 'U The coefficient of proportionality between the shear correction to the pressure tensor and the tensor of velocity derivatives is, up to a factor of 2, the shear viscosity coefficient T 11 {L135}. (4.46) 7]: V As with other results for coefficients, from Appendix E it folltwvs that. the result for 7) above is positive definite, as I.)l'1ysically required [148]. 111 On account of symmetry considerations within the linear theory, the changes in temperature or concentration do not affect the pressure tensor. However, the situation changes if one goes beyond the linear approximation. For a general discussion of different higher-order effects see Ref. [76]. As a next step, I need to find the form factors in (4.30); that requires finding the functions a, b and c in (4.34) by solving Eqs. (4.31)-(4.33). 4.4 Transport Coefficients in Terms of Cross Sections 4.4.1 Constraints on Deviations from Equilibrium Since the. zeroth-order, in derivative expansion, local-equilibrium distributions are constructed to produce the local particle densities, net velocity and net energy, corrections to the distributions cannot alter those InacroscoInc quantities. Thus, I have locally the constraints . (13p - _y 0111' Z W (if, = 0, (4.418.) - (13]) (13]) - (5 V = ———-,- 5 —— (5 = 0, 4.471 - (131) p2 (13p [)2 (5 E = — (i , — (5 = 0. 4.4"? (71") / (27th.)3 21111 f1 + f (271103 21112 b ( H) With driving forces being independent of each other and with form factors in (4.30) being independent of the forces, each of the form factors sets must separately meet the constraints. By inspection, however, one can see that the density and energy constraints are met automatically with the expressions ( 4.34) of form factors. Moreover, the tensorial distortion (4.34c) satisfies all the constraints. At a general level, the ability to meet the constraints while solving Eqs. (4.31)-(4.33) relies on 112 the fact that the linearized collision integrals 1,,- (in Eqs. (4.20) and (4.21)) nullify quantities conserved 111 collisions, so a combination of the conserved quantities may be employed in co1'1structing the form factors (1),, ensuring that the constraints are met. When the transport coefficients get expressed in terms of the brace products, though, ensuring that the constraints are met becomes actually irrelevant for results on the transport coefficients, because the linearized integrals and the corresponding brace products nullify the conserved quantities. Given cross secticms and equilibrium particle distributions, the set of equations (4.31)-(4.33) may be 1.1rincipally solved. However, such a solution is generally complicated and would likely not produce clear links between the outcome and input to the calculations. O11 the other hand. the experience has been that when expanding the form-factor functions, a. b and c in (4.34), in power series in 112, the lowest-order results represent excellent approximations to the complete results and are quite transparent. e.g. [144]. Thus, I adopt here the latter strategy and test the 1.1ccul'1u‘ry of the results in a few selected cases. 4.4.2 Diffusivity If one inserts (4.3411) with c,(p2) : c,- i11to the local velocity ('(‘mstraint (4.4711), one gets the requirement _ .3 (1 (I I) 2 (o) ~(0) + 2 (13]) 2 (0) ~(U) . 1 . . : 0. 4.48 pl (27rh)5‘p l 1 p2 (27rh)3 1) f2 2 ( ) After partial integrations, I find that this is equivalent to the requirement. c1 2 —c2 E c. When (5',- is constant within each species, then C,- is up to a factor equal to momentum and, thus, gets nullified by the linearized collision integral 'unlthin. each species 1,,(C) : 0. To obtain a value for c, I multiply the first of Eqs. (4.31) by C1 113 and the second by C2, add the equations side by side and integrate over momenta. \Vith this. I get an equation where both sides are explicitly positive definite and, in particular, the l.h.s. is similar to the l.h.s. of Eq. (4.48), but with an opposite sign between the component terms. That side of the equation can be integrated out employing the explicit form of [(0) from Eq. (4.17). The other side of the resulting equation represents { C. C} where only the interspecies integrals survive. On solving the equation for c, I find c _ 6 P1 P2 p.112 ‘ (4.49) where 3 Si . _ 2 (I P1 (I P2 ( ,4 (1012 _ I 2 (0) ((1) ~(0)1111); ,_ “2—” / (2.1.)3 (21.1.)3 '1‘2' (.112) (p‘ p‘) f' ‘2 1 '2 ' (4'0”) The i11t1‘1gral stems from a transformed brace product {C, C} and I resurrect here the degeneracy factors g. For the brace product itself, I find {C~C}=:( p > (“FE (4-51) 2 P1 P2 X12. On inserting this into the diffusivity (4.30), I obtain (‘T 11‘5 . ., 2 1).:3— (pm) . (4.52) "112 n X12 P In the above. one sees that the diffusion coefficient depends both on the equation of state. through the factor I16, and on the cross section for collisions between the species, through )(12. The collisions between the species are weighted with the momentum transfer squared. Only those collisions between species that are characterized by large mmnentum transfers suppress the diffusivity and help localize the species. The marginalization of collisions with low momentum transfers is a common feature of all transport ('-(‘)effi(_-ie11ts. 114 At high temperatures the Fermi gas reduces to the Boltzmann gas. In the absence of mean-field effects, one finds 11‘5 ~ 4% for small asymmetries. The integral X12 is then of the order 712 012 p3/m ~ 712 or V'm. T3. Together, these yield no 05 ~ 1121/ §. The precise high—T result for isotropic cross—sections in the interaction of species with equal mass m is [76, 75] _ 3 0 _ 811012 (4.53) The square-root dependence on tenmerature will be evident in the numerical results at high T. With an inclusion of the mean field, with the net energy quadratic in asymmetry, the derivative II“ gets 1111‘)difie(.l into 11‘5 ~ [2 (T + 21’5’,’,’,,,)]/m. Thus, the mean field enhances the diffusion. At low te11‘1}.)eratures. the derivative fl‘5 is simply propm‘tional to the synnnetry energy, 116 ~ (4e,,y,,,)/m.. As to the collisicmal denominator of the diffusion coefficient, at. low temperatures the collisions take place only in the immediate vicinity of the Fermi surface. I can write the product of equilibrium functions in the collision integral as 1 2(i-osh{(.-21% —;_1.,-)/T} , '1 0 0 ”()I*()/ , . ,. . ,. f’ 2‘) [l y 211114.21le15, where 14.: (4.54) and, at low T, K, ~ 27111176022 — 11%,). The integration in (4.50) yields X12 ~ 012 771.2 T3 712/11} In consequence, I find that the diffusion coefficient diverges as l/T2 at low temperatures. For the spin diffusion coefficient, one finds within the low temperature Landau Fermi-liquid theory [90] ,2 " (F (1 {'1' where v; is Fermi velocity, [76' is a. spin-antisyunmetric Landau coefficient and T0 is a characteristic relaxation time that scales as Tn w T”. The isospin diffusivity for symmetric matter should differ from the spin diffusivity in the replacement of the spin—antisynnnetric Landau parameter with the isospin asymmetric parameter, neither of which has a significant temperature dependence. Thus, here consistently I find a T"2 divergence of the diffusivity at low temperatures. Moreover, the factor (1 + (3‘) is nothing else but a rescaled symmetry energy, with F61 being the ratio of the interaction to the kinetic contribution to the energy [150]. Thus, here consistently I find a proportionality of the (,liffusivity to the synnnetry energy at low temperatures. To summarize the above results on diffusivity. I find that the. (.liffusivity is inversely proporticmal to the cross section between species for high 1110111ent.u111 transfers. h'lorecwer, whether at. low or high temperatures. the diffusivity is sensitive to the synnnetry energy in the 111ean—fields. The 111ean-field sensitivity is associated with the factor 89/112 0 (.71 U12 _ 52/1,”, 4 )m, ff 7‘- —‘f* — .~ + ‘.,.,m—~ do ()0 m 1 ")2 ()0 m n15 : inf where the last. equality pertains to the system of nmrtrons and protons and cw", represents the interaction contribution to the symmetry energy at the relevant. density. While I obtained the diffusivity here assuming constant c, in (4.3411), I will show that the next-order term in the expansion of (2,- increases the diffusion coefficient D5 only by 201; or less in the case of my interest. 4.4.3 Heat Conductivity Evaluation of the heat. conduction and shear viscosity coefficients requires similar methodology to that utilized for the diffusivity. \Vhile these coefficients have been obtained in the past for a. one cmnponent Fermi system [144], it can be still important to find them for the two component system. If one assumes (1,012) 2 (1.,- in (4.34b), then, interestingly, one finds that the momentum constraint (4.4711) is automatically satisfied. To obtain the values for (1,, I multiply Eq. (4.33) on both sides by A1 and integrate over momenta and I multiply the equation analogous to (4.33) by A2 and also integrate over momenta. As a. consequence, I get a set of equations for a,- of the form L} =AJ-111.1+.Aj2(12, [=12, (4.56) where A,-, are coefficients independent of (1., 1 A” : T17 ([AA],,' + [A).A,]12) , A12 2 A21 : [A].A2]12 ._ (4.57) (1102 cf. Appendix E. and l .2 25 2 . LJ' :— m<7711§i— 3"] (EJ) ) , (4.58) . 2 . . . where Ell and (E) are. respectively, the average local square k111et1c energy of species j and square average local kinetic energy of the species. The solution to the set. (4.56) is (11 2 (A22 L1 — A12 142) /AA~ (12 =(A11L'2 — A12 L1) /AA , (4.59) where the determinant is a. = A11 A22 — 4%,. (4.60) The brace product. {A, A} for use. in calculating the heat conduction coefficient K. in 117 (4.44) is {A.A} 20.1141 +0210. (4.61) The product {C, A} in (4.44) can be calculated given the values of a. and c, and {C, C} was already obtained before. 4.4.4 Shear Viscosity Evaluation of the shear viscosity coefficient 1) follows similar steps to those involved in the evaluation of H. Thus, I assume. (5(1)?) 2 b,- in (4.34c). To find the coefficient values, I convolute both sides of Eq. (4.32) with pzp and integrate over the momenta and I do the same with the other (onstraint equation for T31. The. l.h.s. integrations produce 1 (13]) ‘= = (0) ~(0) 2 (13]) .4 (0) ~(0) 20 \Vith the above. I get the set of equations for 1),: ‘20 . TIDE} =8111)1+BJ'252. j=1.2, (4.63) where the coefficients 8 are given by. n ., Q Bii : [IT—_Pfiln‘ + [(fib (fiblm, 312 = 321 = [(Ijtfilh (P:)2l12 (4.64) Solving the set for 1). One finds ‘20 ()1 = —3AB (/)1E1322 — P2 E2 812) .~ '20 ., ()2 = .—,_‘ (#2 £2 811 — P1 E1 312) . (4-00) 3438 118 where the determinant is AB = 811322 — 8:122. (4.66) The brace product. for calculating the shear vis<.'osity coefficient 7} = TIMETj} becomes {RE} 2 (b1 {)1 E1 +1);g p252) . (4.67) “IE” 4.5 Quantitative Results 4.5.1 Transport Coefficients I next calculate the transport. coefficients as a. function of density and temperature. using experimentally measured 1nlcleon-nucleon cross sections. The cross sections may be altered in matter. compared to free space, but the modifications are presumably more important. at low than at the high momentum transfers iinpmtant for the transport coefficients. With regard to the diffusivity. I will first ignore any mean-field contribution to the chemical potential difference. between species. This yields a reference (:liff1.1si\.-'ity to which the diffusivity affected by mean fields may be compared. The diffusivity for the experimental cross sections and no interaction contributions to the symmetry energy is shown at. (5 = 0 and different densities n in Fig. 4.1, as a function of temperature T. At low temperatures, the diffusivity diverges due to a suppression of collisions by the Pauli principle. At high temperatures, compared to the Fermi energy, the role of the Pauli principle is diminished and the diffusivity acquires a characteristic x/T dependence. At moderate temperatures and densities in the vicinity and above normal density, the diffusion coefficient turns out to be in the vicinity of my original estimate of D] N 0.2 fmc. 119 It should be mentioned that, for synnnetric matter, the factors for temperature and pressure gradients in the thermodynamic force d12 (4.28) vanish, HP 2 0 and HT = 0, and the brace product in (4.40) vanishes, {A, C} 2 0, yielding 147 = 0 in (4.38). As physically required, the temperature and pressure gradients produce no relative motion of neutrons and protons for the symmetric matter. The diffusivity at normal density at different asynnnetries is next shown in Fig. 4.2 as a function of temperature. Because of charge symmetry, the diffusivity does not depend on the sign of 6. At. low temperatures the diffusivity is generally expected to behave as 73, 2 h 1),, m3 "I”2 (I ‘ while at high temperatures in the manner prescribed by (4.53). \«Vith the respective behaviors serving as a. guidance, I provide a parmnetrization of the numerical results for 1),; as a function of n, T and 6, -. 11.34 "5" D, = (1—01942) (1) T238 "0 1.746 0'5“ . . +__ (11) +(1.0()5857*’-"'3(-@) . (4.69) T no 72 Here, temperature T is in MeV and the diffusivity D, is in fmc. The parametrization describes the numerical results to an accuracy better than 4% within the region of thermodynamic parameters of 1.0 g n/no S 4.0, 10 MeV _<_ T g 100 MeV and (6| _<_ 0.4. This is, generally, the parameter region of interest in intermediate-energy reactions. The heat conductivity is shown for symmetric matter at different densities in Fig. 4.3, as a function of temperature. The results are similar to those in Ref. [144], though there the two component nature of nuclear matter was ignored and the isospin—averaged nucler111-nucleon cross-sections have been used. A ('rloser 120 1.0 .,.. , .. 4 a . . , TIM“ /——-0.ln0 . I, / ————05n0 0.8- |:I\( / ————-—l.0n() - , hi - - - - 2.0 no / l1, / — —3.0no xx 0.6- l\\‘ / —-—4.0n0 // ‘ 9 O p- L L. Figure 4.1: Isospin diffusion coefficient D1 in symmetric matter, for U,- = 0, at dif- ferent indicated densities, as a function of temperature T. In the high-temperature limit, the diffusion coefficient exhibits the behavior D, cx \/T/n. Correspondingly, at high temperatures in the figure, the largest coefficient values are obtained for the lowest densities and the lowest coefficient values are obtained for the highest den- sities. In the low-temperature limit, the diffusion coefficient exhibits the behavior D, (X 713/ 2 / T2 and the order of the results in density reverses. 