Nuclear Science Research Group

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1 UR-NCUR THE UNIVERSITY OF ROCHESTER Nuclear Science Research Group ROCHESTER, NEW YORK , USA Surface Matters - Prospects for Mechanistic Studies at Intermediate Energies W. Udo Schröder Nancy Jurs Triad December 005

2 1 Surface Matters: Prospects for Nuclear Mechanistic Studies at Intermediate Energies W. Udo Schröder Departments of Chemistry and Physics, University of Rochester, Rochester, N.Y. 1467, USA Abstract The phenomenology of nuclear (light-and heavy-ion) reactions is complex and suggests the presence of competing dissipative-dynamical nucleus-nucleus interactions and several m echanism s for the em ission of nucleons and nuclear clusters. An operational decom position of reaction phenomena and their understanding in context requires the development of an overall reaction scenario. This is possible to achieve in studies of surface interactions in peripheral collisions that lead to statistical and non-statistical particle and cluster emission. Exploring nuclear decay prom otes a better understanding of surface structure and interactions. Suggestions are made for a comprehensive new research strategy. I. Introduction and task definition The past two decades have seen vigorous experim ental and theoretical research in the realm of heavy- ion ( HI ) and spallat ion react ions at Ferm i and int erm ediat e bom barding energies, in which nuclear system s are produced close to the lim its of (m eta-) stability [ Lot9, Mon94, Tok95, Mor98, Tra98, Dje01, Sch01, Bor0, Ago0, Hud05, Def05, Gol04]. Much of this activity has been inspired by the opportunity to study the equation of state (EOS) of finite nuclei and nuclear m atter and its isospin (neutron-proton asym m etry) dependence [ Dan73, Ber83, LiK98, LiS01, Bar0, Cho05]. More generally, research has focused on the thermodynam ics of critically hot finite nuclei and a postulated nuclear liquid-gas phase transition. The production of such system s in nuclear reactions has been m odeled m ostly in term s of classical or sem i-classical transport theory such as the Boltzm ann-uehling-uhlenbeck (BUU) or similar approaches [ Col94, LiK98, Dan99, DiT99, Bon00, LiS01], in m olecular dynam ics [ Aic93, Luk93], or int ra- nuclear cascade concept s [ Cug97, Bou04]. At t em pt s at quant al nuclear t ransport descriptions [ Hor91, Fel97] have rem ained incom plete. Theoretical models [ Gro93, Bon95, Bon00, Lop00, Nor00, Shl05, Cho05] for the internal reorganization of initially highly disturbed nuclear systems assum e intrinsic degrees of freedom to equilibrate within the boundaries of the nuclear surface or a hypothetical freeze-out volum e. In spite of impressive theoretical developments, a comprehensive explanation of experiment al react ion and decay data has not yet been achieved. St ill lacking is a consist ent underst anding of the interplay of dynamics and induced nuclear relaxation processes, the approach to equilibrium and the emergence of m ulti-particle saddle shapes, which determ ine the subsequent disintegration of reaction prim aries and the population of the space of final reaction products. Clearly, phenom ena encountered in interm ediate-energy nuclear reactions have turned out to be more complex and interesting than previously imagined. To meet these challenges, new concepts and research strategies ( thinking out-of-the box ) are needed. Consolidation of present knowledge and planning of future research both require elucidation of the 1. Microscopic basis for the dynamic response of nuclei in energetic collisions,. Transport and relaxation phenom ena induced in complex nuclear reactions, 3. Interdependencies between nuclear structure and reactions, e.g., the influence of the nuclear surface and the effects of mechanical and chemical instabilities. Developing a com prehensive underst anding of nuclear behavior requires com pliance wit h accepted general principles of nuclear structure and nuclear phenomena. For exam ple, the facts t hat nuclei are dom inant ly quant al ent it ies wit h very diffuse surfaces and t hat generally collective degrees of freedom, however defined, interact with intrinsic m otion (viscosity) should not be ignored. I deally, a satisfactory understanding of nuclear processes also entails a relation of microscopic theory to models of macroscopic experience while avoiding over-simplification. From future nuclear reaction studies, one hopes to glean answers to question concerning the shapes and density profiles of interacting nuclei, the magnitude of in- medium nucleonic correlations (clusters and effective m asses m n *, m p * ) and scattering, the effects of isospin asymm etry on equilibrium and non-equilibrium transport and on collective m otion. Structure and dy-

