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Review

International Journal of Modern Physics E Vol. 15, No. 5 (2006) 973–1068 c World Scientific Publishing Company

PHYSICS OF HEAVY ION COLLISIONS

R. PLANETA M. Smoluchowski Institute of Physics, Jagellonian University, Reymonta 4, Krak´ ow, 30-059, Poland [email protected] Received 10 May 2006 This review article covers a variety of phenomena observed in heavy ion collisions in full range of available collisions energies. The main reaction channels characteristic of each energy domain are discussed in conjuction with existing nuclear reaction models. Methods used to extract characteristic features of hot nuclear objects are shown. Relations between properties of microscopic nuclear objects and infinite nuclear matter are presented. At the end of this review the transition between hadronic phase and the strongly interacting quark-gluon plasma is discussed. Keywords: Nuclear reactions; heavy ions; energy dissipation; equilibration phenomena.

1. Introduction According to the commonly accepted model of the Universe evolution (see e.g. Ref. 1), at 10−5 seconds after the Big Bang, BB, a quark-hadron phase transition took place. Quarks condensed into protons and neutrons. For a certain period of time the Universe got filled with protons, neutrons, electrons, photons and neutrinos. At the time of about 1 second after BB when the temperature dropped sufficiently down, protons and neutrons started to form atomic nuclei. In this early phase of the Universe evolution mostly hydrogen and helium isotopes were formed. During the next steps the atomic nuclei, microscopic nuclear objects ranging from hydrogen to uranium were created. With few exceptions all these nuclei are being formed up to the present days in stars during their life time. Much bigger pieces of nuclear matter can be found inside macroscopic objects called neutron stars. Their masses are of the order of 1.4 mass of the Sun. The only way to study properties of this form of matter in laboratories in a wide range of densities and temperatures are collisions of heavy nuclei. In spite of the fact that the biggest pieces of nuclear matter not exceeding 500 nucleons can be created in this way the transfer of large amount of energy, linear and angular momentum to the interaction region gives here a unique possibility to study unusual and extreme states of such form of matter. Basic properties of this form of matter can be characterized by a few thermodynamic variables and a nontrivial relation 973

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Early Universe Big Bang Quark-Gluon Plasma

T (MeV)

200

Hadronic Matter Pion Condensate

Liquid –Gas Phase

Liquid

0 0.1

1.0

Superconducting Quark Matter

Neutron stars

10.0

U / U0

Fig. 1. A schematic phase diagram of nuclear matter as a function of the temperature, T , and density normalized to the density of atomic nuclei, ρ/ρ0 .

between these variables called equation of state, EOS. A schematic phase diagram of nuclear matter is presented in Fig. 1. After 40 years of investigations the basic properties of heavy ion collisions are relatively well understood for collisions of symmetric (in number of neutrons, N , and protons, Z) nuclei. This article is an attempt to present the most important phenomena discovered in such collisions. We can list here the following of them: fusion, damped collisions, sequential and simultaneous fragmentation of hot nuclear objects, collective flow of nuclear matter, phase transition between hadron and quark-gluon plasma. Physics of heavy ion collisions at low energies explores the left portion of the phase diagram at lower temperatures and a near normal nuclear matter density. At kinetic energies as low as few MeV/nucleon, close to the Coulomb barrier energy, the elastic and inelastic scattering, Coulomb excitation, one or multi-nucleon transfer reactions or formation of compound nucleus are used as sensitive spectroscopic tools in nuclear structure investigations.2 Nowadays the attention is also focused on superheavy nuclei formation and investigation of their properties.3 At energies of the order of 10 MeV/nucleon the nuclear interaction is much stronger. In this energy region the individual nucleon-nucleon collisions are blocked by the Pauli principle. The reaction is described in terms of mean field models and many aspects of them can be explained in terms of thermodynamics. The energy dissipation can be described by the one-body dissipation mechanism.4 At incident energies close to the Fermi energy the mean field description is significantly modified by nucleon-nucleon collisions. The importance of two body dissipation increases. The preequilibrium particles are emitted from the interaction zone, which at the beginning is much hotter then the rest of the system. Copious

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production of complex fragments related to multifragmentation is observed.5 The liquid-gas phase transition of nuclear matter takes place (see e.g. Ref. 6). At beam energies between 100 and 1000 MeV/nucleon one expects that the colliding nuclei form a system which at some moment reaches density 2 to 3 times the normal nuclear density. The key issue in this energy range is the behavior of nuclear matter at densities greater than normal and at excitation energies high compared to the nucleus binding energy. The high compression at initial stage of the reaction pushes nuclear matter away from the interaction region, producing a flow of particles. The pattern of this flow phenomenon depends on the detailed properties of nuclear matter (e.g. on the incompressibility).7 During many years the nuclear reactions with heavy ions below 1 GeV/nucleon have been extensively studied for the symmetric case, N ≈ Z, of the colliding system. Recent investigations performed with radioactive nuclei8–10 and studies on formation and structure of neutron stars11 have drawn attention to the properties of strongly asymmetric, N > Z, nuclear matter.12–14 At somewhat higher collision energies, nucleons are excited into baryonic resonance states, along with accompanying particle production and hadronic resonance formation. The hadronic resonance matter is created. This region of energies was accessible for heavy ion studies at Bevalac accelerator at Berkeley, at the SIS (GSI), at the AGS facility (BNL), and at the SPS accelerator (CERN). A formation of a quark-gluon plasma, QGP, a deconfined state of quarks and gluons, is the major focus of experiments at highest available energies. At the moment only four experiments BRAHMS, PHENIX, PHOBOS and STAR installed at the Relativistic Heavy Ion Collider (RHIC) in Brookhaven can access the region of temperatures of the order of 150 MeV. Similar conditions occurred at around 10 microseconds after the Big Bang. Even higher temperature region will be available with the ALICE experiment at Large Hadron Collider (LHC) at CERN. Nuclear density in the core of neutron stars exceeds that of nuclei up to the factor of 8. Nuclear matter at such large baryon densities is much less explored. The future FAIR facility at GSI15 will be dedicated to investigations of the superdense baryonic matter at moderate temperatures. Generally, the heavy ion collisions can be classified according to a few parameters. The most important between them are: incident energy of the projectile per number of nucleons, E/A, and the impact parameter, b, or the corresponding angular momentum of the entrance channel. The masses and charges of interacting nuclei are also important in classification of such reactions. Commonly accepted classification in respect of incident energy distincts three energy regimes: (i) low energy regime for energies below about 10 MeV/nucleon, (ii) intermediate energies about 10–200 MeV/nucleon, and (iii) relativistic energies. In the following sections the characteristic features of heavy ion collisions corresponding to all these energy domains will be illustrated by representative experimental data, and the leading dissipation mechanism will be discussed in terms of theoretical models.

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direct reactions

grazing collision compound nucleus formation

b gr

deep inelastic collisions

close collisions

elastic scattering

distant collision

Fig. 2.

Coulomb excitation

Classification of the heavy ion collisions at low energies.

2. Low Bombarding Energies At low bombarding energies (≤ 10 MeV/nucleon) the nuclear reaction picture is rather simple and relatively well understood. For such collisions one can distinguish three qualitatively different types of reactions, according to the value of the impact parameter. The classification is presented in Fig. 2. For grazing collisions b ≈ bgr nuclear interaction between a projectile and a target nucleus is small. Here only few degrees of freedom of the projectile and target are involved (the direct reactions). For impact parameters, which are considerably larger than bgr , nuclear interaction is negligible. Trajectories of such distant collisions are completely determined by the Coulomb field. Only excitation induced by the mutual Coulomb interaction between the nuclei can occur. For impact parameters considerably smaller than bgr practically all nucleonic degrees of freedom are involved. Such close collisions can lead to a formation of a compound nucleus (complete fusion) or a class of the two body channels with the transfer of mass, charge, spin and a sizable loss of kinetic energy (damped collisions). In both cases highly excited reaction products are formed. They can later decay either by the emission of light particles and γ rays or by fission.

2.1. Fusion In complete fusion reaction we can distinguish two well defined phases. In the first phase the compound nucleus, CN, is formed with defined excitation energy and with spin determined by the entrance channel spin distribution. The thermodynamical equilibrium of this object is reached. In the second stage the hot CN decays. For relatively light CN (ACN ≤ 160) after the evaporation of light particles and γ rays

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a cold evaporation residue, ER, is left. For heavier CN the fission is a dominating decay channel. The cross section for nuclear reaction with a stage of CN formation is given by: σ(α, β) = σf us (α)R(β) ,

(1)

where σf us (α) is the cross section for compound nucleus formation in the entrance channel α. R(β) is the probability that CN will decay to the channel β. The basis of this reaction model were established by N. Bohr,16 H. Bethe,17 and F. V. Weissskopf 18 and the later development was done by Wolfenstein19 and Hauser and Feshbach.20 For heavy ion collisions the fusion cross section can be determined by four experimental methods: (i) measurement of angular and energy distributions of light particles, (ii) measurement of the same distributions for heavy ER, (iii) measurement of characteristic γ rays originating from ER, and (iv) measurement of fission fragments for heavy compound nuclei. Detection of ER has been used for the measurement of fusion cross section for relatively light systems. Figure 3 presents an angular distribution of evaporation residues for 16 O + 24 Mg system at two beam energies.21 The fusion cross section is determined by numerical integration of these angular distributions. The resulting fusion cross section is shown as a function of 1/ECM in Fig. 4. The solid line through the experimental points is a fit obtained from the model of Glass and Mosel.22 The dashed line and two dotted lines represent predictions from the model of Bass.23

Fig. 3. Angular distributions of the 16 O + 24 Mg fusion cross sections at the indicated incident energies. The (dσ/dΩ) sin θ is plotted so that the area under the curves is the angle integrated cross section. [Fig. 3 in Ref. 21]

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Fig. 4. The 16 O + 24 Mg fusion cross section as a function of inverse CM energy. The solid circles are the values of σf us from measured angular distributions while the single angle measurements are plotted as open circles. [Fig. 4 in Ref. 21].

For comparison the total reaction cross section, σR , predicted by optical model is presented. The squares indicate the energies at which the optical model fits were made to the elastic scattering data and the line through these squares represents the optical model total reaction cross section. Several features of the fusion process are apparent in Fig. 4: (i) the fusion cross section for 16 O + 24 Mg system rises parallel to the total reaction cross section, σR , with increasing incident energy up to about 30 MeV. σf us exhausts a large fraction of σR in this energy range; (ii) at larger energies the fusion cross section saturates at maximum value of 1100 mb. This value is comparable to the maximum σf us observed for other systems involving 2s-1d nuclei.24–26 Information about the fusion cross section can be also extracted from the inclusive γ spectra. This method does not supply the angular distribution of the reaction residues, but the mass and charge spectra are obtained in a broad range. The range covered includes the region where evaporation residues are expected and the region where products of direct and other reaction mechanisms are to be found. In such analysis the complete fusion cross section is obtained by summing up the contributions for the production of different evaporation residues. An example of the inclusive γ spectrum for the 20 Ne + 24 Mg reaction at 55 MeV is shown in Fig. 5.27 The measured σf us for the above reaction in the energy range of 20 N e projectiles from 45 up to 105 MeV is presented in Fig. 6.27, 28 Similarly

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Fig. 5. The γ spectrum measured for the 20 Ne + 24 Mg at 55 MeV beam energy. lines are shown in the inserts. [Fig. 1 in Ref. 27]

42 Ca

and

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43 Sc

to Fig. 4, the data are shown as a function of 1/ECM . The experimental data are here compared with three different predictions of the Bass model.23, 29 They suppose either no friction or friction and rolling or friction and sticking of the colliding nuclei. Also a curve representing the Glass and Mosel model prediction22 is presented here. Figure 7 presents comparison of fusion cross sections for the 16 O + 40 Ca,30, 31 28 Si + 28 Si,32 and 32 S + 24 Mg33 reactions leading to the same compound nucleus 56 Ni. These data are plotted as a function of the inverse CM energy normalized to their Coulomb interaction, Z1 Z2 e2 /ECM . The bombarding energy dependence of the fusion cross section is generally described by defining three energy domains which clearly emerge from experimental data (Figs. 4, 6 and 7). In the first domain, low ECM energies, the fusion cross section increases as 1/ECM decreases. This behavior depends on the static fusion barrier determined by the interplay of nuclear, Coulomb, and centrifugal forces. As

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Fig. 6. The 20 Ne + 24,26 Mg fusion cross section as a function of 1/ECM . Comparison of the Bass and Glas–Mosel model predictions with the experimental data for the 44,46 Ti compound system formation. [Fig. 9 in Ref. 28]

Fig. 7. Complete fusion evaporation residue cross sections for the 16 O + 40 Ca,30, 31 28 Si + 28 Si,32 + 24 Mg33 systems are shown plotted as solid circles, solid triangles, and open squares, respectively as a function of Z1 Z2 e2 /ECM . [Fig. 24 in Ref. 31]

32 S

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long as the projectile is able to overcome the potential barrier, the fusion exhausts the major part of the reaction cross section (regime I). For bombarding energies well above the fusion threshold, the fusion is seriously limited, although it is still an increasing function of the beam energy (regime II). At higher energies, it is observed that σf us decreases with ECM , due to instability of the composite system (regime III). The excitation function σf us (ECM ) for the complete fusion may be analyzed in terms of semiclassical models (see e.g. Refs. 23, 29, 34, 22, 35, 36). The cross section for the complete fusion is given by: ¯2 σf us (ECM ) = π λ

lf us

(2l + 1)T (l, ECM ) ,

(2)

l=0

where the transmission coefficients T (l, ECM ) are approximated by the transmission of an inverted parabola:37 −1 V (Rl ) − ECM T (l, ECM ) = 1 + exp 2π , (3) ωl with V (Rl ), Rl , and ωl being the barrier height, position, and curvature of the potential for the lth partial wave, respectively. In the sharp cut off approximation the fusion cross section is given by the formula: VB 2 σf us (ECM )) = πRB 1− , (4) ECM where RB is the barrier radius, and VB = V (RB ) is the conservative potential at the barrier radius. Formula 4 describes very well the fusion data at relatively low energies (regime I). This is illustrated in Fig. 8, where σf us (ECM ) versus 1/ECM are collected for 35 Cl plus 48 Ti, 60,64 Ni, 90 Zr, 116,124 Sn systems.35 The information about the ionion nuclear potential has been reported from the analysis of experimental fusion excitation functions based on such simple models. Values of RB and V (RB ) are determined from the slope and intercept of the plot of σf us versus 1/ECM , as illustrated in Fig. 8. It has been shown that nucleus-nucleus fusion cross sections can be also predicted from simple, classical, two-body models (see e.g. Refs. 38, 39 and 40). The basic ingredients of such models are (i) the assumption of a frozen shape of the colliding nuclei during their approach, (ii) the assumption of a two-body potential, and (iii) the assumption of frictional forces which allow the system to be trapped in a region of attractive interaction. In these models a critical distance Rcr is defined, which marks the onset of strong frictional forces. The fusion will then occur for all classical trajectories which penetrate to Rcr . All of these models overpredict the experimental fusion evaporation-residue cross sections in the high energy region (regime III). Part of this discrepancy may reflect the fact that fission channels are not included in the experimental cross sections, yet

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Fig. 8.

Fusion cross sections versus 1/ECM . Solid lines are fits of Eq. (4) [Fig. 6 in Ref. 35].

are implicitly included in the model calculations. The fusion cross section limitation may be also related to the structure of the compound nucleus.41–44 Such a possibility is illustrated in Fig. 9 for the 16 O + 40 Ca,30, 31 28 Si + 28 Si,32 and 32 S + 24 Mg33 reactions leading to the same compound nucleus 56 Ni. Here critical angular momenta deduced from experimental data for these three systems, using the sharp cutoff approximation, are plotted versus the excitation energy in the compound nucleus. The dashed vertical lines correspond to the angular momenta calculated for vanishing symmetric fission barrier (Bf = 0 MeV) and a symmetric fission barrier comparable to the nucleon separation energy (Bf = 8 MeV), respectively.45 The critical angular momentum extracted for the 16 O + 40 Ca system is in agreement with the calculated fission barrier limit. On the other hand, critical angular momenta for the 28 Si + 28 Si and 32 S + 24 Mg are somewhat larger than expected basing on the calculated fission barrier limits. For similar systems 6 Li, 9 Be, 12 C + 40 Ca,46 32 S + 12 C,47 32 S + 24 Mg48 the limitation of the fusion cross section was discussed in terms of the coalescence and reseparation dynamical model.49, 50 It was found that σf us can be limited by the stability of the deformed compound system at high angular momenta with respect of fission (see Fig. 22). Details of the model and his predictions are given in Sec. 2.3. 2.2. Damped collisions For a heavier projectile and heavier target nucleus system the channel of complete fusion is substantially suppressed due to the Coulomb repulsion and the angular momentum. The room left is filled by inelastic reactions with two body exit channels

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Fig. 9. The critical angular momenta extracted from the complete fusion cross sections (Fig. 7) for the three systems forming the compound nucleus 56 Ni under the assumption of a sharp cutoff partial wave distribution are plotted as a function of the excitation energy in 56 Ni. The dot-dashed line corresponds to the statistical yrast line with ∆Q = 10 MeV and r0 = 1.2 fm. The dashed lines indicate the calculated angular momenta for fission barriers of Bf = 0 and 8 MeV using the Sierk macroscopic model.45 [Fig. 25 in Ref. 31]

involving a rather large excitation of both fragments. At incident energies of a few MeV/nucleon above the Coulomb barrier properties of damped collisions can be enumerated as follows: (i) The reaction is binary in a sense that only two massive fragments are observed in the exit channels. They are usually called the projectile like fragment, PLF, and the target like fragment, TLF. The first two-body step of the reaction is followed by a sequential deexcitation of the primary PLF and TLF via evaporation of neutrons, protons, α particles and γ rays or via fission. The validity of the two-body kinematics assumption was tested in some number of experiments, e.g. Ref. 51. The results of such a test for the 40 Ar + 58 Ni reaction at incident energy 7 MeV/nucleon are shown in Fig. 10, where event-by-event sum θ3CM + θ4CM as well as the sum ϕ3 + ϕ4 (the coplanarity test) are presented. Here indexes 3 and 4 denote PLF and TLF respectively. These outcomes verify that the damped collision process is binary. The finite width of both distributions is caused by particle evaporation from excited PLF and TLF fragments.