121 0.40 0.35 — 0.30 - D1 (fm c) O. 15 n l n I 4 l L l a 0 20 4O 60 80 100 T (MeV) Figure 4.2: Isospin diffusion coefficient. D] at normal density n. = no 2 0.16 fm‘3 and different indicated asymmetries (5, for U, = 0, as a function of temperature T. An increase in the asymmetry generally causes a decrease in the coefficient, as discussed in the text. 122 examination of results in Subsections 4.4.3 and 4.4.4 indicates that the use of the isospin-averaged cross-sections is, actually, justified for symmetric matter, when calculating the heat-conduction and shear-viscosity coefficients. Otherwise, however, Fig. 4.3 has been based on a more complete set of cross sections than results in [144]. As in the case of diffusivity, the heat conductivity diverges at low temperatures and tends to a classical behavior at high temperatures, exhibiting there no density dependence and being proportional to velocity, K. oc x/T. As in the case of diffusivity, I next provide a parametrization of the numerical results for the heat conductivity a as a function of 71, T and (5, -. 0.12:5? '2, 0'9"" 7.- : (14.0.1042) T0~rj(_’_) .401) 710 11 0.0816 , 71 0.0171 41.053142 (—) + 002387415627 (—) . (4.70) 'I In 77.0 Here, T is again in MeV and a is in c/f1112. The parameterization agrees with the numerical results to an accuracy better than 4% within the range of thermodynamic parameters indicated in the case of D]. The shear viscosity coefficient 7) is shown for symmetric matter at different. densities. as a function of temperature, in Fig.4.4. Again, the results are similar to those in Ref. [144]. At high temperatures, the dependence on density weakens and the viscosity becomes proportional to velocity. The numerical results for n are well described, to an accuracy better than 4% within the before-mentioned range, by . 856 a "8‘ 240.9 , “2 n=(1+0.1052) (1) ———(1’—) +2.1547‘0-76. (4.71) THO 71.0 T095 72.0 Here, 7) is in l\v*feV/fm‘2 c and T is in MeV. One notes in (4.69)—(4.71), that the diffusion coefficient weakly drops with increasing magnitude of asymmetry 6|. while the viscosity and heat conduction 123 0.3 ,,, 0.2-,\\ K (c/fm2 ) 0.1. —---2.0n0 _\ // — —3.0no ‘ ,// —-—4.0n() 0.0 -/ 1 1 1 1 . 1 . 1 0 20 4O 60 80 100 T (MeV) Figure 4.3: Thermal conductivity a: in symmetric nuclear matter, at different indi- cated densities in units of 7m, as a function of temperature T. The conductivity increases as density increases. 124 T] (MeV frn'2 c-l) -\ //// ——l.0n0 20- \.,/ / —-—-2.0no - // — —i.8no \/ _-_ ' n O 1 a 1 1 . 1 . a . 0 20 40 60 80 100 T (MeV) Figure 4.4: Shear viscosity 7) in symmetric nuclear matter, at different indicated densities in units of no. as a function of temperature T. The viscosity increases as density increases. 125 coefficients weakly increase. Given the weak ('lependencies, the behaviors exhibited in parametrizations represent, in practice, averages over the considered independent-parameter regions. Overall, the drop and rise in the respective coefficients with |6| is characteristic for a situation where the local flux of a component grows faster than the concentration of that component. That type of growth, with the magnitude of asynnnetry, typifies a. mixture of degenerate fermion gases. The general trends can be deduced following the mean—free—path arguments from Sec. 4.2. When the average velocity rises with asymmetry. so do the heat conduction and shear viscosity coefficients. Additional rise for those coefficients, in the. case at hand, can result from the Pauli principle effects and from the difference between cross sections for like and unlike particles. Regarding the diffusion coefficient, though, one needs to consider an irreversible part of relative particle fiux. under the condition of the concentration varying with position. If, starting with a given configuration of concentration gradients. one introduces uniform changes of concentration on top, not just the overall relative flux undergoes change but also the reversible flux of concentration gets altered. The rise in the relative flux associated with the velocity of a dominant component rising with concentration is normally more than con'ipensated by the rise in reversible flux, leading to a reduction in the irreversible flux and producing a reduction in diffusivity with particle asymmetry. A mean-field example where the reversible flux eats into the net flux reducing the diffusivity with increasing asymmetry is the estimate in Eq. (4.10), obtained there without invoking Fermi statistics. As is found in Secs. 4.3.3 and 4.4.2, the dependence of mean fields on species enters the diffusivity through the factor H‘S resulting from the variable change in thermodynamic driving force. from the difference of chemical potentials per mass to asymmetry. The simplest case where one can consider the impact of the mean fields is that of synnnetric nuclear matter, at 6 : 0. In this case. the factor may be 126 represented as H6 : i g + 48”" s y‘m ’ where. g, = Ora/(71,11,- _=_ { (cf. Appendix C). At high temperatures, one has approximately 5, z n, / T , so that n/E z 2T. The naive expectation is that egg", has . V a linear dependence on the net density, em’ = a, (it) , where (1., Z 14 MeV and I/ = 1. The mean-field amplification factor R = H6(c:’;',,,) / 116(6):.3’", : 0) for the diffusion (‘roefficient., assuming the linear and also quadratic density-dependence of ._i n! 6'» . . sym (l/ = 1 and 2) is shown in Fig. 4.5. The quadratic dependence gives higher amplification factors at n > 11.0, than the linear dependence, while the opposite is true at n. < no. At low temperatures and moderate to high densities the amplification is very strong suggesting that the diffusion could be used to probe the synnnetry energy. aside from the Ill-1110(lllllll nei1tron—proton cross sections. 127 4 .,.,-,.,. \ _ \ \ —— Linear , \ — — - Quadratic \ , 3 - \ - \ \ \ m \ \ Q: 2110 1 l n 1 1k 1 1 L 1 Figure 4.5: Mean-field enhancement. factor of the diffusion coefficient in symmetric nuclear matter, R E D,(U,)/D,(U.,~ : 0), at fixed density n, as a function of tem- perature T. The solid and dashed lines represent the factors for the assumed linear and quadratic dependence of the interaction synnnetry energy on density. The lines from top to bottom are for densities n. = 2710, no, 0.5 no and 0.1-no, respectively. At normal density the results for the two dependencies coincide. 128 4.5.2 Testing the Form-Factor Expansion The calculations of transport coefficients above have been performed assuming that the functions (1,, b,- and c, in Eqs. (4.34) can be approximated by constants. In the more general case, the functions can be expanded in the series in p2, e.g. (3(1)?) : c)” + (1:2) [)2 + (753)114 + . .. . (4.73) The coefficients of the expansion can be found by ccmsidering moments of the form-factor equations (4.31)-(4.33). With the more general form of the form-factor functions, the transport coefficients generally increase, but their rise is generally very limited. To illustrate the magnitude of higher—order effects. I provide in Table 4.1 results for the diffusivity obtained in the standard first-order and in the higher-order calculations at sample densities and temperatures. In the indicated cases, the second-order calculations never increase the diffusion coefficient by more than 3% above the first-order calculations. The efficiency of my Monte-Carlo procedure employed to evaluate the integrals for coefficients worsens as the order of the calculations increases and, correspondingly. I provide only a. single third-order result for illustraticm. 4.5.3 Isospin Equilibration To gain a further insight whether the diffusion coefficient. results are. sensible, I will consider the issue of isospin equilibration in a reacting system [151] such as 96RU + 962r at Elmo/.4 = 100 MeV. At this energy, the Boltzmann-limit estimate for temperature, T ~ % E10,, /A ~ 16 MeV, and the degenerate Fermi limit estimate, T ~ W ~ 20 MeV for a. % .4/(8 MeV). produce similar results. The tyincal densities in this reaction are around normal. Based on Figs. 4.1 and 4.5, one can 129 Table 4.1: Diffusion coefficient D, obtained within different orders of calculation, using experimental np cross sections, at sample densities n and temperatures T in symmetric nuclear matter, for species-independent mean fields. The numerical errors of the results on D; are indicated in parenthesis for the least-significant digits. The last two columns, separated by the ’i’ sign, give, respectively, the relative change in the result for the highest calculated order compared to the first order and the error for that change. n T D, Relative 1St order 2Ild order 3rd order Change fm‘3 MeV fm c "/1. 0.016 10 0.29949(15) 0.3055(12) 2.0 i 0.4 0.016 60 2.3891(18) 2.390(14) 0.0 :1: 0.6 0.16 10 0.27964(21) 0.2800(29) 0.2809(25) 0.5 :‘c 0.9 0.16 60 0.29591(24) 0.2965(19) 0.2 :L- 0.7 0.32 10 0.4446(15) 0.4465(26) 0.4 4: 0.7 0.32 60 0.1818705) 0.1827(13) 0.5 :1: 0.7 estimate the streaming contributi<1n to the diffusivity at 0.21 flll c and the mean-field contribution at 0.20 fm c, for a net D] m 0.41 fmc. Considering the direction perpendicular to the plane of ccmtact between the nuclei, with nuclei extending a distance L N (ff/no)”3 ~ 8f1n both ways from the interface, one may use the one-dimensional diffusion equation to estimate the isospin equilibration 06 (926 _, (7t. 2 D, (713 (4.74) where :1: is the direction perpendicular to the interface, cf. Eq. (4.7). With isospin flux vanishing at the boundaries of the region [—L, L]. the solution to (4.74) is (5(33, t.) = 600 + Z (1,, sin kn :1: exp (—D, 1:3, I) + 21),, cos (1,, .1' exp (—D, (1,2,1) (4.75) n=l n=l where kn L = (n — %) 7t and (1,, L = n 77. The coefficients (1,, and 1),, are determined by the initial conditions and, in the case in question, 0,, 2 0. The different terms in the expansion (4.75) correspoml to the different levels of 130 detail in the distribution of concentration, as characterized by the different wavevectors. The greater the detail the faster the information is erased, with the erasure rates proportional to wavevectors squared and with the overall distribution tending towards 600 as t ——1 00. The late-stage approach to equilibrium is governed by the rate for the term with the lowest wavevector, i.e. (1,. Defining the isospin equilibration time t” as one for which the original isospin asymmetry between the nuclei is reduced by half, I get from (4.75) In 2 _ 4 ln 2 L2 ” 1), 1:12 7521), ”7" ( ) for the case above. When I carry out the full respective Boltzmann-equation simulations of the 100 MeV/ nucleon 96Ru + 9‘in reactions, at. the impact. parameter of b = 5 fm Z L/ 2, I find that, indeed, the nuclei need to be in contact for about 40 fin/c for the isospin asymmetry to drop to the half of original value. 4.6 Summary Diffusion and other irreversible transport. phenomena have been discussed for a binary Fermi system close to equilibrium. For weak nonuniformities, the irreversilifle fluxes are linear in the uniformities, with the characteristic transport proportionality-coefficients dependent only on the equilibrium system. It. is hoped that, in analogy to how the nuclear equation of state and symmetry energy are employed, the coefficient of diffusion can be employed to characterize reacting nuclear systems with respect to isospin transport. Following a qualitative discussion of irreversible transport in this chapter, the set of coupled Boltzmann-Uhlenbeck-Uehling equations was considered for a binary system, assuming slow macroscopic temporal and spatial changes. The slow changes allow one to solve. the equation set by iteration, with the lowest—order solution being 131 the local equilibrium distributions. In the next order, corrections to those distributions were obtained, linear in the thermodynamic driving forces associated with the system nonuniformities. These corrections produce irreversible fluxes linear in the forces. The transport coefficients have been formally expressed in terms of brace products of the responses of distribution functions to the driving forces. The considered coefficients include diffusivity, conductivity, heat conduction and shear viscosity. Furthermore, the set of the linearized Boltzmann equations was explicitly solved under the assumption of simplified distribution—function responses to the thermodynamic driving forces. The solutions to the equations led to explicit expressions for the transport coefficients, with the diffusivity given in terms of the collision integral for collisions between the two species weighted by the momentum transfer squared. Besides associated sensitivity to the cross section for collisions between the species, the diffusivity is also sensitive to the dependence of mean fields on the species. The collisions bet-1126671 the species are those that inhibit the relative motion of the species; the difference between mean fields affects the relative acceleration and, in combination with the collisions, the stationary diffusive flux that is established. I calculated the isospin diffusivity for nuclear matter, using experimental nucleon-nucleon cross sections for species-independent mean-fields. At low temperatures and high densities, the diffusivity diverges due a suppression of collisions by the Pauli principle. At high temperatures, the diffusivity is roughly proportional to the average velocity and is inversely proportional to the density. The diffusivity weakly decreases with an increase in the absolute magnitude of asymmetry. I provided an analytic fit to the numerical results. For completeness, I also calculated the heat conduction and shear viscosity coefficients and provided fits to those. Moreover, I calculated the diffuseness mean—field enhancement factor for 132 synnnetric matter, assuming a couple of dependencies of the symmetry energy on density. At low temperatures, the enhancement factor is simply proportional to the net synnnetry energy divided by the kinetic symmetry energy. Considering the expansion of the form-factors in distribution-function responses, I demonstrated that corrections to the Boltzmann-equation transport coefficients, beyond the approximations I employed, are small. Finally, I produced an elementary estimate for isospin equilibration in a low impact-parameter collision. 133 Chapter 5: Transport Simulations As has been ('liscussed, the Boltzmann equation set may be used for simulating heavy-ion reactions, and the results from such simulation can be compared with experimental data to deduce properties of nuclear matter. This chapter is specifically devoted to the reaction simulations. In the first part of this chapter, I will analyze the spectator-participant interaction in peripheral reactions and the relation of spectator olgiservables to the nuclear equation of state (EOS). In the second part, I will discuss isospin diffusion process in reactions of isospin asymmetric systems. 5.1 Spectator Response to the Participant Blast As already stated in Chapter 1, the participant—spectator interaction plays an essential role in the dynamics of heavy—ion reactions. In this section, I will discuss in detail, the impact of spectator shadowing on the development of elliptic flow, and the resulting close relation of the nuclear EOS to the spectator properties following the dynamic stage of a heavy-ion reaction. A brief introduction to elliptic flow within the participz‘int region and the past studies on the. spectator region is given in subsection 5.1.1. An analysis of the evolution of a. reaction system and of the participant-spectator interaction is given in subsection 5.1.2. The sensitivity of the emerging spectator characteristics to the nuclear EOS is investigated in subsection 5.1.3. The predicted speeding up of the spectator during the violent reaction has recently got support from the experiment [152] and a discussion of the spectator velocity increase is given in subsection 5.1.4. The results of this section are sunnnarized in subsection 5.1.5. 134 5.1.1 Introduction As indicated in Chapter 1, the nuclear EOS significantly impacts the development of collective flow in the participant region. Phenomenological parameterizations of the EOS are usually constrained with the properties of nuclear-matter at normal density po and diverge at much higher densities which can be probed in energetic heavy—ion reactions. However, an important complication for heavy-ion collisions results from the fact that the duration of the initial high-density stage of the collision is very short compared to the time scale for the whole reaction process. E. 9., in an 800 MeV/ nucleon I) = 5 fm collision of 124Sn + 12"Sn, the high—density stage with a. central density [1,. > 1.5/1o lasts about 13 fin/c, while the elapsed time from the initial impact to the complete separation of target and projectile is N 40 fni / c. The spectator properties continue to develoj.) well beyond this time [153]. Given the short duration of the high-density stage, signals which carry information about the high-density phase of the collision could be easily washed out by other signals generated at a later stage. As a consequence, reaction simulations are needed to provide guidance for the measurement of signals which not only probe the high-density stage but survive through the entire duration of the collision process. Collective flow of participants has been studied already for quite some time [154, 155, 156]. The flow is believed to result from early stage compression and an expansion [157, 158. 51, 36, 93], and can carry information 011 the initial high-density phase. The relation between the nuclear EOS and the flow phenomena has been explored extensively in simulations and a recent example is the analysis of the transverse-momentum dependence of elliptic flow [12]. Elliptic flow is shaped by an interplay of geometry and the mean field and, and when gated by the transverse momentum, reveals the momentum dependence of mean field at super-normal densities. 135 The elliptic flow pattern of the participant matter is affected by the presence of the cold spectators [93, 12, 53], as will be reiterated. When nucleons are decelerated in the participant region, the longitudinal kinetic energy associated with the initial colliding nuclei is converted into thermal and potential compression energy. In a. subsequent rapid expansion or explosion, the collective transverse energy develops [51, 36, 157, 158] and many particles from the participant region get emitted in the transverse directions. The particles emitted towards the reaction plane can encounter the cold spectator pieces and, hence, get redirected. In contrast, the particles emitted essentially perpendicular to the reaction plane are largely unimpeded by the spectators. Thus, for beam energies leading to a rapid expansion in the vicinity of the spectators, elliptic flow directed out of the reaction plane (squeeze—out) is expected. This squeeze-out. is related to the pace at which the expansion develops, and is, therefore. related to the EOS. On the other hand, since the spectators serve to deflect particle emissions toward the rcactirm plane, their properties may be significantly modified. This suggests an analysis of the characteristics of the spectators resulting from the collision process. In one sense. the spectators can be viewed as probes which were present at the site of the nuclear explosion leading to the rapid particle emission. Thus, a careful study of their characteristics could complement the results from elliptic flow and provide further information regarding the properties of high-density nuclear matter. Long-time evolution of spectators has been studied recently by Gaitanos ct al.[153]. A comprehensive summary of experimental results for spectators produced in reactions at different centralities has been presented by Pochodzalla [102]. In particular, universal features of spectator multifragmentation have been well documented [102, 14]. The transverse momentum change of the spectator during a semicentral collision, to be addressed here, was studied in the past via emulsions by 136 Table 5.1: Parameter values for the different mean fields utilized in the simulations. First three columns refer to Eq. (3.28) and the next two to Eq. (3.33). The last. column gives the Landau effective mass in normal matter at Fermi momentum. EOS a b 1/ c A 'm‘ / 771 (MeV) ( MeV) S 187.24 102.623 1.6340 0.98 SM 209.79 69.757 1.4623 0.64570 0.95460 0.70 H 121.258 52.102 2.4624 0.98 HM 122.785 20.427 2.7059 0.64570 0.95460 0.70 Bogdanov ct (11. [159] (see also [160]). The participant-spectator interaction and spectator physics were discussed in [161]. The systematics of the longitudinal mt‘nnentum transfer to spectators in energetic reactmns induced by light projectiles is discussed in Ref. [162]. The spectator acceleration in certain heavy reaction systems has been observed in exjn‘ériment [152], and further experiments are proposed to study the systematics of the spectator acceleration and its relation to the nuclear EOS. 5.1.2 Spectators and Participants Within transport. simulations, I will investigate here, the spectator-participant. interaction and the spectator shadowing effect on elliptic flow in a. heavy-ion reaction. The Boltzmann equations underlying microscopic transport theory have been already described in Chapter 3. For later reference, I have listed the parameters for the utilized Mean Fields (MP) in Table 5.1. Details of the, momentum independent and momentum dependence MFs have been already given in Chapter 3 (see also Ref. [12]) and shall not be repeated here. Figure 5.1 1:)1‘esents some results from simulations of 197Au + 197An collisions at a beam energy Tum = 1 GeV / nucleon and an impact parameter b = 8 fm. Unless indicated otherwise, the hard 1noment11rn-depeIIdei1t (HM) EOS (cf. Table 5.1) was 137 used. Figure 5.1(a) shows the time evolution of the density for the participant and for the spectator matter. The solid and dashed curves show, respectively, the baryon density pc at the center of the collision system, «i.e., participant density and the baryon density in the local frame at the geometric center of the spectator matter pspec. Here, the operational definition of spectator matter is that the magnitude of the local longitudinal velocity exceeds half of the velocity in the initial state and that the local density exceeds one tenth of the normal density. The solid line in Fig. 5.1(a) clearly illustrates the rapid density build-up (for t S 5 fin/c) followed by expansion of the participant matter. The dashed-line also points to a weak compression of the spectator matter during the expansion phase of the participants. The latter observatirm is ct‘n'isistent with the expected delay associated with the time it takes a. compressimi wave to reach the center of the s1.)ectator matter. starting from an edge. Figure 5.10)) shows the time evolution of the elliptic flow 1:)arameter “(’2 for all rind-rapidity particles. The parameter is defined as 172 2 (cos (2(,”))) . (5.1) where (Z) is the azimuthal angle in the X—Y plane perpendicular to the beam axis Z; the X-Z plane defines the reaction plane. The value of 122 ctmveys information about the pattern of particle emission from the central participant region. The hot. participant region has an initial elliptic shape in the X-Y plane due to the overlap geometry. Since the long and short axes of the ellipse point in the Y-direction and in the X-direction, respectively. the matter starts out with stronger MF and pressure gradients in the X-direction. Given the shape of the emission source and the gradients, the matter is first expected to develop a stronger expansion in the X direction and, hence. to give rise to positive values of eg. If the spectators are nearby 138 m7Au+lwAu Tm=1GeV/nucleon b=8fm (I, .3 \ I °~ i 0.0 V2 1 — - v2 (pt>0.5SGeV/c)‘ N : \ ‘_ , , ’ I ‘0-2 7(an at midrapidity) T ’ . . . . l . . . . r . . . 1+ ‘ I . . . I a . . . I . I j . 0'1 T(c) cascade L l g" 0.0 _ . — ‘ ‘ ~ _________________________ —- _ 1(a11 at midrapidity) ; 0'1 t + + : I : : : + I e : : 4 I 4.9. P I 0.0 C-(d) HM -: + I >01 —o'1 »— \ _________ .— _ __ ’ — ‘ _-: t \— “' i I —o 2 ’— — V2 4 ' i (emitted) ._ _ _ V2 (pt>0.55GeV/c) . . . . . J . . . . l . . . . l m L . . ‘ 0 10 20 30 40 t (fm/c) Figure 5.1: Results from a BUU simulation of the the 197Au + 197An collision at 1 GeV/ nucleon and b = 8 fm, as a function of time: (a) the central densities of the participant pc and the spectator matter p,pm (b—d) the midrapidity elliptic flow parameter v2. The results are from a simulation with the HM mean field, except for those in the panel (c) which are from a simulation with no mean field. The panels (b) and (c) show the elliptic flow parameter for all particles in the system while ((1) shows the elliptic flow for particles emitted in the vicinity of a given time. In the case of the HM calculations, also shown is 222 when a high-momentum gate p, > 0.55 GeV / c is applied to the particles. 139 during the expansion phase, they can serve to stall the expansion in the X direction and a compression wave then develops within the spectator matter (cf. Fig. 5.1(a)). The resulting dominant expansion of participant matter in the Y direction gives rise to negative values of 112. Figure 5.1(b) indicates that this preferential out-of-plane emission pattern begins after ~ 7 fin/c. The time correlation between the change in Sign of 112 and the decrease in the magnitude of the central density should be noted in the figure. The central density of participant matter pc begins to drop at about 7 fm/c and the most rapid declines ends at ~ 16 fm/c; during this time the elliptic flow drops from its maximum positive value to its maximum negative value. A comparison of Figs. 5.1(b) and 5.1(c) illustrates the important role of the MF in shaping the elliptic flow magnitude. Figure 5.1(c) shows the time dependence of U2 obtained when the calculations are performed without the inclusion of a. MF (cascade mode). In contrast to the evolution with the mean field (cf. Figs. 5.1(b)) where ’02 first achieves significant positive and then negative values, Fig. 5.1(c) indicates 1); values which stay close to zero over the entire time evolution of the system. This trend is related to the fact that in the cascade model the transverse expansion is slow compared to the time duration for which the spectators are in close proximity to the participant matter, or compared to the time required for longitudinal motion to stretch the matter to low density. The important role of the MP for the generation of elliptic flow and the sensitivity of this flow to the EOS has been stressed [93, 12, 53]. The temporal difference of v2 for all midrapidity particles in the system, and for those particles that have left the system can be observed by comparing Figs. 5.1(b) and 5.1(d). Figure 5.1(b) shows the change in 122 with time as discussed above. On the other hand, Fig. 5.1(d) indicates little or no change of v2 (over time) for midrapidity particles that have left the system. That is. out-of-plane emission is favored (negative 122) for all emission times. Figures 5.1(b) and 5.1(d) also show e2 140 as a function of time for particles with transverse momentum p, > 0.55 GeV/c; these panels indicate that faster particles are more sensitive to the obstructicms as well as to any directionality in the collective motion. The analyses of elliptic flow and related works have established connections between features of the participant matter resulting from the participant-spectator interaction and the nuclear EOS [51, 157, 158, 12, 93, 53]. On the other hand, it is not known whether the same interaction (during the violent stage of a reaction) leaves any lasting effects in the spectators that could be related to the EOS. Extensive studies of the statistical behavior of spectator matter have been carried out [102, 14] for time scales which are long compared to the collision time. Such studies do not address the dynamical impact of the violent reaction stage on spectators. During the violent stage of a collision, the spectators remain close to the participant matter, so they might. serve as a good sensor for the explosion. Thus, I proceed to take a closer look at the changes which may occur in the spectator matter following their interaction with the participants. In addition I investigate whether or not such changes have a connection to the EOS. Figure 5.2 shows contour plots of different quantities within the reaction plane now from 124811 + l“Sn reaction simulations at the beam energy of Tm, = 800 MeV/ nucleon, at the impact parameter b = 5 fm, carried out with a soft momentum-dependent (SM) mean field. The columns from left to right represent the reaction at 5 fm/c time increments. The top and middle