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Fig. 10. Measured deviations from colinearity for the in plane and out of plane distributions for all inelastic events in the 40 Ar + 58 Ni reaction at incident energy of 7 MeV/nucleon [Fig. 21 in Ref. 51].

(ii) Angular distribution of the reaction cross section is sideways-peaked for heavy systems and forward-peaked for light ones. Both types of distribution are indicative of a fast reaction. The time of nuclear interaction is significantly smaller than the time needed for a complete rotation of the system. An example of such angular distribution of the Xe-like reaction products from the 136 Xe + 209 Bi system at 6.9 MeV/nucleon is presented in Fig. 11.52 (iii) The kinetic energy distribution of the final fragments is very broad. It extends from quasi-elastic energies down to the Coulomb energies of the interacting highly deformed PLF-TLF system. This behavior is illustrated in Fig. 12, where the contours of the double differential cross section (d2 σ/dEdθ)CM in µb MeV−1 deg−1 for the 86 Kr + 139 La reaction at 7.0 MeV/nucleon are shown.53 Here θCM is the observation angle for the cold PLF remnant. Axis y represents the total CM kinetic energy, TKE, in the exit channel: CM CM T KE = Ekin,P LF + Ekin,T LF .

(5)

The upper energy limit is given by the incident energy in the CM system, ECM . The lower one corresponds to the Coulomb repulsion energy of the outgoing fragments. The characteristic correlation between TKE and θCM will be explained later in Sec. 2.3. (iv) The fragment mass and charge distributions are broad and located close to the mass and charge of the projectile and of the target nucleus respectively. The

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Fig. 11. Angular distribution of the Xe-like reaction products for 6.9 MeV/nucleon. The curve is to guide the eye. [Fig. 3 in Ref. 52]

136 Xe + 209 Bi

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system at

Fig. 12. Contours of the double differential cross section (d2 σ/dEdθ)CM in µb MeV−1 deg−1 for 86 Kr + 139 La reaction at 7.0 MeV/nucleon. [Fig. 4 in Ref. 53]

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Fig. 13. Cross section contours in the N versus Z plane for the 74 Ge + 165 Ho reaction at four representative energy losses. Each bin width is ±5 MeV about the centroid. The dot-dashed line shows the line of maximum beta stability. Solid dot is 74 Ge. [Fig. 4 in Ref. 54]

widths of these distributions increase with the decreasing kinetic energy in the exit channel. This feature is characteristic of a statistical process of nucleon exchange between interacting nuclei. An example of such behavior is presented for the 74 Ge + 165 Ho reaction at 8.5 MeV/nucleon (Fig. 13).54 Here the nuclide distribution in the neutron vs. proton number plane is shown for four representative bins of the total kinetic energy loss, ELOSS (ELOSS = ECM − T KE). Only cold remnants of the PLFs, after the evaporation of light particles, are visible on these plots. This is due to the characteristics of the used detection system, which was unable to give the charge and mass information for the target like nuclei. The solid dots represent the projectile (NP , ZP ) values. At ELOSS = 30 MeV, where little energy is available for decay of the primary fragments, it is apparent that the net transfer of one neutron and/or proton from the projectile to the target nucleus is much stronger than the transfer in the opposite direction. With increasing energy loss the contours broaden due to the combined effects of nucleon exchange and statistical decay of the hot primary fragments. Most noticeable in the evolution of these distributions is the strong preference for proton transfer from projectile to the target with increasing energy, and the gradual alignment with the N/Z ratio corresponding to the line of maximum beta stability. A two-dimensional Gaussian fitting procedure55 has been applied to these distributions. From a least-squares fit centroids Z, N , A, were obtained. In Fig. 14

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Fig. 14. Centroids of the Z, N, and A distributions and the N /Z ratio for PLF’s as a function of energy loss. Squares indicate measured post-evaporative values, circles give the primary values reconstructed from the kinematic coincident technique, and the solid line is the prediction of the nucleon exchange transport model.59–61 [Fig. 5 in Ref. 54]

the centroids are plotted along with the N /Z ratio, as a function of energy loss. The circles represent the primary (before evaporation) Z , N , and A distributions reconstructed from the coincidence data, squares show the measured (post-evaporative) results. Primary Z values were obtained from fitting the difference between the primary and secondary masses using the PACE-II statistical evaporation code.56 The experimental primary distributions in Fig. 14 demonstrate that the net transfer of protons from the PLF to the TLF is favored in this reaction. In contrast, the net neutron transfer is small. (v) The problem of the equilibration of the N/Z ratio is strongly related to the evolution of the proton and neutron number centroids. As can be seen in Fig. 14, the average neutron/proton ratio for the primary PLF fragments gradually increases with increasing energy loss and evolves toward, but does not reach, the N/Z of

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Fig. 15. Evolution of net nucleon exchange as a function of energy loss for four systems, as defined on this figure. Measured distributions are indicated by squares, primary distributions by circles and theoretical predictions by solid lines.59–61 Dot-dashed lines show the line of maximum beta stability and dotted lines show the N/Z ratio of the composite system for the respective reactions. The direction and magnitude of the gradient of the potential energy surface for the projectile-target system are shown by the dashed arrow. [Fig. 10 in Ref. 54]

the composite system (239 99 Es: N/Z = 1.41). In a more complex way this behavior is presented in Fig. 15 for the 58 Ni + 238 U, 64 Ni + 238 U,57 56 Fe + 165 Ho,58 and 74 Ge + 165 Ho54 systems. All these measurements were performed at bombarding energy of E/A = 8.5 MeV/nucleon. Here the evolution of the centroids of the nuclide distributions in the N versus Z plane is plotted for successive energy loss bins up to 180 MeV of the energy loss. Squares represent the post-evaporative measured data; open circles are the reconstructed primary data. For reference in each panel the dash-dotted line shows the valley of beta stability and the dotted line is the N/Z ratio of the corresponding composite system. The solid lines are the result of the nucleon exchange transport model59–61 for the systems under consideration. Figure 15 illustrates in graphic fashion the strong differences in behavior between the experimental and transport model centroids. Whereas the transport model strongly favors the transfer of neutrons from the target to the projectile to achieve N/Z equilibration, the experimental data indicate a preference for proton transfer from projectile to target. For each of these systems the vector indicating the direction and magnitude of the gradient of the potential energy surface, PES, at the injection point is also presented. It is expected to define the direction of the Z and N net transfers.

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Fig. 16. Left scale: proton drift (upper panel) and neutron drift (lower panel) as a function of projectile N/Z value for E/A = 8.5 MeV 40,48 Ca, 56 Fe, and 58,64 Ni ions incident on 238 U. All values are taken at an energy loss of 100 MeV. Right scale: values of the gradient in the PES at the injection point (+) for the considered systems. [Fig. 15 in Ref. 57]

Calculations of the potential energy surface were performed according to the relation: LD V = VPLD LF + VT LF + VN + VC + VL ,

(6)

where ViLD is the liquid drop (LD) binding energy with shell corrections but suppression of pairing effects of the PLF and TLF.62 The nuclear potential VN was also based on the liquid-drop model with the modified proximity potential of ´ atecki.63 The effective centrifugal potential, VL , and the Coulomb potential, VC , Swi¸ were those assumed in the Randrup transport model.59–61 The comparison of four systems in Fig. 15 reveals that for the 58 Ni + 238 U system (where the gradient is strongest), the evolution of the neutron and proton centroids as a function of energy loss follows the gradient in the potential quite closely. For the 74 Ge + 165 Ho system, proton exchange is considerably stronger than the one predicted by the gradients. The correlation between the gradients of the PES and the proton and neutron centroids evolution was tested for several systems. Figure 16 shows (squares) the centroids of ∆N and ∆Z for 40 Ca + 238 U, 48 Ca + 238 U,64 56 Fe + 238 U,65 58 Ni + 238 U, and 64 Ni + 238 U57 systems at an energy loss value of 100 MeV as a function of projectile N/Z ratio. Here ∆N = N − NP , ∆Z = Z − ZP . Also shown by crosses are the corresponding values of the gradient of the PSE at the

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Fig. 17. Average of excitation energy divided by primary mass as a function of energy loss for the 74 Ge + 165 Ho reaction at 8.5 MeV/nucleon. Data are averaged over all masses (top). Average percentage of PLF excitation energy relative to total available excitation energy as a function of energy loss (bottom). Solid line is the prediction of the nucleon exchange transport model in both figures.59–61 Dot-dashed line indicates equipartition of excitation energy and dotted line is the equal temperature limit for a simple Fermi gas model, Ap /Atot = 74/239. [Fig. 5 in Ref. 80]

injection point for each system (right-hand scale). One observes that both the experimental charge and neutron drifts behave systematically as a function of projectile N/Z ratio. Similar behavior can be seen for the PES gradient components. One observes a strong correlation between the centroids and the corresponding gradient components. (vi) Process of dissipation of kinetic energy of the entrance channel and its redistribution among the various degrees of freedom is of high significance for understanding of the damped reaction mechanism. A number of experiments and analyses were performed which shed the light on various aspects of excitation-energy sharing in damped collisions. Several early experiments66–72 suggested that thermal equilibrium might be achieved very early in the damping process, perhaps as low as ELOSS = 50 MeV. The subsequent works58, 73–81 indicated a gradual transition from approximate equipartition of excitation energy for small energy losses to thermal equilibrium at large energy losses. As an illustration, the results for the 74 Ge + 165 Ho reaction at 8.5 MeV/nucleon are presented in Fig. 17. In the upper panel the average excitation energy of the

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Fig. 18. Parameter REQ , defined by Eq. (7) as a function of energy loss divided by available excitation energy above the Coulomb barrier. The data for the 56 Fe + 238 U,77 56 Fe + 165 Ho,78 and 74 Ge + 165 Ho79, 80 systems are presented. The solid line is the prediction of the nucleon exchange transport model59–61 and the dashed line is based on the random neck rupture model.82 [Fig. 6 in Ref. 80]

PLF divided by its primary mass (before evaporation) as a function of energy loss is shown. The expected increase in the internal excitation of the PLF, as more energy is dissipated into system, is observed. The representation of partition of the excitation energy, EP∗ LF /ET∗ OT AL , is shown in the lower panel of Fig. 17. Also the corresponding ratios are indicated for equipartition of the excitation energy and for a thermal equilibrium in a simple Fermi gas model. The experimental points exceed the equal-energy sharing ratio at very low excitation energies and then drop below the 50% horizontal line as the amount of dissipated energy grows. For highly damped events, the average excitation energy division ratio saturates at a value of about (41- 44)%, which is considerably higher than the Fermi gas equal temperature value of 31%. Various data sets for asymmetric in mass target-projectile systems are compared in Fig. 18. To facilitate the comparison the factor REQ is defined: E∗

REQ =

E ∗ P LF − T OT AL

0.5 −

AP AP +AT AP AP +AT

,

(7)

where AP and AT are the mass numbers of the projectile and target, respectively. The value of REQ is one for equal excitation energy division and decreases to zero for the equal temperature limit. The data for the 56 Fe + 238 U,77 56 Fe + 238 U,78 and

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Fig. 19. Comparison of the strengths of the correlation between the excitation-energy division and the net mass transfer in terms of the slope parameter R deduced in the straightforward analysis of Ref. 79, 80 (solid boxes) and in Ref. 83 analysis (solid dots). [Fig. 6 in Ref. 83] 74

Ge + 165 Ho79, 80 systems are presented as a function of the ratio of energy loss to available energy above the corresponding Coulomb barrier. Within the limits of the experimental error, all these data sets appear to be self-consistent with one another. It should be noted that in the analysis of 56 Fe + 165 Ho data asymmetric tails to the REQ distributions were eliminated by a Gaussian fitting technique. Thus, the data at the high energy loss limit are not necessarily in conflict. The solid line makes prediction of the nucleon exchange transport model. Also the predicted behavior for the REQ parameter based on the random neck rupture model of Brosa and Grossman82 is shown in Fig. 18 (dashed line). (vii) Some studies have also indicated that the partition of excitation energy is dependent on the net nucleon exchange.74–76, 78–80 Extensive analysis of results obtained by Refs. 79 and 80, done by Ref. 83 confirms the existence of correlation between excitation energy division and net mass transfer in the 74 Ge + 165 Ho reaction at 8.5 MeV/nucleon and estimated in quantitative way the magnitude of the effect. The dependence between the EP∗ LF and the primary AP LF was approximated by a linear function: EP∗ LF = C + RET∗ OT AL (AP LF − A0 ) ,

(8)

where C and R are ELOSS dependent parameters. The AP LF is the true primary mass and A0 is the centroid of the primary mass distribution at a given ELOSS . In Fig. 19 the strength of the correlation is illustrated in terms of slope parameter R. Two sets of points are displayed to demonstrate the importance of the finite resolution effects. Solid squares represent the R dependence on ELOSS when the experimental resolution effects were not taken into account.79, 80 In contrast, the

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solid dots result from the analysis taking quantitative account of finite resolutions effects.83 The analysis of Toke et al.83 suggests also that the acceptor fragment receives on the average approximately 65% of the total kinetic energy dissipated in each individual exchange, whereas the donor receives only 35% of this energy. 2.3. Model description Most of the models describing the interaction of heavy ions at low bombarding energies reduce the complicated many body problem by introducing a few time dependent macroscopic variables. These macroscopic variables which describe the collective motion of the system should vary slowly in time. The chosen set of such macroscopic variables can include: the relative distance of the two nuclei, angles of rotation, shape parameters, excitation energies, mass, charge and others. The remaining variables are then microscopic variables that should vary rapidly in time in an uncorrelated way. For heavy ion collisions the macroscopic variables behave as a good approxima¯ of the tion like classical variables. This is possible in the case when wavelength λ relative motion is much smaller than the geometrical size of the interacting nuclei. For the relative distance r the de Broglie wavelength is given by: − 12 2 2µ ¯ (ECM − V (r)) . (9) λ= Here V (r) is the interaction potential, µ is the reduced mass and ECM is the energy of the system. For the case of 74 Ge + 165 Ho reaction at 8.5 MeV/nucleon ¯ at the interaction barrier (Vint = 228 MeV) is (ECM = 434 MeV) the wavelength λ equal to 0.05 fm. This is small compared to a characteristic length like the nucleus surface thickness of 1 fm. A more general and more precise condition for classical behavior is given by: ¯ |gradλ(r(t))| 1.

(10)

This implies that neither the conservative nor the dissipative force should change ¯ too rapidly within the length of λ. Figure 12 shows an experimental dependence between kinetic energy and decay angle for 86 Kr + 139 La reaction at 7.0 MeV/nucleon. This dependence can be interpreted in the frame of an almost classical picture. In this picture proposed by Wilczy´ nski a monotonic dependence of the deflection angle on impact parameter is postulated (see Fig. 20). For the decreasing value of impact parameter, close to the grazing value, the scattering angle does not increase any more like in a pure Coulomb interaction. The trajectories are bent towards smaller deflection angles because of the attractive nuclear forces. This produces a focusing of quasi-elastic events at an angle close to the grazing angle. For smaller impact parameters the deflection angle decreases, crosses zero, and goes to negative values reaching finally orbiting configuration of the dinuclear system. At the same time more and more

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Fig. 20. Wilczy´ nski plot: (a) sketches in an energy versus scattering angle plot, the contour lines of constant cross section. In a reaction plane defined by the beam axis and the detection angle two components are visible: (i) for impact parameters on the right hand side of the beam axis (solid lines) and (ii) on the left hand side (dashed lines). Part (b) of the figure illustrates the corresponding trajectories leading to the energy-angle correlation of part (a). The impact parameter is denoted by b.

energy is dissipated. Below a certain value of impact parameter the two nuclei fuse and form a compound nucleus. It was recognized by Wilczy´ nski84 and by Beck and Gross85 that features showing up in Fig. 20 can be accounted by introducing friction forces in the classical equation of motion. Improved models have been proposed by Bondorf et al.,86 Gross,87 Siwek-Wilczy´ nska et al.88 and Beck et al..89 Most of these models utilize frictional forces proportional to velocities. The transfer of charge and mass is usually included in such models assuming that: (i) equations of motion in the entrance channel are integrated until the point of closest approach; (ii) transfer of neutrons and protons takes place only at Rmin . At this point the relative velocity is corrected for the mass transfer effect; (iii) in the exit channel equations of motion are solved using the potential of the outgoing system. Variety of models were used to describe specific features of the mass and charge transfer processes, e.g. the minimum transfer model,88 or the random walk

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y

R1

l1

l2

R2 z

r Fig. 21.