rows show the baryon density in the system frame p and the local excitation energy E“ /A, respectively. The bottom row shows the density pbnd of baryons that are bound in their local frame (6 < m). As may be expected, the excitation energies reach rather high values in the participant region but remain low within the spectator region throughout the violent stage of the reaction. Most of the particles in the participant region are found to be unbound, i.e., phnd is low. On the other hand, most of the particles 141 within the spectator region are bound. They move with velocities that are close to each other. and this keeps phnd sizeable throughout the violent collision stage. Figure 5.3 provides next a detailed time development of the selected quantities in the 800 MeV / nucleon l24Sn + l2“Sn system for which the contour plots were given. Figure 5.3( a) displays the evolution of baryon density at the system center, pc, and of baryon density at the center of the spectator region, pspec. The high-density stage for the participant matter in Fig. 5.3(a), characterized by p,. > p0, lasts over a time that is short in comparison to the time needed for a clear separation of the target and projectile spectators from the participant zone, cf. Fig. 5.2. To observe a. stabilization of the spectator properties I needed to follow the 1')articular reaction up to ~ 60 fin/c. Longer-term studies of the spectator development have been carried out within the BUU approach [153]. However. as the spectators approach equilibrium, they may be described in terms of the statistical decay method which at this stage has advantages over the BUU equation. Figure 5.3(b) shows the average transverse momentum per nucleon of the spectator, in the reaction plane, as a function of time. In calculating the average, I include all spectatm‘ particles as specified before (dashed line in the panel) or the subset of particles that are bound in the local frames (solid line). The averages, obviously, approach the same asymptotic value over time, but the approach is faster for the bound-particle average. Note that the extra lines in Fig. 5.3(b) represent the evolution of the average momenta past the 40 fm/c of the abscissa. Calculated in either manner, the spectator average momentum (PX / A) reaches its final magnitude during the high-density stage in the participant matter and only somewhat reduces to stable during the expansion that follows. This suggests that the spectators can, indeed, provide information on the high-density stage of the collision. Figure 5.3(c) shows the average excitation energy per nucleon (EVA) of the spectator as a. function a time. \Vithin the studied time interval, the excitation \ 'g o .D Q W 0. *Tfi' firrrn' '1'rfirv ** fo 'Tfififir'rfii 4 o o ’3 "' \ E o v’ O . C s m .. [3.1 l O 5, a E L0 33 ~ 0 ll .0 O H g I 2 O U H :j A C: \ O o g. > H v Q) N E O O H g I II 0 g H e: (53) l!) O A”: 'C o + v-C C‘. l .5” o a: u—c o 1 ll 0 ...2 O v—a I Figure 5.2: Contour plots of the system-frame baryon density p (top row), local exci- tation energy E‘ /.4 (middle row), and of the density of bound baryons pbnd (bottom row), in the 124Sn + l2“'Sn reaction at Tzab = 800 MeV/nucleon and b = 5 fm, at times t = 0, 5, 10, 15 and 20 fm/c (columns from left to right). The calculations have been carriered out employing the soft momentum-dependent EOS. The contour lines for the densities correspond to values, relative to the normal density, of p from 0.1 to 2.1 with increment of 0.4. The contour lines for pbnd are from 0.1 to 1.1 with increment of 0.2. The contour lines for the excitation energy correspond to the values of E'/A at 5, 20, 40, 80, 120, 160 MeV. For statistical reasons, contour plots for the energy have been suppressed for the baryon densities p < 0.1 pg. Note, regarding the excitation energy, that the interior of the participant region is hot while the interior of the spectator matter is cold. 143 ,rnn " I‘fi I ' 1*— .(a) o? \ Q E 1 a. i , x .1 3:100r / “‘\‘\ 1 A t w I <1: \.. I 7 m t+20fm/c ] V it...ira-.i I I ....I 9 o a A q: \ . , m . V . o: ['(d < 100:- _‘ A 3 ‘ 3 50' - 1""sm‘z‘Sn rm=aooneV/nuc1eon I b=5fmSMEOS t 0'....L....1....i....ini 0 10 20 3O 40 t(fm/c) Figure 5.3: Evolution of selected quantities in the 124811 + 124Sn reaction at 800 MeV / nucleon and b=5 fin, from a calculation with a soft momentum-dependent EOS. The panel (a) shows the density at. the center of a spectator pSpec (dashed line) together with the density at the system center pc (solid line). The panel (b) shows the average iii-plane transverse momentum per nucleon of the spectator (PX /A) calculated using all spectator particles (solid line) and using only bound spectator particles (dashed line). Two extra lines in the panel show evolution of the momenta past the 40 fm/c of the abscissa. The panels (c) and (d) show, respectively, the spectator excitation energy per nucleon (1" //l) and the mass number (A) from all spectator particles. 144 energy rapidly rises and decreases and then changes at a slower pace. During the violent reaction stage, some. particles traversing from the participant into the spectator matter contribute to the excitation of the spectator matter. As time progresses, some of those particles will travel through the matter and leave the spectators. Some other will degrade their energy within the spectator frame. (Note: I consistently continue with definition where the spectator matter is that for which the c.m. local velocity is larger than half the beam velocity and the local density exceeds the tenth of normal.) Figure 5.3(d) shows the mass number of a spectator region as a function of time. The spectator mass number decreases rapidly as particles dive into the }_)articipant region and then the mass recovers smnewhat, around the time of 20 fin/c, as some particles get through the opposite moving corona matter and join the bulk of the spectator matter moving along the beam direction. Later, a gradual deexcitation slowly reduces the spectator mass. I have analyzed in this subsection the interplay between the participants and spectators. I have examined how elliptic flow is generated as a. result of that interplay and how the interplay affects the spectator characteristics. In the next subsection I will explore the sensitivity of spectator characteristics to the EOS for nuclear matter in collision. 5.1.3 Spectator Sensitivity to the Nuclear Equation of State In the light. that. the changes of the spectator properties could probe the compression and explosion of the participant matter, I follow the reaction simulations until a. clear separation develops between the spectators and the participant matter and a. stabilization of the spectator is attained. I explore the sensitivity of the emerging spectator properties to different assumptions on the nuclear EOS. The results could serve to initialize statistical decay calculations for a. complete description of a reaction. In the following, I shall present a sample of my spectator ii’ivcstigations, within the 124Sn + 124811 system in the beam energy range of 250 MeV/ nucleon to 1 GeV / nucleon at impact parameters b = 5 - 7 fm. I shall also quote results from 197An + 197An at 1 GeV / nucleon. I utilized four different EOS explored in the past, of which the parameters are given in Table 5.1. I concentrated on the quantities that could be experimentally determined for the spectator, and thus the average transverse momentum per nucleon (PX / .4), the change in the average c.m. momentum per nucleon A](P / 4)] the average excitation energy per nucleon (E * / l) and the average mass (4) follt')wing the violent stage of the reaction. The change in the average c.m. momentum is A](P/l)] : \/(P—"/.4)2 + (PZ/.4)2 — (P/.4),~. The above mentioned quantities. tmvards the end of the simulations. are shown as a function of the impact parameter at ’11,“, = 800 MeV/ nucleon in Fig. 5.4, by open symlmls, and as a. function of the beam energy at b = 5 fm in Fig. 5.5, respectively. The resulting spectator (PX /.4) exhibits a clear sensitivity to the stiffness of the EOS. I can see in both figures that a stiffer EOS results in a stronger sidewards push to the spectator. However, even more prominent is the sensitivity to the momentum dependence of the mean field. A strong momentum dependence results in a stronger push to the spectator. Recall that the interplay between the spectator and the participant matter also generates elliptic flow for the participant matter and it was possible to exploit the latter in the determination of the mean-field momentum dependence at super-normal densities [12, 53]. The final momentum of the spectator reflects the momentum exchanges with the participant zone throughout the reaction. Initially, the nucleons from the opposing nucleus move nearly exclusively along the beam axis relative to the spectators. As equilibration progresses, the momenta in the participant zone acquire 146 A o b fifi I I Tfif' l I rTv It i I I I r T % 100 L (a) 124’Sn-I-‘z‘Sn Tm=800MeV/ nucleon L a C A l 7 . . re“ AHM i 5O,— . “\‘ USN -. a: b 0.-.‘ 0H V * ' "°‘ ''''' vs ‘ 0’ . . . Y . . :1'1'1';; . . . .1.; ,3 :f...,....,..f.,-...r..‘ g :(b) A : o ‘10.- _——-- T a : ,9 : 2} ‘20? o M- eve" ' ‘3 tE —30Z- 9' -I _V_. : : <'-4o:+v~+: a AW‘ IC 2 s (l . . g 107 _. V b A I i 5— _ o I 1 Ed I V i l l 1 ‘ 0 *+' I'LL l'f ‘ l ' ' ‘ If‘ 4 .(d) . 9: 50. i v T . > 1 L 1 0 91 L. I I1. . 4 5 6 7 8 b (fm) Figure 5.4: Spectator properties in the 800 MeV/ nucleon 124811 + 124811 collisions, as a function of the impact parameter, for four representative EOS: hard momentum- dependent (HM), soft momentum-dependent (SM), hard momentum-independent (H) and soft momentum-independent (S). Panel (a) shows the average in-plane transverse momentum of the spectator per nucleon (PX /4). Panel (b) shows the change in the average net c.m. momentum per nucleon AMP/A) I. Panel (c) shows the average exci- tation energy per nucleon (E‘ /.4), and, finally, panel ((1) shows the average spectator mass (A). Open symbols represent results obtained with reduced in-medium nucleon— nucleon cross sections; filled symbols represent results obtained at b = 5 fm with free cross sections. 147 ’5 r...lr...I...,l....l....1fifd \ - 124 124 ‘ z; 100_ (a) Sn+ Sn b=5fm .. ‘3 AHMj Q 0311‘ _I \u OHI n‘ l V vs. > 4 '3’ J a 4 2:." : \ . '1 "3; * < t 4 ‘3 W. a- - -60“..ij...I.rrfiI...rI.f I. A C 4 :3 O[() . g 1. . V I A h i 5r ‘ E: r V OA,,:1 :l.#fil rmina.i , L l T 1' l' I Md) A 50~ — d u v 4 O....i....i....1..L.i...-1.. 0 250 500 750 1 000 1 250 Tm, (MeV/ nucleon) Figure 5.5: Spectator properties in the 124Sn + 124Sn collisions at b = 5 fm, as a func- tion of the beam energy, for four representative EOS: hard momentum-dependent (HM), soft momenturn-dependent (SM), hard momenturn-independent (H) and soft momentum—independent (S). Panel (a) shows the average in-plane transverse momen- tum of the spectator per nucleon (PX / .4). Panel (b) shows the change in the average net c.m. momentum per nucleon AI(P/A)I. Panel (c) shows the average excitation energy per nucleon (E */ 4) Finally, panel ((1) shows the average spectator mass (.4). 148 a level of randomness. Random exchanges of momentum between spectators and participants generally drive the spectator momentum towards the average for the system, i.e. zero. However, the participant nucleons reach the spectators moving away from the system center, coming with momentum directed on the average outward, delivering an outward push to the spectator pieces. The order of magnitude for the transverse push may be obtained from a simple estimate. Thus, in Sn + Sn at. 800 MeV/ nucleon, estimating the pressure in the compressed region from the nonrelativistic ideal-gas estimate, 2 , I) ”>- p g Yuan/4A (5.2) with p ~ 2;)0 N ‘2/6 flu”, I get 1) ~ 40 I\le\l"/fm3. The size of the high-density region in the X-Z plane for Sn + Sn at medium b is ~ 4 fm. cf. Fig. 5.2. The push to the spectator is then of the order of I” it: pSAI, (5.3) where S is the transverse area pushed by the 1.)articipant matter and A! is the duration of the push. \Vith S = 7r 122/4 ~ 13 fm2 and Al, ~ 5 fin/c, cf. Figs. 5.2 and 5.3, I get PI 40MeV/fm“ x 13f1112 x 5fm/c , l\leV . —— = 2 St) , (5.4) .4 50 c This is within the general order of magnitude as found in the simulations. W hen the impact parameter increases, the fireball pressure decreases while the spectator mass increases. Thus, the momentum per nucleon decreases. With regard to the beam energy variation in the sinmlations. at low energies the pressure in the fireball region drops, resulting in smaller push to the s].)ectators, with some ctmipensation coming from a longer time for the spectators in the reaction zone and a longer fireball 149 lifetime. With the rise in the beam energy from the low energy end, the rise in the transverse fireball pressure is moderated by pion production and an increasing transparency. The spectator time in the vicinity of the explosion continuously drops resulting in a level of saturation in the spectator momentum per nucleon. With regard to the changes in the magnitude of the c.m. momentum per nucleon AMP/AH, one can see in Figs. 5.4 and 5.5 that the results for MD MFs significantly differ from the results for MI MFs for Sn + Sn, with the later MFs giving more momentum loss. The spectator mass and excitation energy, in contrast to the momentum, are rather insensitive to the MP in the present system. While the results discussed until now have been obtained with reduced iii—medium niicleon-nucleon cross sections [12], I also carried out calculations with free nucleon-nucleon cross sections. The latter calculations for the same system at T1,...) = 800 MeV/ nucleon b = 5 fm, are represented by filled symbol in Fig. 5.4. W ith free cross sections, the remnant masses are a bit lower, the excitation energies are higher, and so is the transverse push. The transverse push is more sensitive to the change in the EOS, than to the change in cross section, as evident in the figure. Contrary to what one might naively expect, less momentum per nucleon is lost. in the free cross section case. I will come back to the last issue later. In investigating the differences in results for the different EOS. I olwiously looked at the details in the time development of the systems for the different EOS. Figure 5.6 shows the central participant density as a function of time. For the hard momentum-independent EOS a maximal density is reached earlier and the expansion sets faster than for the soft momentum-independent EOS. The S EOS allows for a higher compression than the H EOS. An MD EOS allows for a lower compression than a corresponding MI EOS. Moreover, the expansion develops earlier for an MD EOS than a corresponding MI EOS. Evidently, the momentum dependence plays a similar role to the stiffness of nuclear matter; it renders the I I I I I I I I I I I I I I —I r I I T 12‘Sn+‘z‘Sn Thb=800MeV/nucleon b=5fm P/Po t (fm/c) F igure 5.6: BarVon density as a function of time at the center of the 12“Sn + ”"811 system at T1,}, = 800 MeV / nucleon and I) = 5 fm, for different MFs. matter less compressible in a dynamic situation. Figure 5.7 shows the spectator transverse momentum in the X-direction as a function of time. As I have already pointed out before, the spectator transverse momentum per nucleon rises within a relatively short time interval. The rise starts about the time when the maximal density is reached at the participant center; the rise stops due to combined effects of the spectator passing by and of the dilution of the participant zone. While there are up to 2 fm/c differences in the start and end of the rise interval in Fig. 5.7, it is apparent that the differences in the final (PX /A) must be due to the differences in magnitude of the transverse pressure (transverse momentum flow) for the different EOS and not in the duration of the rise. In fact, the slopes of the dependence of transverse momentum on time differ considerably more than do the final transverse momenta. A faster dilution for the more incompressible EOS shuts off the momentum rise sooner than for the more 150 p- I I I I I I I I I I I I I I I I I I I I I : : 124Sn-I—lz‘Sn Tm=800MeV/nucleon b=5fm 2 125 _— _- ,3 I ----- HM I > 100 f"__ SM ........ .3 o . . S C — - H ' . . I v __ ‘ 4i A 75 . q < * q N F m 50 —- ‘1 v : s s - — _ g 3 L. ‘ \.~.. 0 l I I l l l I l l l i J J O H O N O CO C .h C t (fm/c) Figure 5.7: Average iii-plane transverse momentum per nucleon of a spectator in b = 5 fill ““811 + 12"Sn collisions at Tia}, : 800 MeV/nucleon, as a function of time, for different. EOS. (oinpressilile EOS and moderates the ('lifferences in the final spectator momenta. Figure 5.8 shows differences in the Landau effective mass, m" = p/v, in cold nuclear matter at different. densities for MI and MD MFs. Lower masses for the MD MP means that particles move out faster at the same momenta. The change in the magnitude of the c.m. momentum per nucleon A|(P/~’l)| is generally dominated by the change in the longitudinal momentum per nucleon. In Figs. 5.4 and 5.5 the net. momentum per nucleon is seen to decrease in the Sn + Sn reactions under all conditions. That change in the momentum might be considered a measure of the friction involved in the interaction of the spectator with the participant zone. The friction is due to mentioned random changes of momenta in collisions between participants and spectators that, besides knocking particles off spectators, over time drive the average momentum towards the system average of zero. When examining the net spectator momentum per nucleon as a function of 1.25 _ I I I I I r I I r TI T I I ‘I I I I I I II I I I I I I4 1 4 I I ' I ‘.'l O 0.75 ’. . I" '. ’ _— E '2 ’ j \ .. .8 L ,. , ” ’ ’, " . O.50'=""'"’., --;.z ’ _. ;-;-_-.-.;'.'.—‘ I , ’ —p/po=1-0 : ; SM — — —p/po=1.5 : 0.25 :- """" p/po=2.0 —: E - ‘ - ‘ 'p/po=2-3 : 0.00 I I I I I I I I I I I I I I I I I l I I I I I L I I II 0.0 0.1 0.2 0.3 0.4 0.5 p (GeV/c) Figure 5.8: Landau effective mass m" : p/e, in units of free nucleon mass, as a function of mmnentnm at several densities in cold nuclear matter for S and SM MFs. time in the Sn + Sn reactions, the momentum is first found to decrease but then found to recover smnewhat. The late increase and part of the early momentum decrease could partly be. attributed to our inability to cleanly separate the spectators from the participants, which intermittedly intermixed and then separate. The above view on the net spectator momentum, however, needs to be revised once the changes in the momentum are examined in the Au + Au system. The change in the net momentum per nucleon is shown for a 1 GeV / nucleon reaction as a function of the impact parameter in Fig. 5.9, by open symbols for the in-medium reduced cross section. For low impact parameters and MD MFs, the average spectator momentum per nucleon increases in the reaction simulations! The mass and the excitation energy of the spectator in Figs. 5.4 and 5.5 do not exhibit a sensitivity to the EOS likely because they are determined by the geometry and the capability of matter to retain the energy, res1_)e.(.-tively. As to the momentum CO C - rIIrIIIr‘IrrIfiIIIfiTTII A 1m'Au+ ’mAu Tm: lGeV/ nucleon N O H O A|| (MeV/c) O IlIIIIjIIIIIjIIIIIIIIIIII / / I i-IIIIIIIIIIIIIIIIIIJIILIIIL ‘ ‘A ‘ A HM —10 v __. DSM gut .... IV— """"" ‘9 OH —20_ V S ' J I I I I I I I I I I I I I I 14 J I I I 5 6 7 8 9 b (fm) Figure 5.9: The change in the net average c.m. momentum per nucleon A|(P/xl)| of spectators in the 197Au + 197Au system at TM, 2 1 GeV/ nucleon. Open symbols represents results obtained with reduced in-medium nucleon-nucleon cross sections; filled symbols represent results obtained at b = 6 fm with free cross sections. A negative value of AMP/4H inditjrates a spectator deceleration, while a positive value indicates a net acceleration. changes, though, I have demmistrated that they can provide information on the violent stage of energetic reactions and constrain the properties of high density nuclear matter. 5.1.4 Spectator Acceleration The acceleration of the spectator during a heavy-ion reaction, is of interest on its own. After all, if we shoot a bullet through a wall, we expect the bullet to slow down, not accelerate. The systematics of the spectator velocity after a heavy-ion reaction has been studied before and the velocity decrease of the spectator piece was found to be proportional to the decrease of the spectator mass in the peripheral collisions (often known as the l\*lorrisse.y systenn‘itics [162]). Such deceleration of the spectator could be attributed to the friction between the spectator and the participant matter. However, the sinnilaticm results shown in Fig. 5.9 point. to an acceleration of the spectator in a heavy-mass system at low impact. parameter. The unusual prediction of the simulation has been confirmed by recent experiment [152]. Below, I will give a qualitative explanation for the spectator acceleration. The speeding up of the spectator at low b in Au + Au may be understood in terms of the explosion of the participant zone. On one hand, the spectator acquires transverse momentum. On the other, in the longitudinal direction the explosion acts more on the rear of the spectator piece than on the front. If the explosion is strong enough, the ordered push may overcome the friction effects, producing a net longitudinal acceleration for the spectator. There is no issue of energy conservation since the work is done by the pa.rti('-i1;)ant on the spectator zone. The difference between Sn + Sn and Au + An is in the equilibration time scale relative to the duration of the fireball. Differences in the net final momentum per nucleon between different MFs for both systems, with significantly higher net momenta for the MD than MI MFs, may be understood in terms of the violence of the explosirm that. accelerates the spectatt )r. An important aspect of the spectator momentlnn per nucleon, underscoring the interpretaticm almve, is its dependence on the nucleon-nucleon cross section. In Fig. 5.9, the results of b : 6 fm Au+Au simulation with the free cross sections are represented by filled symbols. With the larger free cross sections, the spectator remnants emerge even faster from the reaction than the with the lower cross sections! This is because for higher cross sections, the equilibration is faster, which allows the participant to explode more violently when the spectators are still nearby. Quantitatively, in the b = 6 fm HM free cross-section case, the gain in the longitudinal momentum per nucleon contributes as much as 17 out of 24 MeV / c of the gain in the net spectator momentum per nucleon in Fig. 5.9. In the b = (5 fm 155 HM reduced cross-section case. the longitudinal gain contributes about 4 out of 8 MeV / c of the net. momentum gain per nucleon. 5.1.5 Summary W ithin semiclassical transport simulations of energetic semicentral collisions of heavy ions, I have carried out an investigation of the interplay between the participant and spectator regions. The spectators pass by the participant region when the participant matter undergoes a violent explosion. On one hand, the spectators block the expansion of the participant matter in the in-plane direction, producing elliptic flow for the participant matter. On the other hand, the explosion pushes the spectators giving them transverse momentum pointed away from the reaction zone. The momentum transfer to the spectators and the shadow left in the pattern of the participant emission depend on the speed of the explosion. The speed, in turn. depends on the EOS of the dense matter. Due to their nature. the spectators rem‘esent a p(_‘1'f(-)(‘-t.ly timed probe right. at the reaction site. A careful analysis of iii-plane transverse momentum of a. spectator may yield information on the EOS ctmiparable to that provided by elliptic flow analysis. An analysis of the longitudinal momentum transfer may yield information on the momentum dependence of the MF s in the reactirms. The signatures in the spectator mmncnta per nucleon rise with the lowering of the ii'npact parameter, but at the cost of the lowering of a. spectator mass, reducing the chances of identifying the spectator remnants. Significantly. for most repulsive MFs and small impact parameters in a heavy system, spectators may emerge from the reacticm with a higher net average momentum per nucleon than the original momentum. 156 5.2 Isospin Diffusion Process in HIC In this section, I will discuss isospin diffusion process in an isospin asymmetric reaction system. Elementary discussion of isospin diffusion process in realistic heavy-ion reaction system is provided in subsection 5.2.1. The nuclear equation of state based on isospin dependent mean fields (IEOS) and the isospin diffusion coefficient for nuclear matter are. discussed in sul‘)sect.ion 5.2.2. The results from the isospin diffusion simulaticm in a peripheral reaction, including the time evolutions of the spectator isospin asymmetry. are discussed in subsection 5.2.3. The results from the current simulation are compared with data [100] in subsection 5.2.4. The results and the discussions on isospin diffusion are summarized in subsection 5.2.5 5.2.1 Introduction Isospin diffusion process, which results from isospin non-equilibration, generates an isospin flow that transports isospin asymmetry from the higher com-entration region to the lower ctmcentration region. In the limit of small isospin gradient and close to equilibrium, isospin diffusion process could be described in terms of a. transport coefficient (see the discussions in in Chapter 4 “Nuclear Isospin Diffusivity” ). The isospin diffusion coefficient is related to the n-p cross sections 0",, and to the isospin dependence of the nuclear equation of state. As seen in Chapter 4, the different IEOS give rise to different isospin diffusion coefficients, and thus different isospin diffusion time scales. The direct relation between isospin diffusion process and IEOS gives us possibilities to test the different IEOS models in heavy-ion reactions. In reactions of isospin-asymmetric reaction systems, isospin diffusion process is controlled by two competing time scales: the isospin diffusion time scale and the reaction time scale. At lower energies, the reaction time scale is nmch longer than the characteristic isospin diffusion time scale, and the isospin gets close to equilibration in the reaction system. At higher energies, the reaction time scale is much shorter than the isospin diffusion time scale, thus, the projectile-like region and the target-like region will show memories of the initial system. Such a transition, from proximity to isospin equilibration at lower energies to nonequilibrium at higher energies, has been demonstrated in certain reaction systems by Johnston et al. [98, 163]. The isospin non-equilibrium after a heavy—ion reaction is important for the measurement of isospin diffusion process. A complete isospin equilibration would erase the asymmetry in the initial reaction system and make the isospin related observables insensitive to the diffusion process. The isospin nonequilibrium has been first used to measure nuclear stopping power by Rami cf (11. [99]. However, isospin diffusion 1.)rocess is complicated by other processes that affect the isospin content. of the reaction system, such as. the fast. particle emission process, the possible liquid—gas phase transition and the cluster formation process. Impact of the fast particle emission process will depend on the excitation and isospin content of the emitting source; the effects of the liquid-gas phase transition and the cluster formation process have already been discussed in Chapter 2, on a phenomenological basis. Here I will introduce an isospin diffusion ratio that suppresses effects of the non-diffusion processes, by taking the difference between signals from the non-symmetric and symmetric reaction systems. 