Parameterization of nuclear shape.

model.90, 91 Generally such models are able to reproduce the ridge of the Wilczy´ nski plot. More advanced model dealing with dissipation of kinetic energy in heavy ion ´ atecki.49, 50, 92 This model collisions is the coalescence and reseparation model of Swi¸ is an extension of the “chaotic regime dynamics” (see e.g. Ref. 93). In the model a set of three dimensionless time dependent macroscopic variables ρ, λ, and ∆ is used to describe the evolution of shape of nuclear system along the dynamical trajectory: ρ=

r , R1 + R2

(11)

λ=

l1 + l2 , R1 + R2

(12)

∆=

R1 − R2 . R1 + R2

(13)

Their names are: distance ρ, neck λ, and asymmetry ∆ variable, respectively. The shapes are assumed to be axially symmetric and correspond to two, generally unequal, spheres modified by a smooth fitted portion of the third quadratic surface of revolution. The parameters R1 , R2 , l1 , l2 , and r are explained in Fig. 21. The volume of the shape along the trajectory is kept constant. The time evolution of the nuclear configuration is given by dynamical equations of motion. There are altogether six degrees of freedom: three of them defining the shape (ρ, λ, ∆) plus three rotational variables (the orientations in the collisions plane of the two spherical parts and the orientation of the line joining their centers). A set of six second-order differential equations is solved: ∂ ∂ d ∂ − F, (14) L=− dt ∂ q˙i ∂qi ∂ q˙i where L = T –V is the Lagrangian and F is the Rayleigh dissipation function. The

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kinetic energy is given as T =

3 1 1 1 1 2 Mij q˙i q˙j + Irel ωrel + I1 ω12 + I2 ω22 , 2 i,j=1 2 2 2

(15)

where qi = (ρ, λ, ∆) is the set of shape parameters and the rotation of the system is described with two spheres rotating with angular velocities ω1 and ω2 , and the whole system rotating with angular velocity ωrel . Mij is a mass tensor calculated in the Werner-Wheeler approximation to irrotational flow.94, 95 I1 and I2 are inertias tot − I1 − I2 is the inertia of of two spheres taken as rigid bodies and Irel = Irigidbody the relative rotation. For the potential energy V two approaches are used. In the first case a liquiddrop energy approach is used.62 In this case the finite range effects of the nuclear forces are not taken into account. In the more fundamental approach the nuclear part of the potential is calculated according to a double folding procedure developed by Krappe et al..96 It is writen as: exp(−σ/a) 3 3 σ Cs Vn = − 2 d rd r , (16) 8π 2 r02 a3 a σ where σ = |r − r |, Cs = as (1 − κs I 2 ), and parameters r0 , a, as , and κs are taken from the fit done by Krappe et al. in their original paper. The mechanism of the energy dissipation adopted in the coalescence and reseparation model is of the one-body type,97 which means that the dissipation arises due to collisions of independent nucleons with the moving walls of the nucleus. There are two limiting cases in which two different simple formulas for the rate of the dissipated energy are derived. The first one is called the mononuclear regime. In that case the energy dissipation is given by the following wall formula: dE = ρ¯ v dσ(n˙ − D)2 , (17) dt wall S where ρ is the mass density of the nucleus, v¯ is the mean speed of nucleons in the nucleus (equal to three quarters of the Fermi velocity in the Fermi gas mode), and n˙ is the normal velocity of an element dσ of the nuclear surface. The quantity D is the overall drift velocity of the gas of nucleons ensuring the invariance of Eq. (17) against translations and rotations. In the second limiting case, called the dinuclear regime, when two ions are either separated or connected by a thin neck, the wall plus window formula is applied and it reads as follows: dE = ρ¯ v dσ(n˙ − D1 )2 + ρ¯ v dσ(n˙ − D2 )2 dt wall+window S1 S2 1 16 ρ¯ v ˙ V1 . + ρ¯ v (2u2r + u2t )Sw + 4 9 Sw

(18)

The first two terms represent the wall formula dissipations for the two pieces. The last two terms are the windows formulas for the energy dissipation, Sw being

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Fig. 22. (a) Typical trajectories for the 12 C + 40 Ca reaction. (b) For L = 42 and L = 53 the numbers written along the trajectories specify the asymmetry parameter ∆, the collision time τ (in units of 10−22 sec), the number of revolutions of the system nr , and the maximum relative difference between ω1 , ω2 , ωrel . [Reprinted from Ref. 49 (Fig. 3), Copyright 1985, with permission from Elsevier]

the area of the window. The components of the relative velocity of two fragments ut and ur correspond to the velocities parallel and perpendicular to the window. The V˙ 1 being the rate of the change of the volume of fragment 1. In general, when the situation is in between these two limiting cases a smooth transition between formulas (Eqs. (17) and (18)) is used and finally the Rayleigh dissipation function F is expressed as: dE dE dE + (1 − f ) , (19) =f 2F = dt dt wall dt wall+window with a form factor f going to 1 for sphere or spheroid like shapes and going to 0 at scission. ´ atecki model for the 12 C + 40 Ca reacTypical trajectories predicted by the Swi¸ tion at bombarding energy of 186 MeV are presented in Fig. 22. Each trajectory

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represents a collision with a given angular momentum L. The numbers written along some of the trajectories specify the asymmetry parameter, ∆, the collision time, τ , in units of 10−22 sec, the number of revolutions of the system, nr , and the maximum relative difference between ω1 , ω2 and ωrel . As one can see three trajectories cross the scission line (L = 51–53) and correspond to a class of binary collisions. All the other trajectories for L ≤ 50 stop at points located above the scission line and they represent a fusion process. The other class of models is based on the transport equation.59–61, 98–101 Also this class of models is based on the assumption that one can describe the interaction of heavy ions in terms of a few time dependent macroscopic variables q = {qi (t)}. It is assumed that they are coupled to an intrinsic system essentially in chaotic motion. Such coupling leads to transport phenomena reflected in characteristic temporal changes q˙ = {q˙i } of the corresponding coordinates. Assuming an intrinsic system at a local thermodynamic equilibrium at all times, the time evolution of the interacting nuclei is describable in terms of a joint probability P (q, q, ˙ t) for finding the system at time t in the state described by coordinates q and their temporal changes q. ˙ Probability P (q, q, ˙ t) can be calculated by solving the Fokker–Planck transport equation: ∂ ∂2 ∂ ˙ t) = − [vi (q, q)P ˙ ]+ [Dij (q, q)P ˙ ]. + q∇ ˙ q − (∇q U )∇q˙ P (q, q, ∂t ∂qi ∂qi ∂qj i i,j (20) Here U is a potential, vi and Dij are drift and diffusion coefficients, respectively. The left hand side describes the change of the probability distribution P due to the velocity q˙ and the force −∇q U . For the overdamped motion along a coordinate qi , corresponding contributions to the second and third term of the left hand side of Eq. (20) are absent. One of the methods to solve Eq. (20) is to consider the time evolution of average values of q¯i , q¯˙ i and of higher moments of the probability distribution with respect to coordinates qi and velocities q˙i . The average values follow the Lagrange-Rayleigh equations of motion (Eq. (14)). For the overdamped motion along a coordinate, qi corresponding kinetic terms are neglected, such that ∂ ∂ L= F, ∂ q¯i ∂ q¯˙ i

(21)

determines the equation of motion overdamped by frictional forces. In the model proposed by Randrup59–61 the set of macroscopic variables is: {qi } = {r, θ, θP LF , θT LF , ρ, AP LF , ZP LF , TP LF , TT LF } ,

(22)

some of which are defined in Fig. 23. The quantities AP LF and ZP LF denote the mass and atomic numbers of the PLF reaction partner, respectively. The intrinsic temperatures TP LF and TT LF of the reaction partners are, in general, not equal, but each fragment is assumed to be in its own thermal equilibrium at all times. In

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ș RP

șP 2ȡ CM

RT

șT

r

Fig. 23.

Dinuclear shape coordinates assumed in the dynamical calculations.59–61

this simple parameterization it is assumed that neck of radius ρ contains no mass but encloses a free space that allows an unobstructed exchange of nucleons between the reaction partners. In this model, the neck contributes to the surface energy of the dinucleus. In Randrup’s model only the fluctuations in neutron (N ) and proton (Z) numbers of the PLF fragment about the average values are considered. For overdamped motion, the Fokker–Planck equation reduces to: ∂ ∂ ∂2 ∂2 ∂ P (N, Z, t) = − vN − vZ + D + D (23) NN ZZ P (N, Z, t) . ∂t ∂N ∂Z ∂N 2 ∂Z 2 Drift and diffusion coefficients, v and D, respectively, are calculated in this model on the microscopic level. Correlations between neutron and proton exchange are neglected and it is assumed that DN Z = 0. The Randrup dynamical transport model predicts the first and second moments of the proton and neutron number distributions, as well as the excitation energies of the primary reaction fragments. The results of the calculations are shown as a solid lines in Figs. 14, 15, 17, and 18. Apparently, the most significant divergence between experiment and Randrup model predictions is found for the centroids shown in Figs. 14 and 15. The model calculations predict very little net proton transfer over nearly the full energy loss range accompanied by the net pickup of neutrons by the PLF, producing a mass drift towards mass symmetry. In contrast, the data reveal that proton transfer from

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projectile to target dominates the net exchange of nucleons. This produce a mass drift towards asymmetry. Another way to describe the heavy ion interaction in terms of quantum mechanics is to use the Time Dependent Hartree-Fock (TDHF) model.102, 103 It is a one-body approach where no correlations between nucleons are taken into account. The nucleons move in the mean field constructed from elementary interactions between all particles (including the exchange terms). One of its serious limitations is inability to produce variances in relevant variables in agreement with experiment. 3. Intermediate Bombarding Energies The relatively simple reaction picture at low energies — with complete fusion, deep inelastic collisions and quasi-elastic collisions — becomes considerably more complicated above 10 MeV/nucleon. New phenomena appear, including significant preequilibrium emission, production of intermediate mass fragments, IMFs, (Z ≥ 3), projectile and target fragmentation, and incomplete fusion. At high energies (E/A ≥ 200 MeV/nucleon) the mean field is negligible compared to the average kinetic energy and the Pauli principle is less restrictive. As a result the mean free path for nucleon-nucleon collisions decreases, the energy dissipation has a two body character and local equilibration can occur in the interaction zone while the rest of the colliding system plays a role of spectators. The situation is more complicated at the intermediate energy range (20–200 MeV/nucleon), where both the two-body and one-body energy dissipation may be important. 3.1. Linear momentum transfer measurements Many experiments have shown that when the incident energy exceeds 10 MeV/nucleon fusion reaction starts to be incomplete. Only parts of the interacting ions merge into a quasi-compound nucleus and the higher the bombarding energy, the more incomplete fusion. Alternatively, a quasi-compound nucleus is called by some authors a composite system, CS. The simple way to characterize the degree of “incompleteness” of a fusion reaction is to measure the amount of the initial linear momentum transferred (LMT) from the projectile to the quasi-compound nucleus. The incompleteness factor ρLMT , a measure of LMT, is defined as: ppar , (24) ρLMT = pbeam where ppar is a parallel component (to the beam direction) of the linear momentum transferred to the quasi-compound nucleus. The pbeam is the projectile linear momentum. Depending on the fissionability of the total system, two different methods are used to get information on factor ρLMT . For light systems (ACS < 100), the quasi-

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Fig. 24. Velocity spectra of evaporation residues (ER) suggesting that the lighter nucleus loses momentum (mass). Fits are the calculations with Gaussian function. The quoted numbers indicate the fraction of complete fusion (dashed curves). For ρLM T values, see Fig. 29. [Fig. 1 from Ref. 106]

compound nucleus mainly decays by particle evaporation. A relatively heavy evaporation residue, ER, is formed. As was shown by Gomez et al.104 and Morgenstern et al.105, 106 the invariant velocity cross section distribution of the evaporation residues (1/v 2 )d2 σ/dΩdv as a function of velocity v has a Gaussian shape and is centered on v = vR cosθ. Here vR is the velocity of the quasi-compound nucleus from which the corresponding ER originates. In the case of full momentum transfer (complete fusion) we have vR = vCN , where vCN is the compound nucleus velocity in the laboratory reference frame. The angle θ is the detection angle in the laboratory. Representative examples of measured velocity distributions are shown on Fig. 24 for three systems: 40 Ar + 12 C, 40 Ar + 40 Ca, 20 Ne + 27 Al at incident energies of 520,

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Fig. 25. Folding angle, θf old , between the two fission fragments. VR is the velocity of the quasicompound nucleus.

520 and 395 MeV, respectively. As can be seen it is striking that (i) mass symmetry in the entrance channel leads to mean ER velocities vR = vCN ; (ii) mass asymmetry for a light projectile and a heavy target leads to vR < vCN and in the oposite case we have vR > vCN . This means that it is always the lighter nucleus that loses the most of the mass. The principle decay channel for heavy composite systems is fission. In this case the information about incompleteness factor ρLMT can be provided by measurements of the angular correlation between fission fragments. As suggested by Fig. 25, the folding angle, θf old , between the two fragments increases with decreasing LMT. From simple kinematical consideration one can deduce a relationship between θf old and ρLMT : ρLMT =

A4 v4 sin θf old , A1 v1 sin θ3

(25)

where A1 , v1 are the mass and velocity of the projectile, A4 , v4 those of one of the fission fragments, and θ3 is an observation angle of the other fragment. Both velocities are in the LAB system. Typical results obtained by this method are presented in Fig. 26 for the N + U system at various bombarding energies.107 For each incident energy a ppar /pbeam scale has been added. ppar /pbeam = 1 means that complete fusion has occurred. A large folding angle, close to 180 degrees, corresponds to a small LMT (peripheral collisions). Indeed, a uranium target is easily fissionable and even gentle peripheral collisions are sufficient to induce its fission. On the contrary, the small folding angle bump corresponds to violent central collisions for which significant LMT has occurred. From results presented in Fig. 26 two remarks may be stressed: (i) the central collision bump exists for any bombarding energy up to 45 MeV/nucleon and (ii) the LMT is complete up to 10–15 MeV/nucleon and is more and more incomplete above.

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Fig. 26. Fission fragment folding angle distribution for 14 N + 238 U reaction. For each measurement a linear momentum transfer scale, ρLM T , is shown immediately above the data. [Fig. 1 in Ref. 107]

In the case of heavier projectiles a further striking result was obtained for Ar + Th system.108 In this case the incomplete fusion bump vanishes between 39 and 44 MeV/nucleon. The disappearance of a fusion bump does not mean that fusion has disappeared. Still some events are in the region where the fusion bump is expected to be.

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Fig. 27. Inclusive fission fragment folding angle distributions for Ar + Th reactions from 15 to 115 MeV/nucleon. [Reprinted from Ref. 109 (Fig. 3), Copyright 1995, with permission from Elsevier]

The explanation of this behavior was given e.g. by an experiment done by Yee et al..109 The results of this experiment are presented in Figs. 27 and 28. In the first figure the inclusive fission fragment folding angle distributions for the 40 Ar + 232 Th reaction at beam energies from 15 to 115 MeV/nucleon are presented. In agreement with the previous results (e.g. Ref. 108) the fusion bump begins to disappeare around 50 MeV/nucleon. The next figure presents the folding angle distributions for fission fragments emitted in coincidence with other particles. The left column of the figure shows distributions gated by one or more IMFs emitted in the forward direction (θLAB < 15◦ ). The middle column shows distributions gated by IMFs observed at backward angles (θLAB > 68◦ ). The right column shows distributions gated on the top 10% of the total transverse energy impact parameter filter (see Eq. (31)). As one can see the distributions gated by forward IMFs show peaks which are clearly associated with low LMT (left panel). The distributions gated by backward IMFs or small impact parameters show peaks for high LMT even at beam energy equal 115 MeV/nucleon (center and right column). These observations supply the direct evidence that the incomplete fusion persists up to 115 MeV/nucleon.

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Fig. 28. Fission fragment folding angle distributions for Ar + Th reactions gated on IMFs at forward angles (left column), IMFs at large angle (center column), and central collision impact parameter obtained by the total transverse kinetic energy. Solid lines are Gaussian fits to guide the eye. [Reprinted from Ref. 109 (Fig. 3), Copyright 1995, with permission from Elsevier]

Extensive data on the amount of LMT has been accumulated till now. The most striking fact is that values of ρLMT , whether it is extracted from ER or FF measurements, is approximately independent of the projectile and target and decreases roughly linearly with increasing relative velocity of the two incoming ions, vrel (see e.g. Ref. 110). The vrel is defined as: 2(ECM − VC ) , (26) vrel = µ where ECM is the center of mass energy, VC the Coulomb barrier and µ the reduced mass. The compilation of experimental results on LMT vs. vrel is presented in Fig. 29. The decrease of the percentage of LMT when the bombarding energy increases might be an indication that the mean field is no longer strong enough to trap the interacting system together. By several authors (see Ref. 111) the complete/incomplete fusion cross section has been established from the area of the fusion bump. An extrapolation to zero implies in this case that fusion would disappear at around 35 MeV/nucleon. It seems rather dangerous to try to extract limits concerning the fusion process cross section from the LMT distributions as was shown in Ref. 109.

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Fig. 29. Most probable values of ρLM T measured in various reactions as a function of the relative velocity of the incoming ions. The line corresponds to the fit. [Fig. 5 in Ref. 111].

3.2. The 4π detectors During last twenty years some number of complex multidetector systems were constructed. All of them were dedicated to experiments at intermediate energies where the number of ejectiles in the exit channel grows rapidly. Their aim is to detect and identify all outgoing products. The quality of their performance depends on the percentage of the 4π solid angle coverage, granularity, and on detectors used for particle identification. Following devices can be listed: MUR-TONNEAU (GANIL, Caen),112, 113 AMPHORA (INS, Grenoble),114 Multics/MSU Miniball (NSCL, East Lansing),115, 116 Dwarf Ball/Wall,117 INDRA (GANIL, Caen),118, 119 SuperBall (NSCL, East Lansing),120 and CHIMERA (INFN-LNS, Catania).121 As an example the most advanced device CHIMERA is presented here (see Fig. 30). The CHIMERA (Charged Heavy Ion Mass and Energy Resolving Array) apparatus can be schematically described as a set of 1192 detection units arranged in 35 rings in cylindrical geometry around the beam axis. The forward 18 rings cover the polar angles between 1 and 30 degrees and are placed at a distance from the target ranging between 350 and 100 cm with the increasing angle. The remaining 17 rings, covering the angular range 30–176 degrees, are assembled as a sphere 40 cm in radius. Considering the beam entrance and outgoing holes and the detector frames, the overall detection solid angle is about 94% of 4π. Each detection cell consists of a double telescope. The first segment is made of a 300 µm silicon detector, while the second one is a thick CsI(Tl) crystal coupled

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The CHIMERA 4π Detector Array

Fig. 30.