5.2.2 IEOS and Isospin Diffusion In general, isospin diffusion process will be affected by the ist)spin-dependence of the nucleon-nucleon interaction. Isospin dependence of elementary nucleon—nucleon interaction gives rise to different optical potential for protons and neutrons in the mean field description and to different interparticle cross sections, affecting the 158 motion of the protons and neutrons, and enhancing or suppressing the isospin transport process in the nuclear medium. In the limit of small isospin concentration gradient and close to equilibrium, isospin diffusion process may be characterized by an isospin diffusion coefficient which probes, in particular, the isospin dependence of the nuclear EOS (see the discussions on isospin diffusion process in Chapter 4). While assuming the applicability of the Boltzmann equations, the isospin diffusion coefficient for nuclear matter has already been derived in Chapter 4. The different parameterizations of the isospin dependence for the nuclear EOS have been shown, in particular, to give rise to different isospin diffusion coefficients. In the low temperature limit, the isospin diffusion coefficient is proportional to the symmetry energy of nuclear matter. In this section, I will explore. four specific parameterizations of the isospin dependence of the nuclear EOS and the isospin diffusion coefficients for nuclear matter with those IEOS. For the use in later discussions, I will first produce an estimate of the relevant temperature and density in the participant matter during the violent stage for the peripheral reactions to be investigated. Afterwards, I will discuss the characteristics of the four IEOS and of the corresponding isospin diffusion coefficients for the relevant physical region. I will specifically focus on the peripheral reacticms of ‘12'124811+112‘12“Sii at the beam energy of Elab / A = 50 MeV. The central temperature in a fully thermalized Fermi gas is estimated as T = «Em/(2 .40.) ~ 14 MeV, using a z A/(8 MeV). Taking into account the incomplete dissipation of kinetic energy in the participant region and possible collective motion and effects of reduced statistics, the average thermal temperature is estimated to be half of the peak value, Tape ~ 7 MeV. The maximum density at the center of a peripheral collision at such beam energy is around normal density, but since the compression stage is usually shorter than the expansion stage for the participant region and also the spectators partially slide 159 over the participant matter, the average density in the participant region during the violent reaction stage is below normal density pm. ~ 0.7/)0. In BUU simulations, I will use four IEOS as discussed in section 3.5.2, but with slightly different value for the constants. 1452, iso—SH; . 14E, iso-NH; 8.17;; = (5-5) 149/3, iso—NS; 38.5€ — 21.0 £2, iso—SKM. where the reduced density is defined with 5 = p/po and where [)0 : 0.16 fm‘3 is the normal density of nuclear matter. The three power law types of the IEOS (iso—SH, iso—NH, and iso—NS) have the same symmetry energy at normal density; while the iso-SKM type, which was. suggested by Colonna [101], has a larger symmetry energy at the normal density. The symmetry energies for the four different IEOS have very different behavior as a function of density as shown by Fig. 5.10. Following the convention introduced in section 3.5, the iso—SH and iso—NH types belong to the iso—stiff type of IEOS, while the iso—NS and iso-SKM belong to the iso—soft type. At subnormal densities of /)/PO ~ 0.7, the iso-SKM has the largest value for the synnnetry energy, while the iso—SH has the smallest value of the four IEOS. The different density dependencies of the symmetry energy in Fig. 5.10 are expected to give rise to different paces for isospin diffusion process, following the considerations in Chapter 4. The left panel of Fig. 5.11 shows the isospin diffusion coefficients for nuclear matter at temperature T = 7 MeV, with the different lines there corresponding to five different assumptions on the IEOS: the four IEOS in Eq. (5.5) and the free Fermi-gas EOS. The different IEOS, as seen in Fig. 5.11, give rise to different isospin diffusion coefficients as a function of density. In the regions of our interest, at. densities around p N 0.7p0, the iso—stiff type of IEOS (iso—SH and 160 * _iso-SH i 25 _ _—_— iso-NH - A . ___iso-NS . % 20 _ —-—--iso-SKM ,i E . Egls- O) I 10 - 5 ./I/ L/,/’ O / I I L L I I I I I I I 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 P/Po Figure 5.10: The interaction part of the symmetry energy as a function of density for four different IEOS: iso—SH, iso—NH, iso—N S and iso—SKM. The interaction symmetry energy for the. first three of the IEOS yields, by construction, the same symmetry energy at the normal density. while the iso—SKM yields a different value. iso—NH) yields less diffusion as compared to the iso-soft type IEOS (iso—NS and iso—SKM). The softest type (iso—SKM) of the four IEOS gives rise to the highest isospin diffusion coefficient at moderately subnormal densities. The ordering of the isospin diffusion coefficients for different IEOS is seen to be the same as the ordering of the synnnetry energy, cf. Figs. 5.10 and 5.11. Such correspondence is a result of the proportionality between synnnetry energy and the isospin diffusion coefficient at low temperatures (see the discussion on the low temperature limit of isospin diffusion coefficients in Chapter 4). To exhibit the effects of isospin dependence of the MFs, we may normalize the coefficients to the one without such dependence, i.e., the coefficient for the free Fermi gas. The diffusion-coefficient ratio is plotted in the right panel of Fig. 5.11, which now emphasizes the relative difference in results for the different IEOS. In the next subsection, I will explore the relative strength of 161 the diffusion in actual simulations with different IEOS. 2.0...., ..... .....,......m-m~3.0 [ ___iso-SH l ____ iSO-NH 1.5t __.—iso-NS ‘25 ; _-_--iso-SKM ‘ ’8 - ........ Freegas 5 0.5 '1-5 _,,,,,.L,+,, -"--- ~qu.0 00 0 0.5 .0 Figure 5.11: In the left panel, the isospin diffusion coefficients for nuclear matter for MFs with four different dependence on isospin and also without such dependence, plotted as a function of density; in the right panel, the isospin diffusion coefficients are normalized to that obtained with no isospin dependence in the MP. The temperature of nuclear matter is set at T : 7 MeV. 5.2.3 Isospin Diffusion in Reaction Simulations The BUU simulations have been carried out for four different reaction systems: l2"Sn+”"Sn, 12“‘Sn+“28n. ”2872. +124 Sn and ”28n+“28n, at the beam energy of Blah / A = 50 MeV. and a peripheral impact parameter of b = 6.5 fm. I will analyze the isospin asymmetry of projectile-like spectators that, because of peripherity, largely keeps their original identity. The two projectile-target symmetric reaction systems. l2“SnerSn and “2811+”28n, are the most and least neutron-rich systems. These serve as the references for the projectile-target asymmetric reaction systems, 12"Sn+“28n and 112Sn+mSn, where isospin diffusion between the projectile and target regions takes place. The projectile-like (””811 like) spectator in the 162 124Sn+1128n system will have. about the, same excitation as that in the 124811+12"Sr1 reaction system. Since the. fast. particle emission process is most related to the excitation and the isospin asymmetry of the. source, to the lowest order one can assume that the changes of 1;)rciijectile isospin asymmetry due to fast particle emission in the two systems are. the same. Similar ideas apply to the other two systems with ”2811 projectile. The simulation utilized the MI IEOS as discussed in section 3.5.2, and the specific parameters for the IEOS in this simulation are already given in section 5.2.2. The isospin independent mean field is expected to have little impact on isospin diffusion process. and a soft EOS has been utilized for all the simulations. Geometrically reduced iii-medium cross sections. that were shown to yield reasonable description of the stopping in the heavy-ion reactions. has been used in my sinmlations. In the simulations. the projectile-like spectator region was selected according to the phase space selection criteria. with employed velocity gate of higher than half of the beam velocity in the center of mass frame and a low density cut-off of 0.05m). Such selection was shown to give reasonable description of the spectator matter during and after the violent reaction stage [161]. The isospin asymmetry of the projectile-like spectator region is 1:)lotted as a function of time in Fig. 5.12, for the peripheral reactions of 12"*1”Snwaz‘LHZSn. The collision systems were followed till asymptotic large time of t = 150 fm/c, where the two spectator remanent pieces from the reaction were well separated and their properties are quite stabilized. The isospin asymmetry of the projectile-like spectator exhibits clear systematics. The pro jectile-like spectator isospin asymmetry in the most neutron rich system 124Sn+l2”lSn is decreasing with time and is consistently higher than in any other reaction systems throughout the reaction. In the least neutron rich system “2811+”28n, the asynnnetry changes slower than in any other system and is 0.20.,.,.,.,.,.,.. . _\\ 124+124 iSO_SH 0.15- 125+‘11‘2‘~~-__.._ ______ - - 112+124 ______.______.. - ~c‘ -—-—-—-q. o ‘-—_- - _‘fl- 112+11'2 0.20 . . . 4 00 v 0.15 0.10:"-';~._.~ \. _ 112+112‘ ‘-~-—-_ ..... _-__-_. .1 L 4 l I L I l 0 20 4O 60 80 100 120 140 time (fm/c) Figure 5.12: Isospin asymmetry of the projectile—like spectator region is plotted as a function of time. for four different reactions systems of l"“SII+'2“Sn. 124Sn+ll28n. 11‘ZSnanSn and “2811+”28n at beam energy Elab = 50 MeV/nucleon and impact parameter b = 6.5 fm. The top panel is from a simulation with a stiff symmetry energy density dependence (iso—SH) and the lower panel is from a simulation with a soft synnnetry energy density dependence (iso—SKM). consistently lower than any other reaction system. The change of isospin asymmetry in the projectile-target symmetric reaction systems is characteristic of the f. st particle emission process. where the projectile-like spectator is excited by the impact of the target and begins to emit.- protons and neutron. The projectile-like spectator in the. mixed system 12‘1811 +1128n changes differently from that in the synnnetric system 12“SnerSn. reflecting the effect of the different target. The main difference is attributed to the extra isospin diffusion process that transports the isospin asymmetry liietween the projectile—like and target-like regions in the 164 system 124811+”2811. The diffusion process also difl‘erentiates the other two systems 112Sn+'2"Sn and “2811+“28n. The two mixed systems. 1"24811le12811 and “2811+mSn. differ just in the interchange of the projectile and target. The projectile remnant in one system corresponds to the target remnant in another system. The isospin asymmetry for the projectile remnant in the 124811+112311 decreases over time, while the opposite happens to in the counterpart system. Such behavior for the mixed system specifically expected for the process of isospin diffusion, which acts to transport the isospin asymmetry between the two spectator regions. Isospin diffusion process. if allowed to proceed forever. would eventually make the projectile-like and target-like regions to reach the same in isospin asymmetry. As the time evolution in Fig. 5.12 suggests. the scenario of a cmnplete isospin equilibration between the two spectators was not achitwed in any of the mixed systems. By comparing the top and lower panels of Fig. 5.12. I find that isospin diffusion process is affected by isospin dependence of the nuclear equation of state. The top panel shows the simulaticm result assuming a stiff isospin density dependence (iso-SH) for the mean field interaction. and the lower panel shows that assuming a soft isospin density dependence (iso—SKM). The two projectile-target symmetric: systems. l“Sn +12‘lSii and “2811+”23n. exhibit differences in time evolution of asynnnetry and in the asymptotic value for the asymmetry in the top and lower panel. A more significant (ilifference. ccmnected to isospin diffusion process in the reactions. is the separation between the mixed systems in the two panels. \Nhile in the case of iso—SH EOS. the line. for the projectile-like spectator (12"Srl-like) in the 12"Sn+”2Sn system. is always well separated from that for the 112Sn-like spectator in system ”2811+”‘lSn. In the case of iso—SKM EOS. the separation between the lines for the two mixed system is small. signifying an increased isospin diffusion process. The different isospin diffusion process in the two simulations is anticipated 165 from my earlier argument on the relations between the IEOS and isospin diffusion process (Section 5.2.2). Drawing conclusions on isospin diffusion by comparing (:lirectly the asymptotic isospin asymmetry values in the simulations is difficult because the fast particle emission process is also affected by the IEOS. To emphasize the change of isospin in the projectile-target. asymmetric systems as compared to the symmetric systems, however, one may define an isospin diffusion ratio similar as that en‘iployed by Rami et (11. [99]: [ . I (26,- — (5124+124 - 6112+112) (5 6) I (5124+124 — 6112+112) where the (512_,+12_, and 6112+”; are the isosI‘nn asymmetry of the projectile-like spectator region for the synnnetric reaction systems(12"Sii+12"8n and “ZSna-HQSn). and they set the relative scale of the isospin changes in the reactions. By definiticm. the ratio is 11’, = 1 for the most. neutron rich system l2‘JSII+124Sn, and R,- = —-1 for the least neutrcm rich system ”2811+”28n throughout. the reaction. If the spectator isospin were at any time completely equilibrated between the collision partners during the reaction. the ratios of the two asymmetric system would be H, = 0. In Fig. 5.13. the isospin diffusion ratio R,- is plotted as a function of time for the two mixed systems. The center lines and the shaded regions are the average values and the st at istical uncertainties estimated from multiple runs of simulations [164]. As before, the top and lower panel are for the cases of iso—SH and iso—SKM respectively: the upper and lower lines in each panel are for the projectile-like spectators in the 124Sn+ll2Sn and ”2811+er1 systems respectively. The isospin diffusion ratios for the mixed systems start from R,- = 21:1, and gradually decrease in magnitude. The asymptotic values are reached at around 100 fin/c, which is comparable to the spectator separation time scale of 60 ~ 80 fm/ c. After 100 fm/c. the ratios R, do not change much, indicating a stability of the ratio with respect to further particle evapm‘ation process. 166 \~a.-. ~5-,..-.|th- Itll- #‘uh-lI-‘an --.--. .. u - "“ “um-8MP, ‘vr.t-!-t—-.-.-- -- - - - 0.0 ______ -051 ‘ —1.0 . . 0 20 40 60 80‘ 1100‘ 1120‘ 1510 time (fm/c) Figure 5.13: The isospin diffusion ratio as defined by Eq. (5.6) is plotted as a function of time for two IEOS. The top panel is for the most stiff symmetry energy density de- pendence (iso—SH), and the lower panel is for the most soft synnnetry energy density dependence (iso-SKM). Note the stability of the ratio after 100 fm/c. The shaded areas around the lines indicate the statistical error from averaging over multiple sim- ulations. The isospin diffusion ratios in the simulations with different. IEOS could be directly compared to yield information about isospin diffusion process. because effects of the fast particle emission are largely cancelled out in the ratios. In the iso—SH case. the large nuignitudes for the ratios li’,. and the larger difference between the ratios at a. given time for the 12“‘Sii+“ZSn and 112Sn+mSn systems. are reflections of the less isospin diffusion between the projectile-like and the target-like spectators. On the other hand, the results for the iso—SKM case just indicate a more complete isospin diffusion within the mixed system. The degree of completeness for isospin diffusion process. is now simplified to the ratio It. and the effect of the 167 different IEOS in the simulation is more transparent in Fig. 5.13 than in Fig. 5.12. As will be shown in the next section, the isospin diffusion ratio R,- may also be compared to experimental data. and such a comparison can yield information on isospin dependence of the nuclear EOS [100]. 5.2.4 Comparison to Data In this section, I will compare. the. ratios R,- obtained from the experiment and the. simulations, and show that the experimental result favors the iso—SH type of IEOS. I will also discuss the difficulties in comparing the simulation with the. experiment as well as the many factors that could affect the isospin diffusion ratio. The peri1_)heral collision data from the experiment have been selected by gates in charged particle nmltiplicity that. correspond to a reduced impacted parameter of b/bmal 2 0.8 in the sharp cut-off approximation [165, 166]. with the efficiency weighted average impact parameter equal to < b >: 6.5 fm. The isotope yields at rapidities y/ybmm 2 (1.7 were used for the projectile-like region isospin analysis. More. details of the experiment and data selection can be found in [100]. The isospin diffusion ratios from experiment. were extracted for the isoscaling parameter, which describes the change in the yield of an isotope with the change of the isospin content of a reacting system. Specifically. the experimental ratio 321(N, Z) of yields between two systems of similar mass and energy as a. function of isotope N and Z, were found to obey a simple. scaling relation [104,167,168,103.48] may. Z) 2 may. Z)/Y,(N/Z) = (.'(r)xp((i.-‘V + .32). (5.7) where a and .13 are the isoscaling parameters for the two sources. and are related to the free neutron and proton densities in the emitting source. The neutron isoscaling 168 parameter a is used for the experimental isospin diffusion ratio: 2(1,‘ —‘ (1'124 , — (1r ' ~ , +124 — 112-+112 (r12.i+124 — 0112+112 In the Expanding—Emitting-Sourc'e model (EES) as well as in the canonical Statistical Multifragmentation Model (SMM), the isoscaling parameter a is roughly linearly related to the isospin asymmetry 6 [104]. If I assume either the E138 relation or the SMM relation, as derived by Tsang [104], then the isospin diffusion ratio defined by 6 (Eq. 5.6) and by a ( Eq. 5.8) only differs by less than 4% [100]. From the peripheral collision data of the. Sn+Sn systems, the experimental isospin diffusion ratios are found to be H, = 0.48 :f: 0.03 for the system l2"SIHJIZSn. R,- : —0.48 i 0.03 for system “2811+12‘Sn [100]. The experimental result for isospin diffusion ratio R, together with those from the BUU simulation results are plotted in figure 5.14. The experimental value is almost half way between the lines for no diffusion (R. = :tl) and for complete mixing of isospin (R, = 0), which indicates that the isospin diffusion time scale is comparable to the collision time scale. For the two iso—soft type of IEOS (iso—N S and iso—SKM), the simulation suggests almost complete isospin mixing in the projectile-target. asymmetric system. which is just the opposite of the experimental result. Therefore, I could safely conclude that iso—soft type of IEOS induces too nmch isospin diffusion to explain the experimental result. On the other hand, the iso—stiff type of IEOS (iso—SH or iso—NH) introduces much less isospin diffusion between the two spectators, and are much closer to the experimental result. The iso—SH results match the experimental results especially well, indicating a weak isospin diffusion in the reaction system. Another interesting feature is that the experimental R,- for the l2"’Sn+1”Sn and 112811+”“Sn system are of opposite sign but same magnitude. or in another word. 169 0.6 .1 d 0.4 t _ 0.2 .. . i if fcan-pleTe fix — — _ _i — — §— 1 . § . o—n 04 0.0 I l l l r Exp lso-SH ISO-NH lso-NS lSO-SKM Figure 5.14: The isospin diffusion ratios from the simulations are compared to the experimental extracted isospin diffusion ratios. The symbols above the line R, = 0 are for the projectile-like spectators in the l2”‘Sn-l-“23n system, while those below are for the system “2811+124Sn. The error bars reflect the uncertainties in the experiment or in the simulations. they are mirror values against the line at. R,- = 0. Such synnnetry in the experiment, is not seen in the simulations. In the simulation, the pair of R1 for any given IEOS shows deviation from such a synnnetry, the average of the pair is always to the positive side of the line R, z 0. The lack of synnnetry in pair of ratios R,- in the simulations, may be attributed to the effect of fast particle emissions. If the system did not emit any fast particles, all isospin removed from the l2fSn-like spectator would go into ll'zSn-like spectator. and the pair of isospin diffusion ratios from the two mixed systems should average to near zero. However, the emission of fast particles. of which isospin asymmetry need not be linear in the asymn‘ietry of the emitting source, will affect the isospin asymmetry in the spectators directly. F urthermore, the fast particle emission from 170 the hot, participant region will affect the isospin that flows between the two spectators. and thus affect the spectator isospin indirectly. In the mixed system, a reduced asymmetry emission in the 124Sn-like spectator region, an enhanced asymmetry emission in the 112Sn-like spectator region, and a reduced asymmetry emission in the participant region will all shift the average value for the pair of ratios R,- to the positive side. On the other hand, if the experimental data for the projectile-like source happen to incorporate some of the fast particles emitted during the early reaction stage, then the resulting R,- pair will reflect more the values of the initial system. Different experimental source selection criteria are needed for a more conclusive argument. The. assinmition in the comparison that isotope ratios of the projectile—like region reflect the corresponding isospin content after the reaction (t Z 80 fm / c). albeit supported by some correlation analysis [169, 170]. may require further testing. To understand better the effects of the fast particle emission process and isospin diffusion process. one may look in more detail at the simulation results. Table 5.2 shows the average neutron number N, proton number Z, total nucleon number .4 and total isospin asymmetry 6 of the projectile-like spectator at the end of the simulations. The. total nucleon number A for the 12“‘Sn-like spectator in the ”4811+”st system is higher than that in the 124Sn+124Sn system, while the total neutron numbers in both system are quite close. The average difference of the spectators in the two system is about two protons, which are transferred from the “ZSn-like spectator to the l2fSn-like spectator. The same result of two proton difference also applies to the 112Sn+mSn and 112Sn+1128n systems. For simulations with different IEOS, I find differences in the total N , Z, and A, even for the projectile-target symmetric systems. Those differences demonstrate the complex nature of the fast particle emission process. The isospin asymmetries of the 171 Table 5.2: The simulaticm results for the four reaction systems of l24811+12‘1Sn, 12"‘SnJrHZSn, ”2811+12'8n and “2811+”23n, for the four explore IEOS of iso—SH, iso- NH. iso-NS and iso—SKM. The average N, Z, A, 6 values for the spectator-like region at. the end of the simulation t = 150 fm/c are listed here. 12“ Sn. +123 3n N Z A (5 iSO-SH 56.02 40.01 96.03 0.167 is‘O-NH 50.61 40.45 96.06 0.158 iso—NS 55.61 40.00 95.60 0.163 iSO-SKIVI 54.18 40.63 94.81 0.143 121 Sn +11,2 S n. N Z A 6 iSO—SH 56.30 41.70 98.00 0.149 iso—NH 55.79 42.31 98.10 0.137 iso—NS 56.00 42.57 98.57 0.136 iso—SKM 54.20 42.63 96.83 0.120 ”2.912 +172J Sn. N Z .11 (5 iso—SH 48.61 38.05 86.67 0.122 is‘O-NH 48.26 38.06 86.32 0.118 iso-NS 49.35 37.98 87.33 0.130 iso~SKh1 47.49 37.77 85.26 0.114 ”2811. +112 Sn. N Z A (i iso—SH 48.62 39.71 88.33 0.101 iso—NH 48.72 40.22 88.94 0.096 iso—NS 48.86 40.02 88.89 0.099 iso—SKIVI 47.99 40.23 88.22 0.088 symmetric systems. which set the scale of the ratio Ri. do not follow a simple relation with the IEOS. Extrapolating from the two synnnetric systems, I expect that the mixed system is also affected by the fast particle emission process. If a free cross section is used in the above simulations, isospin diffusion process in the mixed systems will. in general, be reduced, but the effect is limited. The two type of cross sections are primarily different at low momentum transfer, i.e., at low energies and / or forward scatterings. As we have learned in Chapter 4. the isospin diffusion coefficient is inverse proportional to a weighted cross section a,,,,, with a 172 weighting factor of the momentum transfer squared (See Eq. 4.50 in Section 4.4.2). This weighting factor suppresses the differences between the two cross sections. As far as isospin diffusion process is concerned, the simulations with free cross section and with in—medium cross section should not yield significantly different results. However, the use of free cross sections might change the dynamical evolution of the system and the fast. particle emissions. In the context of balance energy studies, one already knows that BUU simulations with free cross section give the wrong balance energy for the current system. The almost complete isospin mixing in the simulation with the iso—soft type EOS is unique in the current energy. and was used to differentiate the different IEOS. In general. as the beam energy goes up. the reaction time scale will iiievitaliily be reduced. and the degree, of the isospin diffusion will become more incomplete even with the iso—SKM used here. At much higher energy. isospin diffusion process will be insignificant to be detected in the experiment, and the isospin diffusion ratio will be less useful. 5.2.5 Summary A systematic study of isospin diffusion 1:)rocess in isospin asymmetric heavy-ion reactions has been carried out within the BUU sinmlations. The projectile spectator regions in the peripheral reactions of ”4'112371 +124'112 Sn. at Em, / A = 50 MeV have been studied, and the isospin asymmetry in such regions was found to be influenced by fast particle emission and by isospin diffusion process. The isospin diffusion ratio R,- clefined in Eq. 5.6 reduces the fast-1.>article emission-effects, and improves sensitivity to isospin diffusion process in the projectile-target asymmetric reactions. The ratio R, exhibits long-time stability following a collision. The values at which R,- stabilizes are different. in simulations that rely on EOS with different isospin dependencies. In nuclear matter with weak isospin gradients, isospin diffusion 173 process is expected to be related to the IEOS. and such an a1,)proximate relation is indeed found in the simulations. The isospin diffusion ratios R,- from the simulations have been compared with ratios constructed from data. The two iso—soft IEOS (iso—N S and iso-SKM) were found to induce too much isospin diffusion to explain the data; the simulation results with iso—stiff IEOS (iso—SH and iso—NH), especially the iso—SH type. agree better. However, the pair of experimental isospin diffusion ratios for the mixed systems 12"Sn +112 Sn and “2872 +'124 Sn show a mirror symmetry which is not quite found in all simulations. The effects of fast particle emission competing with isospin diffusion, causing difficulty in comparing simulations and experiment, and the cross section and energy scale issue have been discussed. Specific analysis of the simulations reveals that about two protons are transferred between the spectators in the mixed system. The results in this section demonstrate the new possibilities for the exploratiml of isospin physics in heavy—ion reactions. 