CHIMERA detector set up.

to a photodiode for the light readout: the thickness of the crystal lowers with the increasing angle, from 12 to 3 cm. All the system operates in vacuum and it is thus installed in a dedicated chamber. Three different detection techniques are simultaneously used in CHIMERA. First, the ∆E − E technique is employed for charge identification of heavy ions and for isotopic identification of IMF’s with atomic number Z < 10. Second, mass identification is performed with signals from silicon detectors generating the time-of-flight signal obtained by comparing the timing of the detector signal and the timing of the high frequency signal from the cyclotron. Third, energetic light charged particles, which are stopped in the CsI(Tl) crystal, are identified by applying the pulseshape discrimination method using well shaped amplified signals from a 20 mm × 20 mm photodiode, optically coupled to the crystal. The identification characteristics achieved were described in Ref. 122. 3.3. Impact parameter estimators The introduction of very complex experimental devices gives the possibility for deeper insight into the reaction mechanism. Their detection ability covers high percentage of the total reaction cross section. In order to have the possibility to investigate specific reaction channels it is necessary to use impact parameter estimators. Up to now several global variables have been proposed and used for that purpose. They are supposed to increase (or decrease) monotonically with impact parameter

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b and use the following geometrical prescription:123 Xmax dN bmax dX. best (X1 ) = √ dX Nev X1

(27)

Here X represents the chosen global variable, reaching its maximum value Xmax when b → 0. Nev is the total number of recorded events (corresponding to the total geometrical cross section πb2max ), and best is the estimated impact parameter for events characterized by the value X = X1 . dN/dX is the probability distribution for the measured global variable. The assumption underlying Eq. (27) is that one can assign a single impact parameter to each value of the global variable and vice-versa. In this approximation the fluctuations are neglected. The most common used estimators X are: the multiplicity of charged particles, M ,,123, 124 and the total transverse momentum of charged particles125, 126 defined as: ptr =

M

pi,tr ,

(28)

i=1

where pi,tr denotes the transverse momentum of the ith charged particle. Variables related with the collected charge, Ztot and Zmid are also used as impact parameter estimators. They are defined as follows: Ztot =

M

Zi ,

(29)

Zi (mid) ,

(30)

i=1

Zmid =

M i=1

where Zi is the fragment charge and Zi (mid) is defined only for fragments with rapidity (see Eq. (59)) intermediate between the projectile and target rapidities.127 The total transverse energies for all and exclusively light charged particles are also used as impact parameter estimators X. They are given by formulas: Etr =

M

Ei,tr ,

(31)

i=1

and Etr12 =

M

Ei,tr ,

(32)

i=1(Zi =1,2)

where Ei,tr is the energy of ith particle connected with its transversal motion. Summation in Eq. (32) is taken only over the particles with Z = 1, 2. The advantage of the last estimator is that it eliminates efficiency problems for peripheral collisions for which the PLF or TLF heavy remnants can be undetected.

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3.4. Selection of reaction mechanism and global shape variables Impact parameter estimators are important for midperipheral and peripheral collisions. More direct methods of discriminating between different reaction mechanisms are based on considering how fragments are distributed in the center of mass momentum or velocity space on an event-by-event basis. The simplest example of global shape variable is the isotropy ratio given by Refs. 125 and 126: M CM i=1 Ei,tr , (33) I = M CM i=1 Ei,par CM CM and Ei,tr are the CM parallel (relative to beam) and transverse enwhere Ei,par ergies of the ith particle, respectively. Variable I is equal to zero for the most peripheral collisions and near 2 for the most central ones (creation of composite system). From the Cartesian components of particle momenta in the center of mass, one may construct the tensor:128 (n) p(n) pj i n

Fij =

|p(n) |

(n) | n |p

,

(34)

(n)

where pi is the ith Cartesian momentum component of the nth particle, and p(n) is the nth fragment momentum vector. Both summations go over all detected or selected particles. For the ordered eigenvalues t1 < t2 < t3 of the tensor F one defines the reduced quantities: t2 qi = i

2 j tj

.

(35)

Then sphericity and coplanarity parameters are defined as: 3 S = (1 − q3 ) , (36) 2 √ 3 C= (37) (q2 − q1 ) . 2 The major axis of the ellipsoid, corresponding to the largest eigenvalue t3 , defines the principal axis of the event (flow axis). The deviation of the principal axis from the beam axis is called the flow angle, θf low . In the C versus S plane all events are located inside a triangle. The most elongated √ (rod-like) shapes are concentrated in the vicinity of the (0,0) point. The ( 34 , 43 ) point corresponds to flat (disc-like) shapes, and the most spherical shapes are located close to the (1,0) point. Peripheral events “remember” entrance channel kinematics, what is reflected in their elongated shapes, located close to the (0,0) point. On the contrary, the most central collisions “forget” entrance channel kinematics, and no direction in the linear momentum space is privileged. Such events have spherical shapes and are located close to the (1,0) point. For limited multiplicities of observed particles the picture of reaction is smeared over quite large region,

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Fig. 31. Map of the well characterized events (M ≥ 16) in the (S,C) plane for the 40 Ca + 40 Ca reaction at 35 MeV/nucleon. Location of the AB line is given by a distance d from the (1,0) point. [Fig. 11 in Ref. 129]

but not the same for both reaction scenarios. Figure 31 shows a map of events for the 40 Ca + 40 Ca reaction measured at 35 MeV/nucleon on the (S,C) plane.129 In this presentation only events with charged particles multiplicity M ≥ 16 were taken into account. It was proposed by Pawlowski et al.129 and tested with 40 Ca + 40 Ca data measured at 35 MeV/nucleon to use a triangle A,B,(1,0) cut to isolate events corresponding to formation of composite system (see Fig. 31). In the same paper the ρ filter dedicated to central collisions was also proposed. The variable ρ is built from the momentum of the heaviest fragment, p1 , the relative √ velocity of the two heaviest fragments, v12 , and the event elongation, = 1 − S, in the momentum space. The ρ is defined as:

(38) ρ = x2 + y 2 + z 2 , where x = p1 /pP , y = v12 /vpt , and z = . Here pP is the CM projectile momentum, and vpt is the projectile-target relative velocity. Other method for selection of events corresponding to formation of the composite system, based on multivariative technique, was proposed by Desesquelles et al.130 and tested in some number of analyses (see e.g. Ref. 131).

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3.5. Central collisions (“single source” events) Increasingly exclusive experiments with 4π detector arrays have shown that the so-called “single source” events are observed for very central collisions of heavy ions around the Fermi energy (Pb + Au at 29 MeV/nucleon,132 Au + Au at 35 MeV/nucleon,133 Xe + Sn at 50 MeV/nucleon,134 Ca + Ca at 35 MeV/nucleon,135 Gd + U at 36 MeV/nucleon,136 Sn + Ni at 35 MeV/nucleon131 ). In these events all emitted fragments and particles, apart from preequilibrium component of light particles, seem to originate in the multifragmentation of a single nuclear system containing almost all of the mass and energy of the entrance channel. Gd + U 36 A MeV INDRA

Fig. 32. Logarithmic contour plots showing event-by-event correlations between the total detected charge Ztot , total parallel momentum normalized to the incident projectile momentum Ptot /Pproj , and multiplicity of charged particles Nc . The numbered boxes correspond to the three classes of events discussed by the authors. [Reprinted from Ref. 136 (Fig. 2), Copyright 2001, with permission from Elsevier]

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As an example of such an analysis for a very heavy system, results for the Gd + nat U reaction at 36 MeV/nucleon are presented. An experiment for this reaction was performed at the GANIL accelerator using the 4π multidetector INDRA.118, 119 As the first step of the analysis of the calibrated data the selection of well measured events was performed. The manner of selection is presented in Fig. 32. Here the event-by-event correlations between the total detected charge, Ztot , versus total detected parallel momentum, Ptot , normalized to the incident projectile momentum, Pproj , (upper part), and versus multiplicity of charged fragments, NC , (lower part) are presented. Events located in box 3 are considered as a class of well measured events. For these events two conditions are fulfilled: Ztot ≥ 120 (ZP + ZT = 156) and 0.8 ≤ Ptot /Pproj ≤ 1.1. The measured cross section corresponding to these complete events is equal to 93 mb and is only a small fraction of reaction cross section σR = 6.5 b. As was shown by the authors the well measured events cover quite broad range of impact parameters. One could say that the selection of the “single source” events for central collisions requires application of some impact parameter estimators. However a more complex analysis shows in this case that a selection based on different estimators is not satisfactory for “single source” events. Also an analysis in terms of global shape variables does not make an accurate job. Finally, it was shown that the event classification using the modified “Wilczy´ nski diagram” (see Fig. 20) is able to select interesting events (Fig. 33). The modification introduced here is related to the x

155

155 Gd + nat U, 36 A MeV, complete events

Fig. 33. “Wilczy´ nski” diagram for complete events: logarithmic intensity scale representing measured cross section as a function of total measured CM kinetic energy normalized to the available energy and “flow” angle θf low . For events in zones 2 to 4 the mean value of energy is indicated (points) for each θf low . [Reprinted from Ref. 136 (Fig. 7), Copyright 2001, with permission from Elsevier]

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Fig. 34. For the most dissipative collisions (zones 2–4 of Fig. 33): (a) “flow” angle distribution; (b) mean sphericity of the events as a function of fragment multiplicity Nf ; (c) distribution of relative angles θrel between pairs of fragments for zone 4 events; (d)–(g) evolution of fragmentfragment relative angle distribution with θf low , each distribution was divided by the distribution for zone 4 events. [Reprinted from Ref. 136 (Fig. 5), Copyright 2001, with permission from Elsevier]

axis. The angle defining the direction of the event principal axis with respect to the beam direction, θf low , is used. The axis y represents the total kinetic energy in the exit channel normalized to the entrance channel CM energy. Only fragments with charge greater than 5 were taken into the construction of the event ellipsoid. Properties of events are discussed for four zones indicated in Fig. 33. In zone 1 the less dissipative collisions are located. The direction of the principal axis for CM = 9.8◦ ,137 ). These events these events remains close to the grazing angle (θgr correspond to binary collisions for which PLF remnant, TLF fission fragments and some light particles are detected. In zones 2 to 4 the very dissipative collisions are located, most of them concentrated in zone 2, at θf low < 30◦ . The deeper inspection of these events is presented in Fig. 34. The cos(θf low ) distribution becomes flat for large flow angles (Fig. 34(a)). The mean event-shape sphericity as a function of fragment multiplicity and the flow angle is shown in Fig. 34(b). Here one observes an increase of sphericity with a number of fragments which is obvious, and an increase of sphericity with the “flow” angle. This last observation gives a strong signal of the evolution of fragments kinematics towards single source emission. On Fig. 34(c)–34(g) distributions of relative angles θrel between pairs of emitted fragments are shown. Fragments emitted by a single source should have an isotropic distribution in the rest frame of the emitter, except small angle Coulomb distortion. Figure 34(c) shows exactly this type of distribution. Figures 34(d)–34(g) show the evolution of the relative angle distri-

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Fig. 35. The CM angular distributions of light charged particles emitted in coincidence with the single source formed in central 155 Gd + nat U collisions at 36 MeV/nucleon. Histogram are raw data, while triangles correspond to corrected values. [Reprinted from Ref. 138 (Fig. 7), Copyright 2001, with permission from Elsevier]

butions with the flow angle of the events. These distributions are divided by the distribution shown in Fig. 34(c) in order to highlight differences. This is especially visible in Fig. 34(d) (θf low < 10◦ ), where small and large relative angles between fragment pairs are favored indicating the dominance of emission from two separate sources moving apart in the CM reference frame of these events. The emission from two sources is strongly reduced for events with larger θf low values. For the same reaction 155 Gd + nat U reaction at 36 MeV/nucleon the properties of light charged particles emitted by the single source are presented in Figs. 35 and 36.138 In Fig. 35 the angular distributions of p, d, t, 3 He, and α particles as a function of the center of mass angle are displayed. The angular distributions for all of them are very similar, being flat between 60◦ and 120◦ . The energy spectra of p, d, t, and α particles emitted in the range 60◦ –90◦ and 90◦ –120◦ in the center of mass are presented in Fig. 36. These spectra are, as expected, identical for the two defined angular ranges. Such behavior is compatible with the picture of the source having reached thermal equilibrium at the moment of its desintegration. The properties of a single source system are derived from the comparison of experimental data with statistical models. The aim of this procedure is to determine whether a thermodynamical equilibrium of this source has been reached at some stage of the collision. These models are purely static, they describe the system at a freeze-out moment, defined as the time when the nuclear interaction between fragments vanishes (see Sec. 3.7.1). Fragment kinetic energy comes only from their mutual Coulomb repulsion and thermal motion. Any deviation of the measured

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Fig. 36. The CM light charged particle energy spectra for two angular ranges. [Reprinted from Ref. 138 (Fig. 7), Copyright 2001, with permission from Elsevier]

kinetic energy from the predicted values is then attributed to some extra collective energy (e.g. expansion energy, rotational energy). In the analysis of Frankland et al. the SIMON139 event generator was used. This highly simplified algorithm for generating partitions reasonably accounts for the measured charge and multiplicity distributions. Such approach permits a rigorous treatment of the space-time correlations between all emitted particles. The comparison of SIMON simulations with experimental data is shown in Fig. 37. In panel (a) the average CM fragment kinetic energies vs. their charges (the heaviest fragment is excluded) are presented. The calculations based on a sequential scenario underestimate the experimental values (solid line). The simulations based on a simultaneous multifragmentation scenario reveal the need for a radial collective motion (radial flow) at the freeze-out whose energy is r = 0.5 ± 0.5 MeV/nucleon. Panel (b) shows the experimental spectrum for Z = 7, and the calculated spectra with and without expansion energy. Clearly the hypothesis with expansion energy is the best, the option r = 0 leading to a too broad spectrum. This indicates that expansion energy begins to appear for central collisions between very heavy ions around 30 MeV/nucleon. The phenomena related to collective motion (flow) are discussed more extensively in Sec. 3.8.

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Fig. 37. (a) Average CM fragment kinetic energies versus their charges (the heaviest fragment is excluded). Experimental data (points) are compared to different SIMON simulations. (b) Kinetic energy spectra for Z = 7; the filled histogram represents experimental data, and the solid and dashed lines the result of SIMON simulations. r is in MeV/nucleon and Tlim is in MeV. [Reprinted from Ref. 138 (Fig. 8), Copyright 2001, with permission from Elsevier]

3.6. Noncentral collisions The relatively simple picture of two-body reactions for noncentral collisions at small energies is replaced by the more complex one. With increasing incident energy the importance of the intermediate velocity source, IVS, increases, while the PLF and TLF sources change into relatively low-excited spectators. 3.6.1. The IVS source The intermediate velocity sources, primarily suggested by BUU140 and BNV141, 142 calculations, have been observed in some number of experiments. This observation is done by a reconstruction-subtraction procedure,143, 144 by inspecting the shapes of the velocity distributions of charged particles projected on the beam direction,145, 146 by inspecting invariant velocity plots,142, 147–150 by use of a three source model,151 by rapidity and transversal energy distributions,152, 153 or by observing the so-called “aligned break-up”.154 As an example of IVS observation in the case of reaction with relatively light projectiles results obtained for the 40 Ca + 40 Ca, 197 Au systems, measured at 35 MeV/nucleon are presented.145, 146 Experiments for these reactions were performed at the Grenoble SARA facility using the 4π multidetector AMPHORA.114 In order to lower the detection energy thresholds for IMF’s additional 30 ionization chambers were installed.155

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For the IVS search the class of well defined events was selected. It was done by the condition ptot > 8 GeV/c, and additionally the detection of at least one intermediate mass fragment was required. Such conditions were sufficient for the detection of most PLF decay fragments and the IVS fragments. The TLF heavy remnants could not be detected by the AMPHORA system. Figure 38 presents the laboratory velocity distributions of different ejectiles from the 40 Ca + 40 Ca reaction projected on the direction parallel to the beam. In Fig. 38(a) the spectra of protons, deuterons, tritons, α particles, and lithium ions are collected. Figure 38(b) shows the spectra for some IMF’s from the Z = 4– 14 range. For particles with Z = 2–5 the distributions exhibit some kind of plateau which is difficult to explain as a superposition of the particle emission from the two most obvious sources, the PLF and the TLF. For protons, deuterons, tritons the distributions resemble a Gausian shape. The analogous spectra for the 40 Ca + 197 Au reaction are presented in Fig. 39. Here we observe much less emission of charged particles from the heavy Au target-like nuclei. The elucidation of the data displayed on both figures is done by a comparison with the predictions of a stochastic model proposed by Sosin156 (see Sec. 3.9.3). The events generated by this model are filtered by the software replica of the AMPHORA detector. The generated spectra of particles are presented in Figs. 38 and 39 by red, blue, green and violet lines for IVS, PLF, TLF and CS sources respectively. The black line gives the total emission. The relative contribution of the IVS source in emission of different ejectiles (model prediction) is presented in Fig. 40 for the 40 Ca + 197 Au reaction (a), and also for the 40 Ca + 40 Ca reaction (b). The IVS contribution is given as a percent of the total emission for peripheral collisions taking place above some threshold angular momentum. Three values of the threshold angular momentum are considered: L1 , L2 and L3 . For the 40 Ca + 197 Au reaction L1 = 460, L2 = 517, and L3 = 240. For the 40 Ca + 40 Ca case L1 = 220, L2 = 250, and L3 = 110. For L1 the entrance channel cross section σ(L) reaches its maximum value for both systems. As can be seen in Fig. 40 in peripheral collisions deuterons, tritons and 3 He particles are preferentially emitted by the IVS source. The difference between the ntritons /nhelium3 ratios (ni denotes here the respective IVS contribution) varies between 4 and 5 for 40 Ca + 197 Au and between 1.4 and 1.5 for 40 Ca + 40 Ca system, respectively. The isospin dependence of the IVS contribution of mass 3 particles seems to be properly correlated with the N/Z ratio of the system under consideration. For heavier systems the IVS source is substantially bigger. Very extensive analysis for the 129 Xe + nat Sn reaction at energies between 25 and 50 MeV/nucleon was presented in Refs. 143 and 144. In order to have an overall view of the kinematical properties of particles emitted in this reaction at 50 MeV/nucleon the Galilean invariant cross section contours d2 σ/vtr dvtr dvpar in the vtr vs vpar plane are presented in Fig. 41 for d, t, Li, C and O fragments. The numbers located in the upper-right corner of each subfigure are associated with Etr,12 impact parameter

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Si

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Fig. 38. Velocity distributions (LAB) for the 40 Ca + 40 Ca reaction at 35 MeV/nucleon projected on a direction parallel to the beam. (a) light particles; (b) intermediate mass fragments; black dots: experimental data. Model predictions for IVS, PLF, and TLF sources: red, blue, and green lines, respectively. Black line: predicted total emission. Violet line: CS contribution. [Fig. 1(a) and 1(b) in Ref. 146]