174 Chapter 6: Conclusions In this thesis. I have discussed various aspects of transport. in heavy-ion reactions. Much of the discussion has been devoted to microscopic transport theory. which underlies the transport reaction simulations. The simulations are essential for understanding the mechanisms of central heavy—ion reactions. The phenomenological analysis helps one to understand the physical process as well as the cause-effect relationship in the simulations. Analytical solutions of the transport equations can not be developed for realistic reaction system, but can be for some very simplified cases. However. when such analytical solutions are available. they provide important insights into the physical process beyond the pure numerical understanding from transport simulations. The validity of the theories. and of the assun'iptions in the theories. is tested when the analytical and / or simulational results are (onfronted with the experimental data. Chapter 1 introduces the general background and some of the active areas of research for nuclear transport theory and transport phenomena. Heavy-ion reaction represent an important tool for studying the properties of the hot dense nuclear matter. The possible nuclear liquid—gas phase transition and the nuclear equation of states have been extensively studied in heavy-ion reactions. The isospin related transport theory and phenomena. have recently raised quite some interest. Chapter 2 is devoted to a discussion of neutron enrichment in the midrapidity source in the heavy-ion reactions. A phenomenological phase transition model is introduced for nuclear matter. and the neutron enrichment. is explained in terms of the phase equilibrium condition between the liquid and the gas phase. The cluster formation process in the neck region, when viewed as droplets in the gas phase. counteracts the. trend of neutron enrichment in the midrapidity source. When the nonequilibrium nature of the. heavy-ion reaction is taken into account. the phase equilibrium conditions give the direction of isospin flow in the reaction system. A reversal in the direction of isospin flow is proposed to occur during the reactions. Chapter 3 introduces microscopic transport theory. The reacting nuclear system may be viewed in terms of the transport and interaction of quasiparticles. The Boltzmann equation set, which is the center of transport theory, contains two essential ingredients: the mean field dynamics and the inter-particle collisions. The mean field is introduced through the energy-density functional; both the momentum independent and momentum dependent parameterizations of the functional are discussed. Isospin physics has raised much interest in recent years spurred by the development of the experimental facilities. After a brief introduction of the isospin related phenomena, I discussed isospin dependence of the mean field in detail. Both the density dependence and the momentum dependence of the neutron and proton optical potentials have been discussed. A general T—matrix argument. is used to justify the high momentum behavior of the isospin dependent optical potentials. The isospin dependent mean fields have been used to access the Urca cooling process in neutron stars. Some practical issues for reaction simulations have also been discussed, including the initialization of a reaction system. test particle method for integrating transport equations, the lattice Hamiltonian method for improved accuracy and n‘iodification of cross sections inside a nuclear media. Chapter 4 was devoted to the (.lerivation of the isospin diffusion coefficient for nuclear matter. A systematic expansion of the Boltzmann equation gives self-consistent equations for the variations of the distribution function, from which the flux and transport coefficients could be derived. The isospin diffusion coefficient, shear viscosity and heat cmiductivity have been all calculated using the free space N-N cross—secticms. The isospin diffusion time scale have been also estimated for a heavy-ion reaction system. and the result was cmnpatible with that from a simulation. 176 Chapter 5 was (.levoted to transport sinmlations for heavy-ion reactions. In the first part of that chapter, the interplay of the participant and spectator zones in high energy reactions was examined. The interplay, on one hand, produces the elliptic flow pattern in the participant region, and on the other hand impact the properties of the spectator remnants. In transport simulations. the properties of the spectator remnants after a collision turn out to be directly linked to the features of the nuclear equation of state (EOS) in the participant zone. An acceleration of the spectator piece is found in a heavy system at low impact parameters in the simulations. The acceleration may be explained in terms of the blast of the exploding participant matter impacting the spectator. Finally. isospin diffusion process is studied in heavy-ion reaction sinuilations. The process is expected to be sensitive to isospin dependence of the. mean fields. and such sensitivity is indeed found in transport. simulations. The results from the simulation have been compared to data from the same reaction system. The experimental results are better explained with a synnnetry energy (liaracterized by a stiff dependence on the nuclear density. 177 APPENDICES 178 Appendix A: Macroscopic Quantities I shall consider different types of macroscopic quantities, either not or for sel‘xu'ate components, either in the general frame of observation or in a local frame. For a single component i in the observation frame, the density 72,-, mean velocity 31,-, mean kinetic energy e1, momentum flux tensor 1:),- and kinetic energy flux q,. are given in terms of the distribution f,, respectively. as 12.-(w) ——— (7’5 [dammit (Ala) "in = (2731);; /d3p;%f,(p,r.l). (A.1b) = (2,3,), [cpgfggmpma (Alc) 51 = (25ft)3/d31)f—;§fi(p,r,t). (A.1d) (12' = (23h1)3/d3p% ;—:—if,j(p.r, t). (A.1e) The net quantities result from combining the component contril’mtions. Thus. the net density is n. = 11.1 + 71.2, the net mass density is p = [)1 + [)2 = 7711721 + T712712 while the net velocity y is obtained from pg 2 pl 31 + p2 X2. The kinetic energy g averaged over all particles is given by 71g 2 n1 5] + 72.2 532, the net i’nmncntum flux is 1:) -— 1:), + 52 and the net kinetic energy flux is q = q1 + (12. Local quantities are those calculated with momenta transformed to the local mass frame, i.e. following the substitution p —-) p — m, y. To distinguish local quantities from those in the observation frame, when the frame matters. the local quantities will be capitalized. The local momentum flux tensor f is the, kinetic pressure tensor and the local kinetic energy flux Q is the heat flux. 179 Appendix B: Continuity Equations The collisions in the Boltzmann equation set (4.13) conserve the qumiparticle momentum and energy and the spe( ies identity. T his leads to local conserv ation laws for the corresponding macroscopic quantities. Let x J(p) represent one of the quasiparticle quantities conserved in collisions, x'j(_p) —— 60. p 01 112/2111. For those quantities the integration with collision integrals produces Z/(1311xj J]- : 0. (8.1) 1 As a consequence. from the Boltzmann equation set. I obtain (1 11 (if, p {if} (if,- Eff—“)1111XJ—Ih—( ()f + III} iir + FJ 0p (B ) After a. partial integration. I get from the above 6) t) p ('ix , — . _—— - — , — F- =0 BB ()1, (”1) + fir (n m X) n __iip ’ ( ) where the averages are defined with ”X: Z/(dril 27th) —'—)_3Xjfj(pvr 1) (BA) Substituting for x,- the conserved quantities (xJ-(p) = 15,-). p or 112/2111]). I get. the respective continuity equations: 7 'i (n + —(-— - (n, L) : 0 (H.511) ()1 (if i ci _ 5([1y)+$-fi—HIF1 —712F2=0, (B.5b) f”) i) .. (7)7('Izg)+ 5;“(1’17'1X1'F1—"2X2'F2 =0. (Boc) 180 _J __' Here. I made yet no use of the local frame. The local frame is useful when wants to make use of the assumption of local equilibrium that. imposes restrictions on local quantities. On representing the average velocities as y, = X, + X in the equations above, I obtain the following set, {in- (‘i 0 (if) (i [)1 + (T); ' (pi) = 03 (36b) (9 ii ('i = 57(pV)+I;'(/)VV)+‘0—r"[)“anl“‘nQFQZOv (36“) if) ii = (‘i ('i minfil+ $.(71Ev)+P: 5v+ 5;.Q—n1XlFl—712X2F2 =0B.6d) The equation for mass density in the set above follows from cmnbining the equations for particle densities. The above equations significantly simplify when the assumption of a strict local equilibrium is imposed. Under that assumption, the local species velocities and the heat flow vanish. V,- : 0 and Q = 0. and the kinetic pressure tensor becomes diagonal, P : €— 11 E 1. The. equations reduce then to the Euler set (in, ('i T». a: ‘ ("1!) — 0* ‘8'“) ii ('i __ 2 (i 711? 501v) + ('i—r - (pvv) + 3 (0:) — 711 F1 — 712 F2 2 0, (B.7b) 6) ('i 5 a , (Whig)+Vo5;(n£)+gn_flar-v:0. (B.7c) 181 Appendix C: Space-Time Derivatives for an Ideal Fluid In an ideal fluid. all local quantities can be expressed in terms of the local temperature ’1‘ and the local kinetic chemical potential 11,. If I consider changes of the densities 71., or of the local kinetic energies E, with respect to a parameter 1‘ representing some spatial coordinate or time. or their combination, I find (in, _ , (in, 3 (i ,1 ()J' _ " (ii 2 n, (if i ()(n, 1?) 3 (in, 5 (i3 —I Z _ i T_ + _ l E __ 5 (7.1 11.1’ ‘2 n ()J' ‘2 n —' (if ( ) where (1, : ;(,/'I'. J : log '1' and E, : (i)n,~/(')/(,)T. \Vith the trace derivative defined as (1 ('i + (i _ : _ v . __ ‘ (11 (it (if a particular version of the above relations is (1n, _ E T (1(1, + 3 _ (1.} (l1 _ ’ (11 2'1’ (11 ’ (101,1: ) 3 (1(1, 5 (13 —————'— : -',~T -— .,-Ee—. C2 (11 2 n (11 + 2 n *4 (It ( ) A combination of the above trace-derivative relations with the Euler equations from Appendix B yields the following simple results. (1(1 _l. : (1. C32- (11 ( I) (1.3 ‘2 ('i —‘ Z —— — ' ~ C31 (11 3 01‘ v i ( )) the consistency of which with (C2) and (B7) is easy to verify. The results ((7.3) 182 express basic features of the isentropic ideal-fluid evolution of a. mixture. The entropy per particle in species 2' depends only on (1,, while the ratio of the densities of species (1.1/72.2 depends both on (.11 and (12. The conservation of a, for both species is equivalent to the conservation of entropy per particle and of relative concentration. Finally, the density for species 2'. is proportional to Ti/2 multiplying a function of (1,, which is equivalent to the second of the results above, given the continuity equation for species and the conservation of (1,. 183 Appendix D: Variable Transformation The driving forces for diffusion are 1'1aturally expressed in terms of the gradients of temperature and of chemical potential difference per unit mass [132. However, given the typical constraints on systems, it can be convenient. to express the chemical potential in terms of other quantities, that are easier to assess or control, such as the differential concentration 6, temperature T and net pressure P’ . A transformation of the rariables for the driving forces has been employed. at a formal level. in Sec. 4.3.3. Here, I show, though. how the transf(1rn'1ation can be done in practice for the interaction energy per particle specified in terms of the particle density n and concentration 6, E" : 13"(71, (5). With the nuclear application in mind, I limit myself to the case of ml : 1712 : m. The transformation can exploit straightforward relations between different ('lifferentials. One of those to exploit is the Gibbs-Duhem relation m n 6 ‘ (11" = n. (1/1'I + nz (In:2 + n s (11" = 11 (1/1.' + (1113.2 + n 5 (1T. (D.l) Here. .9 is the entropy per particle and [1’ = (11’l + 1112) / ‘2 is the median chemical potential. 'I‘wo other relations stem from the differentiations of equilibrium particle distributions. already utilized in Appendix C. 3 . Eur — £2 11: (1771‘ (1n, = 5, (111,~ + ,1, (1T E (E,- (111, + 37-,- (I. With it? : ('i(n E")/('i'n.,-, on adding and subtracting the two (i = 1, ‘2) relations side 184 by side, I find I (5 +5 ) I t (7/1.“ I (711," IS. + m (E 5 ) :2 I , — _._ f I - p — ' (Tl 1 ‘2 ([l ()n 6(7 ()(5 "((- 2 _l 2 . . . f . .- (lp'fi til/L". - (3711 ()712 l ’. ~— _ 12 1' — ~—'—2 1) —._——- ——- . .‘ x [([112 ( (7n )5 (n ( 00 n (t + 0'11 ,1] + 0T M dID 3) - and - ‘ ‘) "v ) v , , edn + 'I?.(/(5=({1 — £2) [(1;1'— (bl—)6 (In, — (55%)" d6] + 21(61 +52) 71 'l/t’l 0/1‘1 c (971 1 071.2 1.'..— (_‘Z 1u— —.¥ 1) — — .. x [I [112 (0” )5 (n (06 )n u] + [(W)m (07,112] (M4) Those two equations have the structure (I’m (In : (1%“pr + Ch! (l/l’]:2 + (I'M (15 + ("kT (1T . (D5) where k = 1,2 and where the coefficients 6' can be worked out from (D3) and (D4). 011 multiplying the sides of the first (1" : 1) equation by G2,, and the sides of the second (k = ‘2) equation by (I In and on subtracting the equations side by side. I can eliminate the (In differential obtaining 0 ((1'2,2(r'1l1 — (1‘1" (12”) (Ill! + ((1.21, (1'1“! — (1'1” ("2(1) (“£112 +(c:2,, (2,6 — G." (12,5) (16 + (02,2 or, — (:m (:27) (1T 3 If“ (1/1' + Rddp'm + R; (16 + HT (IT. (D.6) ()n eliminating next the (In, differential using the Gibbs-Duhem l‘(‘l'd.ll(,)ll HI) _ (W’iz) __ If” 12 — Opt 7:6 — 72 (Ru 1'29 - Rd) ‘ HT : ((91312) __ RVs—RT 12 _ Pro ——— a'r ,‘ ltd—I{#¥?‘ n6 _ (1)]le __ R6 12 _ 76 .—I{’—’fl—R' ‘ 1",? u 2 d 186 , I find (D.7a) (D.7b) (D.7c) Appendix E: Brace Algebra T he brace products are employed in finding the transport coefficients within linear approximation to the Boltzmann equation. The brace product of two scalar quantities .4 and 8 associated with the colliding particles is defined as (13') (13‘) {.4,[3} Z /(§7—g)—§.41111(B)+/'(—2;g)—3'.41112(B) d3) (13.) +/(—2#A2121(B)+/(7%.‘12122U3l —_— “3],, + {A, 8],, + [AB].22 . (El) where, in the last step, I have. broken the brace prm'luct into stmare-bracket products representing contributions from collisions within species 1. from collisions between species 1 and ‘2 and from collisions within species 2, respectively. I will first show that the square-bracket product is synnnetric. Thus. I have explicitly 1 (13]) (131)!) (10* (0) (0) ~(0’), an); A. B .. 2 — ", , 192' —” . -. , ' l . ll: 2 / (27f)3 (271-)3 ( U ( (10 ) -fl(l lb fm flb xAm (Bm + Bib — B’- ‘ :5) HI 8 (2fi)3 (2W)3 ([9 X(."lm + All) — ill-a -— .‘ 21))(Bm + [3,}, — B, — B, ), (E12) ' m if) where, to get the last result, I have first utilized an interchange of the particles in the initial state of a collision and then an interchange of the initial and final states within a collision. It is apparent that the r.h.s. of (E2) is synnnetric under the interchange of A and B. Moreover, one can see that a square bracket for B = A, [.4, AL“ is nonnegative and that it vanishes only when A is conserved in collisions. 187 I next consider the. contribution from collisions between different species. (13 [)1 d3 I___)__2 (1012 ()0 1(0) ()0 A.B . = (1520 l ' 1” /<2T> (27) (T 0)")30) ’2 ><(Al + A2) (31 + 32 — Bi - Bi) _ 1 (13171 ‘13 P2 (1012 (0) (0): (0)/ — - (lflv — fl 2 (27r)3 (27r)3 dQ ><(Al + A.) —A’1—A§_)(Bl + 32 — B; — g). (13.3) Here, I again utilized an interchange between the initial and final states and I again observe a symmetry between A and B on the r.h.s. Thus, indeed. all square brackets are symmetric. Moreover, for B = A, One sees that [A,A]12 2 0 and that, the zero is only reached if A is conserved. Combining the results, I find that the brace product (E1) is synnnetric. Moreover, I find that the brace product of quantity A with itself is nonnegative, {:‘l. A} Z 0, and vanishes only when A is conserved. As the brace product has features of a pseudo-scalar product, a version of the Cauchy-Schwarz-Buniakowsky (CSB) inequality [171] holds, {A,A} {3, B} 2 ({A,B})‘2 . (B4) All the results from this Appendix remain valid, in an obvious manner, when the brace product (13.1) is generalized to the pairs of tensors of the same rank associated with the particles, when requiring that the tensor indices are convoluted between the two tensors in the brace, as e.g. in (4.36). 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