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Ne

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Fig. 39. Velocity distributions (LAB) for the 40 Ca + 197 Au reaction at 35 MeV/nucleon projected on a direction parallel to the beam; black dots: experimental data. Model predictions for IVS, PLF, and TLF sources: red, blue, and green lines, respectively. Black line: predicted total emission. Violet line: CS contribution. [Fig. 3(a) in Ref. 145]

selector. The value 1 corresponds to most peripheral collisions (b ≈ 9–10 fm), value 5 corresponds to impact parameters in the range (b ≈ 3–4.5 fm). In order to quantify the importance of the IVS emission one has to estimate the velocities of the two main sources (PLF and TLF), and to subtract components, which can be attributed to a sequential decay of PLF and TLF. These last components are expected to be a forward-backward symmetrical emission in the source frame. Two different methods are used to estimate the velocities of PLF and TLF sources. Method I assumes that the most probable velocity of the heaviest fragment detected in a sample of events for a given impact parameter bin is a reasonable estimate for the mean source velocity of that sample. The analysis is performed on the PLF side for which detection efficiency is the best. Since the system is almost symmetric, the results are easily extrapolated to the whole system. Method II

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(a)

(b)

Fig. 40. Relative contributions of the IVS source in the emission of different ejectiles — model prediction with no experimental limitations. (a) 40 Ca + 197 Au; (b) 40 Ca + 40 Ca. Calculations performed for different angular momentum thresholds: L1 , L2 , L3 — see text. [Fig. 5 in Ref. 145]

utilizes the thrust concept.128 In this method one attributes all fragments (Z ≥ 3) to PLF or TLF sources in a way that maximizes the “thrust” value: | i∈P LF pi | + | j∈T LF pj | , (39) T = k |pk | where the summation in the denominator includes all observed fragments. In L ukasik et al.143 this reaction was studied only at 50 MeV/nucleon. It was found that up to 25% of the total charge is located in the IVS source for the midpheripheral collisions (b ≈ 5 fm). The results of this analysis are presented in Fig 42. The lower row of Fig. 42 presents the IVS contribution in the total emission of different ejectiles. As can be seen, up to (60–80)% of light IMS’s (Z = 3–6) detected in most peripheral collisions (bins 1 and 2) originate from the midvelocity region. Also very large fraction of tritons (65–70)% comes from this zone. This

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Fig. 41. Invariant velocity plots for deuteron, triton, lithium, carbon and oxygen fragments detected in the most peripheral collisions (top row, bin 1; middle row, bin 2) for the 129 Xe + nat Sn reaction at 50 MeV/nucleon. The presented projections (bottom row) refer to the plots for bin 2. [Fig. 3 in Ref. 143]

Fig. 42. Upper row: mean multiplicities of p, d, t, 3 He, α particles and fragments with Z ≤ 15 detected in the forward CM hemisphere for the first five Etr12 bins. Lower row: contribution of the specified above fragments to the midvelocity component. The solid and open squares correspond to methods I and II, respectively. All presented results correspond to the 129 Xe + nat Sn reaction at 50 MeV/nucleon. [Fig. 5 in Ref. 143]

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Fig. 43. (a) The percentage of charge emitted in the forward hemiellipsoid of momentum tensor (see Eq. (34)) for midvelocity emission (squares) and evaporation process (circles) obtained with method I for eight Etr12 bins. The numbers in boxes label the incident energies. The dashed lines specify the Etr12 region for which the subtraction method gives unreliable results. (b) The percentage of charge in midvelocity component as a function of incident energy for the four most peripheral Etr12 bins. (c) Same as (b) but for statistical emission. [Fig. 3 in Ref. 144]

observation is similar to that for reactions with much lighter, e.g. 40 Ca, projectiles.145, 146 The upper row of Fig. 42 presents the mean multiplicities of p, d, t, 3 He, α, and fragments with Z ≤ 15 detected in the forward CM hemisphere for the five Etr,12 bins. If one integrates the multiplicity distribution for IMFs and calculates the IVS contribution, one ends up with one or two IMFs emitted dynamically per event for bins 2–4 (after extrapolating the value to the whole PLF, TLF, IVS system). The analysis done by L ukasik et al.143 was extended to an energy range from 25 to 50 MeV/nucleon by Plagnol et al..144 Results of this analysis are summarized in Fig. 43. They were obtained by the method I of IVS extraction. The size of the midvelocity and evaporative components is presented as a function of impact parameter normalized to the bmax value at each incident energy (Fig. 43(a)). In this representation, the largest fragment is not associated with either of the above processes (Ztot = Zmidv + Zevap + Zheaviest ). For this reason, the sum of the two processes does not add up to 100%. The difference is corresponding to the charge of the largest fragment. The numbers given measure the amount of charge removed from the PLF by either of these two modes. Figure 43(b) shows a percentage of charge in midvelocity component as a function of incident energy for four most peripheral

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bins. We see that the midvelocity component evolves from a very small value at 25 MeV/nucleon incident energy to a significant contribution at 50 MeV/nucleon. Figure 43(c) shows the energy dependence for the statistical emission component. For a given impact parameter this component is observed to be insensitive to the incident energy. These observations indicate that with increasing incidence energy the importance of the IVS source grows, while the PLF and TLF evolve in the direction of cold spectators.

3.6.2. The PLF source In many works done in the field of heavy ion collisions the attention was focused on properties of the hot PLF (see e.g. Refs. 125, 157–159). As an example of such attempts results obtained for reactions 40 Ca + 40 Ca measured at 35 MeV/nucleon are presented.158 Some details of these experiments were already shown discussing the IVS issue. As before the class of well defined events with ptot > 8 GeV/c was used. In order to get information on PLF properties the reconstruction of such object was performed on an event by event base. In the first step of the reconstruction procedure the velocity vector of the center of mass of the PLF is derived from the momentum vectors of the decay products. The PLF velocity vector is approximated by calculating the CM velocity of fragments with Z ≥ Zmin = 3 and is given by: Z ≥Z vP LF = i min

Zi vi

Zi ≥Zmin

Zi

.

(40)

Only fragments with velocities larger than CM velocity of the reaction are considered. Particles with smaller Z values are excluded from this procedure, since they may be coming from other sources. Figure 44 presents the Galilean invariant velocity distribution for different groups of particles (Z = 1, Z = 2, Z = 3–5, Z > 5) in the reference frame of the reconstructed PLF. The x-axis of this reference frame, v , is oriented along the reconstructed PLF velocity vector in the CM of the total system. The presented distributions are generated only for events with the total transfer momentum, ptr < 4.5 GeV/c, which is used here as a measure of the dissipated energy. The emission from the hot PLF nucleus is better observed for heavier ejectiles. Some contribution from other sources is seen at negative velocities. This admixture is more pronounced for light particles. For the class of IMF fragments, Z ≥ Zmin = 3, a projection of the distribution of Fig. 44 on the v axis is presented (Fig. 45). This plot was made for consecutive windows in the transverse momentum. For all values of ptr , we observe a strong maximum located at the center of mass velocity of the reconstructed PLF. An additional, weaker maximum at negative values of v originates mostly from the TLF. Its contribution increases with increasing value of ptr .

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Fig. 44. Galilean invariant velocity plots (linear scale) in the rest frame of the reconstructed PLF with the parallel velocity axis oriented by the PLF CM velocity, for different groups of particles (ptr < 4.5 GeV/c) in the 40 Ca + 40 Ca reaction at 35 MeV/nucleon. [Fig. 1 in Ref. 158]

Fig. 45. Projection of the distribution of Fig. 44 for the class of heaviest fragments (Z > 5) on the v axis for consecutive windows of ptr . [Fig. 2 in Ref. 158]

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Fig. 46. Mass (a), and excitation energy (b) distributions of the reconstructed PLF of consecutive ptr windows. Experimental points and model predictions156 for the 40 Ca + 40 Ca reaction at 35 MeV/nucleon. [Figs. 3(a) and 3(b) in Ref. 158]

In the next step of the reconstruction procedure, the PLF charge is calculated, event by event. The charges of all those IMFs for which the parallel velocity component is in the rest frame of the PLF larger than −0.1c are summed up. The IMFs with lower velocity values are assumed to come from other sources. For Z = 1 or 2 only particles emitted in the forward hemisphere are taken into account and their contribution is multiplied by two. In this way the contribution of light particles coming from other sources is minimized. In order to estimate the PLF mass, a mass equal to 2Z is assumed for each IMF. For Z = 1 or 2 particles the masses of all particles emitted in the forward hemisphere are summed up and multiplied by two. It is also assumed that in each event the number of emitted neutrons is equal to the number of emitted protons. This approximation is expected to be reasonable for the 40 Ca + 40 Ca symmetric system.

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Fig. 47. Angular distribution of Z > 5 decay fragments in the reference frame of the reconstructed PLF. [Fig. 5 in Ref. 158]

To estimate the excitation energy the calorimetric method is used. The kinetic energies of the fragments in the rest frame of the PLF are summed up with the same restrictions as for the reconstruction of charge and mass. The contribution of light particles (Z < 3) emitted in the forward hemisphere is multiplied by two. The summed kinetic energy of neutrons is assumed to be equal to that of protons, minus the Coulomb energy. Finally, the involved Q value is included. The distribution of the reconstructed PLF mass is presented in Fig. 46(a) for consecutive ptr windows. The average value of this distribution is located for the first two windows slightly below the projectile mass, which may imply the existence of the IVS. The width of the distribution increases with the ptr . The reconstructed excitation energy distributions are shown in Fig. 46(b) for the same ptr windows. The average excitation energy and the width of its distribution increase for larger ptr . For bins with higher ptr the excitation energy distributions cover practically the full range of energy available in the CM systems. Figure 47 presents the angular distribution of the decay fragments in the reference frame of the reconstructed PLF, for heavier fragments (Z > 5) which are less contaminated by other sources. If the memory of the initial direction of the projectile is lost, the angular distribution should exhibit forward-backward symmetry, which can indeed be observed in Fig. 47. Figure 48 presents charge distributions of the decay products for consecutive bins of the PLF excitation energy, E ∗ /A. The group of light particles is about 10 times stronger than the IMFs, which form some kind of plateau up to about Z = 15 for lower E ∗ /A, and up to Z = 11 for higher E ∗ /A values.

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Fig. 48. The charge distributions of the decay products for consecutive bins of the PLF excitation energy per mass unit, E ∗ /A, for the 40 Ca + 40 Ca reaction at 35 MeV/nucleon. Experimental points and model predictions:156 with the experimental filter (solid lines) and without the experimental filter (dashed lines). The model distributions have been normalized to the same surface. [Fig. 4 in Ref. 158]

Figure 49 shows energy distributions of protons and α particles emitted from the PLF for different bins of the PLF excitation energy. Systematic variation in the distribution slope parameter with excitation energy is observed for both ejectiles. The data displayed on Figs. 46, 48 and 49 are compared with the predictions of a stochastic model of Sosin156 (solid lines). The good description of the PLF source decay characterists is consistent with a thermalized source picture. The forwardbackward symmetry of the angular distribution observed in the reference frame of the reconstructed PLF also suggest a loss of memory of the reaction entrance channel.

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Fig. 49. The energy distributions of protons and α particles emitted from the PLF for different bins of the PLF excitation energy in the 40 Ca + 40 Ca reaction at 35 MeV/nucleon. The solid lines are the model predictions.156 [Fig. 6 in Ref. 158]

As presented in Figs. 38–40 and in Figs. 46, 48 and 49 the model of Sosin satisfactorily describes both the PLF and IVS sources contribution to the reaction picture. Basic features of this model are presented in Sec. 3.9.3. The overall agreement between the Sosin model predictions and data give the possibility to test the efficiency of the PLF source reconstruction procedure as a function of the total transverse momentum, ptr . Also the assumption on the charge threshold of particles used for PLF velocity determination was tested. Table 1 presents the average percentage participation of the PLF source in the reconstructed PLF mass AP LF (rec), and the excitation energy (E ∗ /A)P LF (rec) predicted by the model for different ptr windows. The calculations are made for Zmin = 3, and for Zmin = 6. As shown in Table 1, the reconstructed PLF source for Zmin = 6 and ptr = 0.5– 1.5 GeV/c contains 99% of the primary PLF mass and 96% of the primary PLF

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Table 1. The average percentage participation of the PLF source in the reconstructed PLF mass AP LF (rec), and the excitation energy (E ∗ /A)P LF (rec) predicted by the model for different ptr windows.

ptr (GeV/c)

APLF (rec)

0.5–1.5 1.5–2.5 2.5–3.5 3.5–4.5

99% 98% 95% 88%

ptr (GeV/c)

APLF (rec)

0.5–1.5 1.5–2.5 2.5–3.5 3.5–4.5

99% 98% 94% 86%

Zmin = 6 (E ∗ /A)PLF (rec) (MeV/nucleon) 96% 92% 84% 74% Zmin = 3 (E ∗ /A)PLF (rec) (MeV/nucleon) 94% 90% 83% 72%

excitation energy. These numbers drop to 95% and 84%, respectively, in the 2.5– 3.5 GeV/c window. For Zmin = 3, the contamination by other sources is slightly worse. Clearly we may reasonably speak of a reconstructed PLF source only for an upper limit of ptr . 3.7. Caloric curves The liquid-gas phase transition in nuclear matter is often explained by the van der Waals type of nucleon-nucleon interaction: repulsion at short distance and attraction at long distance. In order to get information about this phase transition in nuclear matter the collection of experimental data concerning the identification and reconstuction of hot nuclear objects was used. In such analysis one can use data for which a derivation of the initial temperature of deexciting nuclei with reasonably well characterized masses and excitation energies was performed. A compilation of the measured caloric curves defined as temperature, T , versus excitation energy, E ∗ /A, is plotted in Fig. 50.160 The temperatures obtained were extracted using the method of double ratios of isotope yields .161 Here Tiso (He-Li) and Tiso (He-dt) are denoted by circles and squares, respectively. All plotted temperatures are the experimental apparent temperatures. No corrections for sequetial decay were applied. In the left panel, all curves are plotted on the same scale. In order to view each succesive curve better, they are replotted in the right panel. Here each curve is offset from its predecessor by 2 MeV and the corresponding reactions are labelled close to the symbols. If the liquid-gas phase transition is of the first order, one would expect enhanced specific heat corresponding to a plateau region in the caloric curve. Such simple picture with a plateau in the temperature assumes that the pressure is constant,162

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Fig. 50. Collection of measured caloric curves. The curves in the right panel were offset by 2 MeV for each successive curve, starting with Ar + Ni. [Fig. 13 in Ref. 160]

but there is no experimental evidence that such a condition is met in nuclear collisions. As one can see only the curve obtained from the spectator decay of the Au + Au reaction at E/A = 600 MeV/nucleon shows nearly a plateau between 3 and 10 MeV/nucleon and a rapid increase above 10 MeV/nucleon of the deduced excitation energy.163 The data presented in Fig. 50 were reanalysed by Natowitz et al..6 To obtain the initial thermal excitation energies and initial temperatures the corrections to the raw observed values were applied. The analysis of these corrected data shows that the existing body of experimental results provides rather a consistent picture when the mass dependence of the caloric curve is taken into account. Figure 51 presents the caloric curves for the mass regions 30–60, 60–100, 100–140, 140–180, and 180–240. For comparison, each subfigure also includes the calculated Fermi-gas model predictions for level density parameter a = A/13 and A/8 MeV−1 . For the lightest mass window (30–60) one observes a flattening of the distribution near 8 MeV/nucleon excitation energy. For the higher windows this feature appears near 4 MeV/nucleon excitation energy. For the highest mass window (180– 240) the flat region of the distribution starts near 3 MeV/nucleon. Inspection of Fig. 51 shows that the value of limiting temperature reached in the plateau decreases with increasing mass. This limiting temperature represents

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Fig. 51. Caloric curves for five selected regions of mass. Measurements of temperature versus excitation energy per nucleon are represented by symbols. The solid and dashed curves correspond to the Fermi gas model predictions for level density parameter a = A/13 and A/8 MeV−1 , respectively. The line located below correspond to total vaporizaion model. [Fig. 4 in Ref. 6]

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Fig. 52. Limiting values of T (upper panel) and E ∗ /A (lower panel) at which Tlimit is reached are indicated by solid diamonds. The critical temperature and excitation energy derived from the Fisher droplet model analysis of Elliott et al.169 are represented by solid circles. The lines in the top panel represent the calculated Coulomb instability temperatures from168 (dashed line) and165 (solid line). [Fig. 6 in Ref. 6]

the “Coulomb instability” temperature, which was calculated with a temperaturedependent Hartree-Fock model employing a Skyrme interaction164 and later with other models (see e.g. Ref. 165). In these calculations, the limiting temperature, which represents the limit of the equilibrium phase coexistence between the liquid and vapor, was designated as the point of Coulomb instablility because in the absence of the Coulomb forces the coexistence is possible up to the critical temperature of nuclear matter.166 Values of the temperature and excitation energy at which this transition takes place as a function of mass system are plotted in Fig. 52. One observes that both values appear to decrease with increasing mass. For comparison lines in the top panel present the Coulomb instability limiting temperatures calculated by Zhang et al.,165 employing both a relativistic chiral symmetry model167 and the Gogny GD1 interaction.168 Additionally, the critical temperature and excitation energy derived from the Fisher droplet model analysis of Elliott et al.169 are represented by solid circles. From limiting temperature values obtained in five different mass regions a critical temperature of 16.6 ± 0.6 MeV for symmetric infinite nuclear matter was determined by Natowitz et al.170

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3.8. Flow In a system in a complete equilibrum, particles are moving thermally. There is no net current of particles at any space point of the system at any time. If the current exists, collective motion is present on the top of the thermal motion, which is called “flow”. The concept of flow comes from hydrodynamics. It was introduced to nuclear collisions by Scheid et al..171 Following hydrodynamical considerations, they anticipated that nuclear matter, if stopped in central collisions, is compressed and heated, and may subsequently expand. As was already mentioned in Sec. 3.5, the analysis of experimental data for central collisions at intermediate energies indicates that a thermal motion of reaction products does not describe the data properly. There exists a need to include also a collective motion in the reaction models (see Fig. 37). The experimental studies to test these ideas began in 1978 at Bevalac (LBL) and are presently being pursued in heavy ion collisions from intermediate energies (e.g. at NSCL Michigan State University) to ultrarelativistic energies (RHIC, BNL). Quantifying collective motion in nucleus-nucleus collisions begins with determining the impact parameter. The impact parameter vector (chosen to be in the x direction) and the beam direction (in the z direction) define a reaction plane for the collision. The method for determining the reaction plane of an observed collision was developed by Danielewicz and Odyniec.172 The method is based on an assumption that the net transfer momentum p⊥ normal to the reaction plane averages to 0 by symmetry. If collective flow exists and p⊥ = 0, then the vector Q=

M

ωi pi,⊥

(41)

i=1

will lie in the reaction plane. Here pi,⊥ represents momentum of particle, i, normal to the reaction plane, ωi is a weighting factor usually defined as +1 for particles going forward (along the z axis) in the center of mass reference frame and −1 for particles going in the backward direction. When the reaction plane is established, in the next step of the analysis the transverse momentum vectors of particles are projected on this plane. (The investigated particles must be excluded when determining Q, to avoid autocorrelation effects). The average transverse momentum per nucleon projected on the reaction plane, < px , is then determined as a function of a variable in the direction parallel to the beam, such as rapidity y (see Eq. (59)). Traditionally, the center of mass rapidity is scaled by the projectile rapidity, yproj , to remove trivial scaling with the incident beam energy.173 The distribution of px versus y/yproj shows a characteristic curve centered at midrapidity, the average of the projectile and target rapidities (see Fig. 53174 ). The solid (open) squares are for particles with Z = 2 from semicentral 58 Fe + 58 Fe

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Fig. 53. Mean transverse momentum in the reaction plane versus the reduced CM rapidity for Z = 2 fragments from semicentral collisions (b/bmax = 0.48) at 105 MeV/nucleon. The solid (open) squares are for 58 Fe + 58 Fe (58 Ni + 58 Ni). The straight lines are fits over the midrapidity region −0.5 ≤ (y/yproj )CM ≤ 0.5. [Fig. 1 in Ref. 174]

(58 Ni + 58 Ni) collisions at 105 MeV/nucleon. The vertical offsets from the origin are visible because no recoil correction was applied in the reaction plane calculations, but this does not affect the final values of the flow observables. Both spectra shown in Fig. 53 are fit with a strait line over the midrapidity region −0.5 ≤ (y/yproj )CM ≤ 0.5, and the slope of each line is a measure of the amount of collective flow and is known as the flow parameter, F (Flow). The extracted values of flow are shown in Fig. 548 as a function of the reduced impact parameter for Z = 1, 2, 3 particles from 58 Fe + 58 Fe and 58 Ni + 58 Ni collisions at 55 MeV/nucleon. The neutron rich system 58 Fe + 58 Fe systematically exhibits larger flow values than 58 Ni + 58 Ni system for all three particles types at all reduced impact parameter bins displayed (except for Z = 3 in the most peripheral bin). The largest difference in the magnitude of the flow between these two entrance channels occurs for heavier mass fragments in semicentral collisions. By studying beam energy dependence it has been found that the transverse collective flow angle changes from negative to positive at an energy Ebal (defined as the balance energy) due to the competition between the attractive nuclear mean field at low densities and the repulsive nucleon-nucleon interaction at small separations (high densities).

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Fig. 54. Transverse flow as a function of the reduced impact parameter for three different particle types from 58 Fe + 58 Fe and 58 Ni + 58 Ni collisions at 55 MeV/nucleon. The lines are included only to guide the eye. [Fig. 3 in Ref. 8]

Figure 55 shows the transverse flow plotted versus the incident beam energy. The solid (open) squares are for fragments with Z = 2 from semicentral 58 Fe + 58 Fe (58 Ni + 58 Ni) collisions. Because the measurements are unable to distinguish the sign (+ or −) of the flow one expects that minimum observed in measured distribution corresponds to the balance energy Ebal . To extract the balance energy, the data were fit with a second-order polynomial. The curves do not pass through zero at Ebal because no recoil correction was used in the reaction plane determination. As can be seen in Fig. 55 the balance energy is larger for the more neutronrich system. This observation is valid at all measured impact parameters as was shown in Ref. 174. These observation confirms the predictions of the BUU transport model incorporating an isospin dependent potential and isospin dependent nucleonnucleon scattering cross sections. The details of the BUU model and his predictions are discussed in Sec. 3.9.2.

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Fig. 55. Excitation functions of the measured transverse flow in the reaction plane for Z = 2 fragments from semicentral collisions. The solid (open) squares are for 58 Fe + 58 Fe and 58 Ni + 58 Ni systems. [Fig. 2 in Ref. 174]

3.9. Models for intermediate energy heavy ion collisions Several approaches have been developed to describe the heavy ion collisions at intermediate energies. All of them have only restricted area of applicability. Generally the models can be classified either as static or dynamic ones. 3.9.1. Static models The static models take the initial system, e.g. the composite system created in central collisions, for granted. Usually they assume the initial system to be in a statistical equilibrium. The deexcitation process is treated in statistical way as a simultaneous multifragment decay in microcanonical, canonical, or grand canonical ensembles175–186 or as a sequential binary decay of hot nucleus.187–190 The models based on the percolation concept191–194 can also be qualified into the static models class. An example of the models, which treat deexcitation process as simultaneous decay is the Statistical Multifragmentation Model SMM.178–180 The SMM is based

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on the assumption of statistical equilibrium at a low density freeze-out stage of the nuclear system (e.g. composite system) formed during the collision. Equilibrium partitions are calculated according to the microcanonical ensemble of all breakup channels composed of nucleons and excited fragments. The statistical weight of decay channel j is proportional to exp[Sj (Es , Vs , As , Zs )], where Sj is the entropy of the system in the channel j and Es , Vs , As ,and Zs are the excitation energy, volume, mass and charge numbers of the fragmenting source. The breakup configurations are initialized according to their statistical weights. The fragments are then accelerated in their mutual Coulomb field and allowed to undergo secondary decay. The SMM model has already been shown to reproduce properties of the composite systems formed in central collisions of heavy ions (see e.g. Ref. 131). 3.9.2. Dynamical models Such models describe the heavy ion reactions on the microscopic level taking into account nucleon-nucleon collisions. They treat the input channel more realistically than the static models do. These models, however, suffer some limitations too. The models which use the classical approach either loose some quantum effects or try to simulate them. Dynamical models can be divided into two main classes, namely the one-body models and the N-body models. The former type of model is often referred as a mean-field model, while the latter is also dubbed as molecular dynamics. One-body models base on the observation that at sufficiently low “temperature” the nuclear properties, static as well as dynamical, can be understood in terms of individual nucleons moving in a one-body mean field and subjected to residual two-body interaction. The main drawback of these models is that they enable studying the one-body observables only. The most reliable one-body model is the Time Dependent Hartree–Fock (TDHF) model (see e.g. Refs. 102, 195–197). Its main advantage is that it is a fully quantum model, however it may be applied only to very low energy collisions for which the residual interactions are negligible (see Sec. 2.3). The classical analogue of the TDHF is the Landau-Vlasov (LV) model (see e.g. Refs. 198–200). As energy increases, the effect of residual interactions grows up because the available phase space increases and the Pauli exclusion principle becomes less effective in blocking the final states for direct two-body collisions between constituent nucleons. Models that incorporate these direct nucleon-nucleon collisions propagate the one body phase space density using the classical transport equation supplemented by the Uehling–Uhlenbeck blocking factor accounting for the exclusion principle.201 These models are known under the names of the Boltzmann–Uehling–Uhlenbeck or Vlasov–Uehling–Uhlenbeck (BUU/VUU).202–206 The BUU transport equation for the nucleonic one-body density distribution function f = f (r, p, t) is given by: 1 ∂ dσ d3 p2 d3 p2 dΩ v12 + v∇r − (∇r U )∇p f (r, p, t) − 6 ∂t (2π) dΩ {[f f2 (1 − f1 )(1 − f2 ) − f1 f2 (1 − f )(1 − f2 )(2π)3 δ 3 (p + p2 − p1 − p2 )} . (42)

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Here dσ/dΩ and v12 are the in-medium nucleon-nucleon cross section and the relative velocity for the colliding nucleons, and U is the mean-field potential consisting of the Coulomb potential and a nuclear potential with isoscalar and symmetry terms. The potential field is approximated by (see e.g. Ref. 204): σ ρ ρ ρn − ρp U (ρ, τZ ) = A + (1 − τz )VC + C τz , (43) +B ρ0 ρ0 ρ0 where ρ0 is the normal nuclear matter density. ρ, ρn , and ρp are the nucleon, neutron, and proton densities defined by the distribution of nucleons, respectively. The parameters σ, A and B are adjusted to give the correct saturation density, the binding energy and a preset value of nuclear matter incompressibility parameter K (see Eq. (49)). τz equals 1 or −1 for neutrons or protons, respectively. The parameter C is the strength of the symmetry potential. In the paper of Bertsch et al.205 the test-particle method to solve the BUU equation is presented. As in the cascade model,207 one has to initialize nucleon positions. Each nucleon is replaced by N test particles. Thus, in the case of collisions between the nucleus A with NA nucleons and nucleus B with NB nucleons we have together (NA + NB )N test particles. The density is defined as ρ(r) = N /[N (NA + NB )](δr)3 , where N’ is the number of test particles in a small volume (δr)3 around the point r. The test particles hit each other with a cross section σnn /N . In this model a single nucleus with NA nucleons is represented as a collection of N ∗ NA test particles distributed within a volume of radius R. The Fermi motion to these test particles is also assigned. The momenta up the radius pF are available. The radii R and Fermi momentum pF are not independent. If we take into account the spin-isospin degeneracy the relation 4(4π/3)2 R3 p3F = h3 NA is fulfilled. In the BUU model also Pauli principle is observed (see the right hand side of Eq. (42)). When two test particles collide they change their coordinates from (r1 , p1 )(r2 , p2 ) to (r1 , p1 )(r2 , p2 ). If the phase-spaces around (r1 , p1 ) and (r2 , p2 ) are essentially empty the scattering is allowed. In the opposite case the scattering is prohibited. The implementation of this has varied in detail from one work to another (see e.g. Refs. 202 and 205) but this is the basic procedure. If the right hand side of Eq. (42) is set equal to zero, one obtains the Vlasov equation. The test particles between collisions move classically according to Hamilton’s equations: p˙ i = −∇r U (ρ(ri )) ,

(44)

r˙ i = vi ,

(45)

where, depending upon the original beam velocity, vi can be calculated relativistically or non-relativistically. If the beam energy is higher then about 150 MeV/nucleon the inelastic nucleon-nucleon scattering channels are taken into account. As an example of BUU model predictions a isospin dependence of transverse flow in heavy ion collisions is presented.204 These calculations were done for two

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Fig. 56. Transverse momentum distributions in the reaction plane as a function of rapidity for reactions 48 Ca + 58 Fe and 48 Cr + 58 Ni at an impact parameter of 2 fm and beam energies of 50, 60, and 70 MeV/nucleon. [Fig. 3 in Ref. 204]

reaction systems, 48 Cr + 58 Ni and 48 Ca + 58 Fe, which have the same mass number but different neutron to proton ratios of 1.04 and 1.30, respectively. As an input to these calculations the neutron and proton density distributions calculated from the relativistic mean field theory were used. The standard flow analysis was performed (see Sec. 3.8). Results for central collisions (b = 2 fm) at beam energies of 50, 60, and 70 MeV/nucleon are shown in Fig. 56. At beam energy of 50 MeV/nucleon the transverse flow in reaction 48 Ca + 58 Fe is still negative, while that in the reaction 48 Cr + 58 Ni is already positive. To be more quantitative the flow parameter F was extracted. The beam energy dependence of the flow parameter is shown in Fig. 57. It is seen that in both central and peripheral collisions the neutron rich system 48 Ca + 58 Fe shows a systematically

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Fig. 57. The flow parameter F as a function of beam energy for reactions 48 Ca + 58 Fe and 48 Cr + 58 Ni at impact parameters of 2 and 5 fm. The lines are the fits using linear functions. [Fig. 4 in Ref. 204]

smaller flow parameter indicating a stronger attractive interaction during the reaction. Consequently, the balance energy in the 48 Ca + 58 Fe reaction is higher than that in the reaction of 48 Cr + 58 Ni by about 10 MeV/nucleon (central collisions) to 30 MeV/nucleon (peripheral collisions). Molecular dynamics models for nuclear reactions were adopted from chemical analyses. These models use the classical equations of motion approach and describe individual nucleons as interacting directly via specified two-body forces. The simplest models that describe the many-body dynamics in this regime can be classified as Classical Molecular Dynamics (CMD) models.208–212 In the CMD model, particles are treated as point particles and their transport is governed by a classical equation of motion in a given mean field. They use the phenomenological nucleonnucleon potential adjusted so as to reproduce the bulk properties of nuclei. However, most of them neglect the quantum effects, e.g. Pauli blocking, stochastic scattering and particle production. The models that take into account these quantum features and simulate the fermionic nature of the nucleons are called the Quantum Molecular Dynamics (QMD) models (see e.g. Refs. 213 and 214). In the QMD each particle is described by a Gaussian wave packet. Initial nuclei are constructed by ensuring that there is less than one nucleon in each phase space cell of 1/h3 . During the time evolution of the wave packets the Pauli principle is respected only by the Liouville theorem of classical mechanics. In the model nucleon-nucleon collisions are allowed and the Pauli blocking is treated in an approximate manner.

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There are several attempts to respect the Pauli principle more strictly during the propagation of the wave packets within the classical mechanics. The introduction of a Pauli potential is one of possible methods.215, 216 However, when it is applied in heavy ion collisions, the Pauli potential operates as a spurious repulsive force to increase the nuclear stopping, especially during the early stages of the collision.217 The other procedure to remove overlaps of the wave packets in the phase space, without introducing the Pauli potential, was proposed by Papa et al..218 The model is called the Constrained Molecular Dynamics (CoMD). In order to resolve problem in terms of quantum mechanics, the fermionic molecular dynamics model (FMD),219 and the antisymmetrized molecular dynamics (AMD) model220–222 or its improved version AMD-V (Vlasov)223 have been proposed. In these models the total wave function of the system is antisymmetrized and described by a Slater determinant of Gaussian wave packets. The time evolution of the centroid of the wave packets is treated in a classical manner. In FMD the width of the wave packets is treated as variable in time and nucleon-nucleon collisions are treated as potential scatterings. Until now the calculations have only been made with a harmonic oscillator potential and no application to heavy ion collisions has yet been made. In AMD-V the probabilistic nature of the wave packet is taken into account as a diffusion process during the wave packet propagation. The diffusion process is formulated in a manner to take into account the quantum branching to many final states.221–223 AMD-V has been applied for intermediate energy heavy ion collisions and found to reproduce reasonably well the experimental results.217, 223, 224 As an example, the triton energy spectra for violent collisions for 64 Zn + 92 Mo system at three incident energies are shown in Fig. 58.223 However, the time necessary to work out AMD-V calculations for systems with total mass larger than 200 is very long for practical studies.

3.9.3. Hybrid models The microscopic models are frequently replaced by simpler models which are quite successful in reproducing results of various measurements related to noncentral collisions (see e.g. Refs. 90, 139, 156, 225–227). One of such models is a two-stage, stochastic model of heavy-ion reactions of Sosin.156 In this model a two-stage reaction scenario is assumed: (i) in the first stage, a number of nucleons becomes reaction participants as a result of the mean-field effects and/or nucleon-nucleon interactions; (ii) in the second stage, participating nucleons are transferred to definite states, creating finally a PLF, a TLF, or clusters. They can also escape into continuum. The competition between the mean-field process and the nucleon-nucleon process which produce participant nucleons is illustrated by Fig. 59, which represents distributions of the number of nucleons participating in the 40 Ca + 40 Ca reaction,

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Fig. 58. (Color online) The triton energy spectra for the violent collisions for 64 Zn + 92 Mo system at three incident energies (indicated at the top of each column). The observation angles are indicated in the left column. Experimental results are shown by dots. Thick and thin solid lines correspond to calculated results for soft and stiff NEOS, respectively. [Fig. 22 in Ref. 223]

versus the impact parameter b, and for different collision energies. As can be seen, the mean-field effects (dashed lines) are more important at lower energies, while nucleon-nucleon collisions (solid lines) dominate at higher energies. The maxima observed for mean-field nucleons result from orbiting effects at peripheral collisions. At higher collision energies the mean-field nucleons disappear for central collisions because of competition with the nucleon-nucleon interaction. In the second stage, participating nucleons are transferred to different objects of the system consisting of the projectile remnant, the target remnant, and clusters. The creation of clusters begins from a coalescence of two participating nucleons. The nucleon transfer is treated as a stochastic process with the number of steps equal to the number of participating nucleons. The nucleon transfer probabilities are governed by the state densities. An application of this model is presented for the 40 Ca + 40 Ca, 197 Au reactions at 35 MeV/nucleon. A satisfactory agreement and a consistent picture of the reaction mechanism were achieved. Both the properties of the PLF145, 158 and of the

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60

10 AMeV 40 20 0 60

2

4

6

8

10

Number of participants

20 AMeV 40 20 0 60

2

4

6

8

10

35 AMeV 40 20 0 60

2

4

6

8

10

50 AMeV 40 20 0

0

2

4

6

8

10

Impact parameter (fm) Fig. 59. Numbers of nucleons participating in the 40 Ca + 40 Ca reaction versus impact parameter, for different collision energies. The nucleon-nucleon mechanism (solid line); mean-field mechanism (dashed line). [Fig. 1 in Ref. 156]

intermediate velocity source146 were properly reproduced. Results of the calculations are shown as lines in Figs. 38, 39, 46, 48 and 49. The model predictions for the IVS source emission are shown in Fig. 40. 3.10. Nuclear equation of state In the theoretical considerations nuclear matter is treated as an idealized system of equal number of neutrons and protons. It is assumed that the Coulomb interaction between protons is switched off and the considered system is very large. A successfull description of many aspects of nuclear collisions in terms of microscopic models has given the possibility to get information about properties of nuclear matter.

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Fig. 60. The equation of state of asymmetric nuclear matter from the Skyrme-Hartree-Fock (left panel) and relativistic mean-field (right panel) model calculations. The solid curves correspond to proton-to-neutron ratios ρp /ρn of 0., 0.2, 0.4, 0.6, 0.8 and 1 (from top to bottom). [Fig. 2.1 in Ref. 12].

From the other side the properties of nuclear matter can be deduced from theoretical considerations. One can mention the following approaches: (i) nonrelativistic Brueckner approach,228–230 (ii) relativistic Brueckner approach,231, 232 (iii) Thomas–Fermi approximation,233, 234 (iv) relativistic mean-field theory.235 Figure 60 shows predictions for the EOS of asymmetric matter from the nonrelativistic Skyrme-Hartree-Fock using the parameter set SHI (left panel) and the relativistic mean-field (RMF) model using the parameter set TM1 (right panel).12 The solid curves correspond to different neutron to proton ratios. Commonly the EOS of asymmetric nuclear matter i.e. the density of energy per nucleon e(ρ, δ) = E/A as a function of density ρ and isospin asymmetry parameter δ is parametrized as: e(ρ, δ) = TF (ρ, δ) + V0 (ρ) + δ 2 V2 (ρ),

(46)

where δ is given by δ=

ρn − ρp ρ

(47)

and where TF is the kinetic energy of Fermi gas, V0 and V2 are the isospinindependent and isospin-dependent potential terms. The relation between pressure and density is given by a thermodynamical relation: ∂e(ρ, δ) . (48) P (ρ, δ) = ρ2 ∂ρ A very important characteristics of nuclear matter is given by compressibility parameter, K. It is defined by: K(ρ, δ) = 9

∂P (ρ, δ) . ∂ρ

(49)

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In most models the compressibility K decreases as the matter becomes more neutron rich, but the predictions varies widely among models, in particular at high densities. Equation (46) is often written in the form of: e(ρ, δ) = e(ρ, 0) + δ 2 esym (ρ) ,

(50)

where e(ρ, 0) represents the equation of state for symmetric matter and esym (ρ) is the symmetry energy. When we consider the nuclear matter not far away from normal density e(ρ, 0) can be expanded around the normal nuclear density ρ0 : 2 K ρ − ρ0 + ··· , (51) e(ρ, 0) = e(ρ0 , 0) + 18 ρ0 where compressibility parameter for symmetric matter K is given by: ∂ 2 e(ρ, 0) K = 9ρ20 . ∂ρ2 ρ=ρ0 The symmetry energy, esym (ρ), is given by: 1 ∂ 2 e(ρ, δ) 5 esym (ρ) = = TF (ρ, 0) + V2 (ρ) . 2 2 ∂δ 9 δ=0

(52)

(53)

Using the empirical parabolic low one can easy extract the symmetry energy esym (ρ) from microscopic calculations done for two extreme cases of pure neutron matter and symmetric nuclear matter. The esym (ρ) is given by: esym (ρ) = e(ρ, 1) − e(ρ, 0) .

(54)

Furthermore, the symmetry energy can be expanded around the normal nuclear matter density ρ0 : 2 L ρ − ρ0 Ksym ρ − ρ0 esym (ρ) = esym (ρ0 ) + + ··· (55) + 3 ρ0 18 ρ0 In the above approximation L and Ksym are the slope and curvature of the symmetry energy at normal density: ∂esym L = 3ρ0 , (56) ∂ρ ρ=ρ0

Ksym =

∂ 9ρ20

esym . ∂ρ2 ρ=ρ0

2

(57)

Using the above expansion the model prediction for esym (ρ0 ) can be compared with mass formula, which gives a value 27–36 MeV.236 Ussually models are tuned to give esym (ρ0 ) within this range. The L and Ksym characterize the density dependence of nuclear symmetry energy around normal nuclear matter density, and thus provide important information about the properties of nuclear symmetry energy at both high and low densities. The theoretically predicted values for Ksym ranges from 460 MeV to −700 MeV (see e.g. Ref. 230). The data from giant monopole

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resonances indicate that Ksym is between −560 ± 1350 MeV to 34 ± 159 MeV depending on the mass region of nuclei.237 The value of Ksym = −500 MeV was extracted from isospin diffusion measurements.238 The Ksym data thus still cannot distinguish these model calculations. There are also no experimental data for the value of L, and different theoretical models give its value from −50 to 200 MeV.239 4. Relativistic Energies Collisions of heavy nuclei at relativistic energies are expected to probe the history of the early Universe by reproducing it on a microscopic scale. One of the main goals of experiments is to create a quark-gluon plasma, to study properties of this phase and to characterize the phase transition between quark-gluon plasma and hadron phase. Two major experimental programs have produced vast amount of data for dif√ ferent systems measured at center of mass energies per nucleon pair, sN N , from 2 GeV to 200 GeV. One at Brookvaven National Laboratory, where AGS and RHIC accelerators are located and measurements are done for Au + Au system in the energy regions 2–5 GeV and 63–200 GeV, respectively. The other program was performed at CERN using the SPS accelerator for Pb +Pb system in the energy range 9–20 GeV. In this section different ascpect of these experiments are presented. 4.1. Geometry of the collision Figure 61 illustrates a schematic view of two nuclei approaching each other in the center of mass frame of these nuclei.240 The incoming direction is taken as the z axis. The nuclei are Lorentz contracted in their direction of motion due to their relativistic speed. The “white” nucleons that interact with nucleons in the other nucleus are called participants and the grey nucleons are called spectators. The number of participating nucleons depends strongly on impact parameter b. It is only in the very central collisions b ∼ 0 fm that all nucleons interact. The spectator-participant picture is used to characterize the centrality of collision by the centrality parameter C. This parameter is related to the overlap surface

b

Fig. 61. The two nuclei collide at impact parameter b. The “white” nucleons that interact with nucleons in the other nuclei are called participants and the grey nucleons are called spectators. [Fig. 1.5 in Ref. 240]

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in the transverse plane of the colliding nuclei and is defined as: bc dσin (b ) db db C= 0 ∗ 100% , σin

1047

(58)

where σin is the total inelastic cross section and bc the impact parameter cut-off. From this definition, C is the probability that a collision occurs at b ≤ bc . For two identical solid spheres, dσin (b)/db = 2πbdb. Therefore, since bmax = 2R, where R is the sphere radius, one obtains C = b2c /(4R2 ). For two Au nuclei the centrality range (0–5)% corresponds to impact parameters ranging from 0 to 3.1 fm. As a measure of velocity component parallel to the beam direction the rapidity variable is used. It is defined as: y=

1 1 + β cos θ 1 E + pz ln . = ln 2 E − pz 2 1 − β cos θ

(59)

Here E, pz , β, and θ denote the total relativistic energy, longitudinal momentum, velocity in units of light velocity and angle relative to the beam axis, respectively, of particle. Under Lorentz transformation the rapidity of a particle in the CM and in the LAB frames of reference are related through a simple addition. 4.2. Stopping in heavy ion collisions In heavy ion collisions among other quantities the total relativistic energy and the baryon number are conserved. Initially, all the energy is carried by baryons, but after the collision they carry only a fraction of this energy. The baryons are stopped in the collision. The energy difference is used to create new particles and generate flow. The energy loss of baryons occurs in three ways: (i) initial interaction — nucleons of the projectile interact with nucleons in the target; (ii) rescattering — the momentum and energy of baryons is modified by elastic and inelastic partonic and hadronic rescattering with created particles leading to further particle production; (iii) decays — excited baryons decay and their energy is distributed among the decay products. There are two extreme cases that are used to define maximal and minimal stopping: (i) in the ideal case of full transparency (Bjorken picture241 ) the baryons loose little of their initial kinetic energy during the interpenetration of the colliding nuclei. In that case, as original nucleons move away from the interaction zone at a barely altered rapidity, the midrapidity zone is characterized by a net baryon density Nnet = NB − NB¯ ,

(60)

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Fig. 62. Proton rapidity distributions for Au + Au interaction at three beam energies for different centrality classes as indicated. The curves represent double Gaussian fits to the data, the centroids of which are indicated by arrows. [Fig. 2 in Ref. 243]

the number of baryons minus the number of antibaryons close to zero due to baryons number conservation. Practically all baryons and anti-baryons detected at midrapidity are produced; (ii) in the ideal case of full stopping the baryons loose all their kinetic energy in the collisions. This implies that the original nucleons are mostly distributed at midrapidity, with maximum at y = yCM , where yCM is the rapidity of the CM in the laboratory frame. The stopping can be quantified as was proposed by Videdaek and Hansen.242 The average rapidity loss δy is then defined as δy = |yp − yb | ,

(61)

where yp is the incoming rapidity in the CM system and yb is the mean net baryons rapidity after the collision. These average value in symmetric collisions is calculated as

yp yp dNnet dNnet dy dy + yCM , (y − yCM ) (62) yb = dy dy yCM yCM where dNnet (y)/dy is the net baryon rapidity density.

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In Fig. 62 results from a study on the centrality dependence of stopping for Au + Au collisions at 6, 8, and 10.8 GeV/nucleon are presented in Ref. 243. At these low energies the number of produced antiprotons is small and the net baryon distribution is approximated by the proton distribution. The data are shown relative to the rapidity of the center of mass of the system which equals 1.35, 1.47, and 1.61 for corresponding beam energies. The solid points (data) reflected about midrapidity are shown as open symbols. For each beam energy a common trend is observed in the evolution of the shape of the rapidity distribution as a function of centrality. For the most peripheral event class the distribution has a minimum value of dN/dy ≈ 6 at midrapidity. This concave shape persists to the next event class, corresponding to a centrality cut of (23–39) %, consistent with the expectation that most of participant protons reside at beam rapidities following these relatively peripheral collisions. For more central collisions, the rapidity distribution becomes progressively flatter and begin to develop a broad maximum at midrapidity for all beam energies. A closer inspection of this maximum suggests, however, that for the most central collisions the distribution does not evolve to a single peak centered at midrapidity but rather is consistent with two components each displaced from midrapidity, or with a set of sources spread throughout the rapidity range. The evolution of the net proton rapidity distribution for the most central collisions (top 5%) with the incident energy is presented in Fig. 63. Here the data for √ √ Au + Au at sN N = 5 GeV (AGS),243–245 Pb + Pb at sN N = 17 GeV (SPS),246

dN/dy net-protons

80

AGS

AGS yp

(E802,E877, E917)

SPS

60

SPS yp

(NA49)

RHIC

RHIC yp

(BRAHMS)

40 20 0

-4

-2

0

yCM

2

4

√ Fig. 63. The net-proton rapidity distribution measured at AGS (Au + Au at s = √ √ NN 5 GeV),243–245 SPS (Pb + Pb at sNN = 17 GeV),246 and RHIC (Au + Au at sNN = 200 GeV).247 The data are all from the top 5% most central collisions. The data were symmetrized. [Fig. 3 in Ref. 247]

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4

Rapidity loss < δ y>

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dN/dy net-Baryons pol6 momgaus

80

3

60 40 20 0

0

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1

0

1

2

E917

BRAHMS maximum

E802/E866

BRAHMS minimum

NA49

BRAHMS estimate

3

4

5

yp Fig. 64. The inserted plot shows the net-baryon distribution (data points) with fits (represented by the curves). The full figure shows the rapidity loss as a function of projectile rapidity (in the CM). The hatched area indicates the unphysical region and the dotted line shows the phenomenological observed scaling δy = 0.64ybeam . The data for lower energies are from Refs. 243 and 242. [Fig. 4 in Ref. 247]

√ and Au + Au at sN N = 200 GeV (RHIC)247 systems are collected. These distributions show a strong energy dependence, the net-protons peak at midrapidity at AGS energy, while at SPS energy a dip is observed in the middle of the distribution. At RHIC a broad minimum has developed spanning several units of rapidity, indicating that at RHIC energies collisions are quite transparent. The dependence of rapidity loss as a function of projectile rapidity is shown in Fig. 64.247 Linear scaling observed at low energies is broken at RHIC. The small increase from SPS to RHIC could be an indication that the rapidity loss saturates for such high energies. The RHIC data show that at this energy a high degree of transparency is observed.

4.3. Particle production and energy density Very useful data concerning the particle production are obtained from multiplicity detectors. These detectors supply information only about the number of charged particles at given observation angle. In this case the pseudorapidity variable η is used. It is defined as: θ 1 1 + cos θ = − ln tan , (63) η = ln 2 1 − cosθ 2

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where θ is an angle between the particle momentum p and the beam axis. For the case of particles moving with velocities close to the light velocity we have y ≈ η. Figure 65 shows the measured dNch /dη distributions for charged particles for √ several centrality regions for sN N = 200 GeV data.248 For the most (0–5)%, dNch /dη|η=0 = 625 ± 1(stat) ± 55(syst). This gives a scaled multiplicity value of (dNch /dη|η=0 )/Npart /2 = 3.5 ± 0.3 charged particles per participating nucleon pair. The integration of the charged particles pseudorapidity distribution corresponding to the (0–5)% most central collisions gives us that 4630 ± 370 particles are produced. The study of pseudorapidity distribution of emitted charged particles, dNch /dη, is a fundamental observable in ultrarelativistic collisions in respect of kinetic energy dissipation mechanism. In such collisions charged particles can be produced by hadronic (soft) as well as partonic (hard) collision processes. The study of dNch /dη versus η, and centrality enables to disentangle the contributions of different mechanisms.249 Figure 66 shows particle densities normalized to the number of participating √ √ pairs for SPS data at sN N = 17 GeV,250 and RHIC data at sN N = 130 GeV251 and 200GeV for two selected centrality cuts.248 Data at different beam energies are plotted as a function of the pseudorapidity, respectively, shifted by respective beam rapidity. The presented results appear to be independent of both collision centrality and beam energy over a pesudorapidity range from 0.5 to 1.5 units below the beam rapidity. This observation is consistent with a limiting fragmentation picture in which the excitation of the fragment baryons saturates already at moderate collisions energies. The increased projectile kinetic energy is utilized for particle production at rapidities below beam rapidity, as evidenced by the observed increase in the scaled multiplicity for central events at midrapidity. After the primary collision, the matter and energy distributions can be conceptually divided into two main parts, a fragmentation region consisting of excited remnants of the colliding nuclei which experienced an average rapidity loss of about 2.2 and a central region in which few of the original baryons are present but where significant energy density is collected. This energy can be estimated using the Bjorken formula:241 3 ET dNch = , (64) 2 πR2 τ0 dη η=0 where dNch /dη|η=0 is density of emitted particles at η = 0. The τ0 is the initial formation time, usually taken to be the thermalization time-scale 1 fm/c. The transverse area of the interaction zone πR2 is here calculated for R = 6 fm. The mean transverse energy ET = 0.5 GeV. With these input data the equals 3.5 √ and 4.5 GeV/fm3 for sN N = 130 GeV and 200 GeV, respectively. The last value exceeds the energy density of a nucleus by a factor of 30, and the energy density of a baryon by a factor of 10. These both values are well above the lattice QCD prediction (crit ≈ 1 GeV/fm3252 ) for hadron gas to QGP phase transition.

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700 600

dN ch /d η

500 400 300 200 100 -5

-4

-3

-2

-1

0

η

1

2

3

4

5

√ Fig. 65. Distributions of dNch /dη for Au + Au system at sNN = 200 GeV. The data sets correspond to the centrality ranges, from top to bottom, (0–5)%, (5–10)%, (10–20)%, (20–30)%, (30–40)%, and (40–50)%. Statistical errors are shown for all points where they are larger than the symbol size. [Fig. 1 in Ref. 248].

dη

dN ch

/(0.5)

3

0-5% Au+Au,

S NN =130 Gev

0-5% Au+Au,

S NN =200 Gev

30-40% Au+Au, 9.4% Pb+Pb,

2

S NN =200 Gev

S NN =17 Gev

1

0 -3

-2

-1

η -ybeam

0

1

Fig. 66. Charged particle multiplicities normalized to the number of participant nucleon pairs for the (0–5)% centrality (open circles) and (40–50)% centrality (open squares) Au + Au results at √ √ sNN = 200 GeV,248 the (0–5)% Au + Au results at sNN = 130 GeV (closed circles),251 and √ the 9.4% central Pb + Pb data at sNN = 17 GeV (closed triangles).250 The data are plotted as a function of the pseudorapidity shifted by relevant beam rapidity. Representative total uncertainties are shown for a few Au + Au points. [Fig. 2 in Ref. 248].

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Ratio

1.0

0.5

π-/π+ -

K /K

+

p /p 0.0

0

1

2

3

4

y Fig. 67. Antihadron to hadron ratios as a function of the CM rapidity. Error bars show the statistical errors while the caps indicate the combined statistical and systematic errors. [Fig. 3 in Ref. 253]

4.4. Antihadron to hadron ratios Ratios of yields of antihadrons to hadrons are also significant indicators of the dynamics of high energy nucleus-nucleus collisions. Figure 67 shows the π − /π + , √ K − /K + , and p¯/p ratios as a function of rapidity for Au + Au reaction at sN N = 200 GeV for the central collision events.253 The π − /π + ratio is consistent with the unity over the considered rapidity range. The K − /K + ratio shows a decrease from 0.95 ± 0.05, at y = 0, to 0.67 ± 0.06, at y = 3.05. Similarly, the p¯/p ratio shows a decrease from 0.75 ± 0.04, at y = 0, to 0.23 ± 0.03, at y = 3.1. The ratios for protons and kaons are the highest observed in nucleus-nucleus collisions. These ratios are constant in the interval y = 0–1, as expected for boost invariant midrapidity plateau dominated by particle production from the color field. Away from midrapidity the net baryon content originating from the initial nuclei is significant and production mechanisms other than particle-antiparticle pair production play a substantial role. The measured set of particle ratios were analyzed in terms of the statistical model based on the assumption of a system in chemical and thermal equilibrium (see e.g. Refs. 254). Figure 68 shows the ratios of negative kaons to positive kaons as a function of the corresponding ratios of antiprotons to protons for various rapidities. The figure also displays ratios for heavy ion collisions at AGS and SPS energies.255–257 There is a striking correlation between the BRAHMS kaon and proton ratios over 3 units of rapidity. It was shown that this correlation can be described assuming the chemical and thermal equilibrium at the quark level, the ratios can be written then as: 6µu,d Np¯ = exp − Np T

(65)

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and 1 NK − 2(µu,d − µs ) 6µs Np¯ 3 = exp − , = exp − NK + T T Np

(66)

where N and T denote number of particles and temperature, respectively. Parameters µu , µd and µs are chemical potentials for u, d, s quarks, respectively. Eq. (66) tells us that for a vanishing strange quark chemical potential we would expect a power low relation between the two ratios with exponent 1/3 (broken line in Fig. 68). The observed correlation deviates from this assumption suggesting a finite value of the strange quark chemical potential. A more elaborate analysis assuming a grand canonical ensemble with charge, baryon and strangeness conservations was carried out by fitting these and many other particle ratios observed at RHIC. Values of the chemical potentials and the temperature were obtained. It was found that a very large collection of such particle ratios is extremely well described by the statistical approach. The results of the fitting procedure are shown as a solid line in Fig. 68.254

µB 1.0

255

137

(T=170MeV)

78

43

19

0

0.8 -

K + K 0.6

BRAHMS (200 GeV) BRAHMS (130 GeV) NA44 (17 GeV) NA49 (9,12,17 GeV)

0.4

E866 (5 GeV) Becattini et. al K -/K+ = (p /p)

0.2 0.0

0.2

0.4

0.6

1/3

0.8

p /p Fig. 68. Correlation between kaon and baryon ratios. Solid points data from Refs. 251 and 253. The open symbols show lower energy data255–257 at the listed center of mass energies. The line shows the statistical model prediction of Becattini et al. 254. The top scale for the baryon chemical potential µB is in MeV. Error bars present statistical and systematical errors. [Fig. 4 in Ref. 253]

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d 2N /(2 π mT dmTdy) (c /GeV 2 )


10

(a)

2

1055

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(b) 10 1

10

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1

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0

10

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-

K K+

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0.3

0.4

2

mT - m K ( GeV/c )

-2

0

p p 0.2

0.4

0.6 2

mT - m p ( GeV/c )

Fig. 69. Invariant yield of the transverse mass for π ± , K ± , p and p¯ at midrapidity (|y| < 0.1) for p +p (bottom) and Au + Au events from (70–80)% (second from bottom) to the (0–5)% centrality √ bin (top) at the nucleon-nucleon center of mass energy of sNN = 200 GeV. The curves shown are explained in the text. [Fig. 1 in Ref. 258]

4.5. Flow at ultrarelativistic energies The properties of the expanding matter in the later stages of the collision up to the moment when interactions cease (kinetic freeze out) can be studied from the momentum distribution of the emitted particles. The slopes of spectra of emitted particles depend in general on the temperature of the source from which they were created and on kinetic effects that may alter the expected Maxwellian distribution, such as a velocity component resulting from an overpressure leading to an outwards flow of the matter (see Sec. 3.8). Figure 69 shows transverse mass, mT , spectra for π ± , K ± , p and p¯ particles for p + p collisions and all centrality bins of Au + Au data within |y| < 0.1 range √ measured at the nucleon-nucleon center of mass energy of sN N = 200 GeV.258 Here the transverse mass is defined as:

(67) mT = (p2T + m2 ) , where pT is the transverse momentum and m is the rest mass of particle. For clarity, proton spectra are scaled by 0.8. Particle and antiparticle spectra shapes are similar for each centrality bin. While the π ± spectra shapes are similar for p + p and Au + Au systems, the K ± , p and p¯ spectra show a progressive flattening from p + p to central Au + Au events. The blast-wave model — a hydrodynamically motivated model with a kinetic freeze-out temperature Tf,kin and a transverse flow velocity field259 can simultaneously fit the K ± , p and p¯ spectra and high pT part (pT > 0.5 GeV/c) of the π ± spectra. In the analysis the velocity profile of n r β = βs , (68) R

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was used. Here r < R and the term r/R accounts for the change in the velocity as a function of a radial distance. The parameter βs is the surface velocity and n is treated as a free parameter. The value of n ranges from 1.50 ± 0.29 in peripheral to 0.82 ± 0.02 in central events. The fit results are superimposed in Figs. 69(b) and 69(c). The obtained fit parameters for the (0–5)% Au + Au events are Tf,kin = 89 ± 10 MeV and βs = 0.84 ± 0.07. The low pT part of the pion spectrum deviates from the blast-wave model description, possibly due to the large contribution from resonances at low pT . The pion spectra were fitted with the Bose–Einstein distribution (1/ exp( mT⊥ − 1)), the results are superimposed in Fig. 69(a). Another powerful tool to study the thermodynamic properties of the source is the analysis of the azimuthal momentum distribution of the emitted particles relative to the reaction plane. This distribution is usually parameterized as: dN ∼ 1 + 2v1 cos φ + 2v2 cos 2φ + · · · , dφ

(69)

where φ = ϕ − ϕr , and ϕ, ϕr denote the azimuthal angles of the particle and of the reaction plane, respectively. The coefficient v1 measures the so-called directed flow and the coefficient v2 measures the elliptic flow. In Fig. 70 the transverse momentum dependence of v2 for identified particles is √ shown for Au + Au reaction at sN N = 200 GeV (PHENIX experiment).260 The v2 values were generated for particles measured at midrapidity for minimum-bias events. The top-left panel shows negatively charged particles, while the top-right panel shows positively charged particles. The combined results for positive and negative particles are shown in the bottom left panel. The lines in that panel represent a hydrodynamical calculation including a first order phase transition with a freezeout temperature of 120 MeV.261 The data show that at lower pT (< 2 GeV/c), the lighter mass particles have a larger v2 value at a given pT , which is reproduced by the model calculations. A striking feature observed at higher pT is that the v2 of protons are larger than for pions and kaons. This is in sharp contrast to the hydrodynamical picture, which would predict the same mass ordering for v2 at all pT . The v2 values for mesons begin to show a departure from the hydrodynamical prediction at pT of about 1.5 GeV/c, while the baryons agree with the prediction up until 3 GeV/c. Such behavior is predicted by the quark-coalescence mechanism,262 as shown in the bottom-right panel where both v2 and pT were scaled by the number of quarks. This could be an indication that the v2 of the measured hadrons is already established in a quark matter phase. 4.6. The high pT hadron suppression In order to extract any effects of a nuclear medium on the particle production in nucleus-nucleus collisions the corresponding production yield is compared to that for nucleon-nucleon collisions. For this purpose the nuclear modification factor is

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−

+

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2

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π π + − K K p pbar

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0.05 0.1

0

hydro π hydro K hydro p 0

1

2

3 4 pT (GeV/c)

0

0

0.5 1 1.5 pT/nquark (GeV/c)

Fig. 70. (Color online) Transverse momentum dependence of v2 for identified particles. The circles show p and p¯, the squares show K + and K − , and the triangles show π + and π − for minimum bias events. Statistical errors are represented by error bars and overall systematic error due to all sources by the solid lines in the top two panels. The combined positive and negative particles are shown in the bottom left panel, and the lines there represent the result of a hydrodynamical calculation for π, K, and p from upper to lower curves, respectively. The bottom right panel shows the quark v2 values as a function of the quark pT by scaling both axes with the number of quarks for each particle, as motivated by a quark-coalescence model.262 [Fig. 2 in Ref. 260]

constructed. Its definition is as follows: RAA

1 = Nbin

d2 NAA dpT dη d2 NN N dpT dη

,

(70)

where Nbin is the average number of binary nucleon-nucleon collisions as estimated by the Glauber model. The d2 NAA /dpT dη and d2 NN N /dpT dη are the charged production yields in the A + A and N + N collisions respectively, taken at the same transverse momentum and pseudorapidity. If there are no medium effects, i.e. if a nucleus-nucleus collision could be viewed simply as individual nucleon-nucleon collisions, then we would expect a RAA factor of unity above a certain pT threshold and decreasing smoothly below. The later is at low pT where we expect a scaling of particle production with the number of participating nucleons Npar than Nbin .

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pT [GeV/c]

Fig. 71. (Color online) Top row: Nuclear modification factors RAuAu as a function of transverse √ momentum for Au +Au collisions ( sNN = 200 GeV) at η = 0 and η = 2.2 for the (0–10)% most central collisions. Middle row: as top row, but for centralities (40–60)%. Bottom row: ratio of the RAuAu factors for the most central and most peripheral collisions at the two rapidities. The dotted and dashed lines show the expected value of RAuAu using a scaling by the number of participants and by the number of binary collisions, respectively. Error bars are statistical. The grey bands indicate the estimated systematic errors. The grey band at pT = 0 is the uncertainty on the scale. [Fig. 2 in Ref. 263]

Figure 71 (upper two rows) shows the ratios RAuAu as a function of pT for two √ centrality cuts for Au +Au collisions at sN N = 200 GeV.263 Left panels correspond to measurements at η = 0 and the right panels correspond to measurements at η = 2.2. The RAuAu rise from values of 0.2–0.4 at low pT to a maximum at pT ≈ 2 GeV/c. The low part of the spectrum is associated with soft collisions and should therefore scale with the number of participants. Thus the applied scaling with the (larger) Nbin value reduces RAuAu at the lower pT . Above pT ≈ 2 GeV/c the RAuAu distributions decrease and are systematically lower than unity for central collisions, while they remain near 1 for more peripheral collisions. For the most central collisions at both pseudorapidities, RAuAu is only about 0.4 at pT ≈ 4 GeV/c. As compared to the SPS results at lower energies246, 264 the high pT part is suppressed by a factor of 3–4. Because of the lack of independent measurements of the p + p reference spectrum at forward rapidity, the systematic error on RAuAu at η = 2.2 is estimated to

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pT [GeV/c]

Fig. 72. Nuclear modification factor for charged hadrons at psudorapidities η = 0, 1, 2.2, and 3.2. One standard deviation statistical errors are shown with errorbars. Systematic errors are shown with shaded boxes. The vertical shaded band around unity indicate the error on the normalization to Ncoll . [Fig. 2 in Ref. 265]

be σsys ≈ 30% at high pT . To avoid this problem, the p + p collisions yield is periph approximated by peripheral nucleus-nucleus collisions yield scaled to 1/Nbin . The ratio of central to peripheral collisions RCP is defined as: RCP =

Nbin (P ) Nbin (C)

d2 NAA dpT dη |C d2 NN N dpT dη |P

,

(71)

where “C” and “P” denote the most central and peripheral bins, respectively. The bottom panels of Fig. 71 show the ratio RCP using the centrality classes (0–10)% and (40–60)%. Again we see a clear saturation of the modification factor at a value below unity and a subsequent further drop as pT increases. These observations show that there is an attenuating mechanism present in central collisions. √ For comparison the results for d + Au system at sN N = 200 GeV are presented in Fig. 72. The ratio RdAu is shown for minimum bias events at pseudorapidities η = 0, 1.0, 2.2 and 3.2.265 For these collisions at midrapidity, where we do not expect the same degree of “heating” of the nuclear matter as in central Au + Au collisions, we can see enhancement of RdAu , very similar to the Cronin effect,266 rather than suppression. However as we move to more forward rapidities, this effect vanishes and at η ≥ 2.2 we can once again see indications of suppression. The absence of the high pT suppression around midrapidity for d + Au may be taken as direct evidence for the fact that the strong high pT suppresion seen in Au + Au collisions around y = 0 is not due to particular conditions of the colliding nuclei (initial state effects). It was concluded that the suppresion seen in the central Au + Au collisions is due to the final-state interactions with the dense medium generated in such collisons (see e.g. Ref. 267). The midrapidity enhancement of RdAu can be explained as broadening of the momentum distributions due to multiple scattering (Cronin enhancement) as for SPS data, indicating that processes responsible for bulk particle production at √ midrapidity in d + Au collisions at sN N = 200 GeV are similar to those in Pb + Pb √ collisions at sN N = 17 GeV.

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When we move to more forward rapidities for the d + Au system one observes a high pT suppresion starting already at η = 1 (see Fig. 72). It was proposed that this effect at forward rapidity is related to the initial conditions of the colliding d and Au nuclei, in particular to the possible existence of the Color Glass Condensate.268 5. Summary The heavy ion reactions are inseparately related to the conversion of the incident kinetic energy into internal excitation of the reaction products. At low collision energies loss of kinetic energy is related with Coulomb excitation, a one- or multinucleon transfer processes or formation on the compound nucleus. In the later case all kinetic energy is totally converted into the excitation energy of compound system. In central collisions at energies close to the Coulomb barier the superheavy nuclei up to Z = 118 are formed and their basic properties may be investigated. At incident energies of the order of 10 MeV/nucleon the dominant dissipation mechanism is related to the collisions of nucleons with moving walls of the interacting system. The nucleon-nucleon collisions are blocked by the Pauli exclusion principle. Most of the models used in this energy range reduce the complicated many body problem by introducing a few macroscopic time dependent variables. The characteristics of experimental data (see e.g. broadening of the mass and charge distributions) support the use of Fokker–Planck transport equation as way to describe such processes. As was shown in many analyses this approach is very successful in describing many aspects of the damped collisions, the dominant reaction channel. Some problems have remained unresolved till now. These models are unable to describe properly the N/Z equilibration process (see e.g. Fig. 15) and the correlation between the excitation energy partition and net mass transfer (see e.g. Fig. 19). They predict no such a correlation. At incident energies above 10 MeV/nucleon the nucleon-nucleon collisions are more important because the available phase space increases and Pauli exclusion principle becomes less effective. Such behavior is visible in experimental data. For central collisions the complete fusion is replaced by incomplete fusion process. The mean field potential is no longer strong enough to trap all the nucleons and some number of nucleons is emitted before the composite system reach thermodynamical equilibrium. Composite systems were observed in some number of experiments. It was shown that thermodynamical equilibrum is reached in such objects (see Figs. 35 and 36). For noncentral collisions the two-body reaction picture with PLF and TLF fragments evolve to the more complex one. With increasing incident energy the importance of the IVS source increases, while the PLF and TLF sources change into relatively low-excited spectators (see e.g. Fig. 43). For colliding systems with relatively light projectiles the deuterons, tritons and 3 He are preferentially emitted by the IVS source (see e.g. Fig. 40). The isospin dependence of the IVS contribution of mass 3 particles seems to be correlated with the N/Z ratio of the system under consideration.

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In some number of analyses attention was focused on PLF source reconstruction and its properties. It was shown that the decay characteristis of the reconstructed PLF source are consistent with a thermalized character of this source (see e.g. Figs. 47 and 49). Data from a number of experiments were used to construct caloric curves for hot nuclear objects. It was shown that the limiting temperatures represented by the plateaus decrease with increasing nuclear mass (see e.g. Fig. 51). These temperatures values are in good agreement with the results of recent model calculations. This agreement favors a soft equation of state and a critical temperature of 16.6 ± 0.6 MeV for symmetric nuclear matter. As incident energy increases above 30 MeV/nucleon the collective motion of reaction products is observed. This behavior is induced by a significant compression of nuclear matter in the initial phase of the reaction. Measurements showed that observed flow is stronger for the more neutron rich systems (see e.g. Fig. 54). These observation is in agreement with BUU model predictions when isospin dependence of the mean field potential and nucleon-nucleon cross section is taken into account. The compression of nuclear matter in the initial phase of the reaction is also related to stopping phenomenon observed in central collisions at energies of the order of few GeV/nucleon (see Fig. 62). Going in the region of ultrarelativistic collisions one observes that stopping of nuclear matter is weaker, the colliding objects are more transparent(see Fig. 63). The first results of RHIC experiments showed clearly that we moved in high energy nucleus-nucleus collisions into a qualitatively new physics domain characterized by a high degree of reaction transparency and formation of a nearly baryon free midrapidity region. In this region a significant energy is collected. The estimated energy density is of the order of 4.5 GeV /f m3 for the highest RHIC collision energy. This value is well above the lattice QCD prediction for hadron gas to quark-gluon plasma phase transition. The picture of the baryon free midrapidity region emerges also from analysis of particle ratios in terms of the statistical models. Analyses assuming a grand canonical ensemble with charge, baryons and strangeness conservation suggest chemical equilibrium at temperature at temperature T = 175 MeV and near-zero light quark chemical potential µB of the order of 20 MeV. The dependence of the elliptic flow coefficient v2 on pT indicates that the observed flow of particles is already established in a quark matter phase. The observation of a very large suppression of high momentum particles originating from hard scattering in the very early stage of reaction is consistent with a significant energy loss of high momentum protons moving in a dense medium of unscreened color charges. There is no doubt that the experiments at RHIC revealed that the matter created in these experiments differs from anything that had been known before. The body of information obtained at RHIC in conjunction with the availabe theoretical

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results suggest that the matter created at midrapidity region cannot be characterized only by hadronic degrees of freedom but requires a partonic description. The future investigations of heavy ion physics will be focused on: (i) the limits of nuclear existence, formation of the heaviest elements; (ii) the dependence of nuclear structure and dynamics on the asymmetry in neutron-proton composition; (iii) new forms of nucleonic matter at extremes of N/Z ratio; (iv) effective interactions in proton-neutron asymmetric media; (v) the search for new states of matter at highest baryon densities (≈ 8ρ0 ). These research will be performed at the planned radioactive beam facilities FAIR (GSI),15 SPIRAL2 (GANIL),269 and RIA (USA).270 Information collected in planned experiments will be very important for astrophysics. Many of the nuclear reactions in stellar environments, e.g. proton and neutron capture processes, are sensitive to both the structure of the unstable nuclei involved and to the temperatures, densities, and timescales of explosive stellar events. Studying unstable nuclei thus affects both our understanding of the synthesis of the elements and the nature of the stars and their evolution as well. Acknowledgment The author is indebted to Professor Kasimir Grotowski for a critical reading of this manuscript, many years of collaboration and inordinate number of hours spent on discussions. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

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