3 nam ics of the surface of hot nuclei represent an interesting new challenge, for exam ple, the evolution and decay of neck-like m atter bridges between projectile and target nuclei and the m elting of the surface in m ulti-fragm ent decay. A priority exploration of the nuclear surface is important, feasible, and timely. To derive a realistic reaction scenario, in which to place specific phenom ena, a plausible strategy first addresses peripheral (surface) nuclear int eract ions of different inelast icit y, following their evolution with decreasing im pact param eter and increasing bom barding energy. I n cases where projectile/ target fusion in its degree of com pleteness (m om entum transfer) can be charact erized, st udy of com pound nuclear fusion, from sub- barrier t o high- energy fusion, presents a viable alternative. Specific surface properties, such as promotion of cluster correlations, are probed in transfer reactions of particles and clusters, their prompt non-statistical emission, as well as in m easurem ents of neck breakup particles. Finally, the influence of the surface and its therm odynam ic fluctuations on nuclear stability and decay m odes can be studied through sequential evaporation of nucleons and clusters from well characterized reaction primaries, such as fast-moving projectile-like fragments (PLFs). The initial projectile/ target isospin asym m etry provides an im portant new tool in the study of transport processes, whose thorough understanding is a prerequisite for the deduction of the nuclear EOS. I n this context, the im portance of studies em ploying secondary beam projectiles cannot be over emphasized. The following section provides som e illustrations of how pieces of experimental evidence conflicting with popular expectations can help in generating a viable overall reaction scenario. Exam ples are then given of prospects of surface reaction and decay studies. Conclusions and outlook in Section I V represent an appeal to construct a m ore com petent microscopic reaction and transport theory. II. Experiment and illustration of current model understanding A com m on feature of interm ediate-energy reactions induced by heavy or light ions consists in the fact that large am ounts of kinetic energy ( 1 GeV) of relative m otion can be transferred within short tim es ( s) into intrinsic nuclear excitation (typically = E*/A 5 MeV). Depending on density ( and tem perature (T), therm al equilibration occurs on even faster time scales (t rel ), e.g. [ Abu93], T trel ( T, ) 1 0.1T T 1 160/ T I n Fig. 1, relaxat ion t im es given by Equ. ( 1) for various relat ive densit ies and com poundnucleus particle evaporation tim es (t evap ) are com pared to t ypical heavy- ion reaction tim es (t rxn ) [Sch87] based on the 09 Bi+ 136 Xe 40AMeV one-body nucleon exchange m odel NEM Reaction Time t rxn t [Ran87]. I t suggest s t hat nuclear system s produced in energetic collisions t rxn evap equilibrate and disintegrate prom ptly, t rel i.e., before proj ect ile- like ( PLF) and t arget- like ( TLF) react ion part ners can reseparate in the exit channel. However, experim ental observations of m assive reaction fragm ents and interm ediate m ass clusters clearly disagree wit h such expect at ions. Several HI reactions, including the system s 09 Bi+ 136 Xe = 0 ( = 7-6 MeV) and 197 Au+ 86 Kr ( = Lab Lab =0.5 0 Figure 1 : Thermal relaxation, particle evaporation, and reaction times vs. nuclear temperature. (1) MeV) dem onstrate unexpected resilience of dissipative, reaction features for essentially the entire reaction cross section and the entire range of bom barding energies. This feature is illustrated in Fig. for the reaction 09 Bi+ 136 Xe [Gaw05] at two bom barding energies. Experiments quoted here and below were

4 3 Figure : Scatter plot of experim ental PLF lab energy (E plf )/ angle ( plf ) correlations from the reactions 09 Bi+ 136 Xe at Lab = 8 and 6 MeV. [Gaw06] performed at the MSU/NSCL cyclotron laboratory with a 4 setup consisting of position-sensitive Si telescopes for the sam pling of PLF fragm ents, the DwarfBall/ Wall charged-particle detector, and the SuperBall 4 neutron calorimeter. The greatest surprise of the experiments was that m assive PLFs (and TLFs) are observed at all. Furtherm ore, the m easured PLF lab energies (relative to the beam energy), plotted in Fig. vs. lab angle, dem onst rat e a classical dissipat ive orbit ing pat t ern ( Wilzcy ski plot). The superim posed solid lines represent average trajectories predicted by the above NEM simulation calculations. Decays of hot reaction prim aries have been simulated with the statistical code GEMI NI [Cha99]. Since dissipative orbiting prevails up to the highest energies m easured so far, one is led to conclude that mean-field conservative and dissipative forces change only in a gradual fashion from near-barrier to intermediate bombarding energies. Good agreem ent between data and NEM predictions has also been obtained for other correlations, such as between PLF energy and m ean atom ic number. A high-efficiency 4 m easurem ent of the joint m ultiplicities of neutrons and light charged particles (LCP) provides an opportunity to reconstruct the total therm al excitation energy E* carried by evaporated neutrons and LCPs. The correlation is one between functionals P, i.e., P E* P P Elab, mn, mlcp f ( Elab, mn, m LCP ) () Figure 3 : Scatter plot of the multiplicity of neutrons vs. that of light charged particles. The dots mark the ridge line of the most probable multiplicities for three bombarding energies. [Gaw05] rather than a one-to-one function f. I n Fig. 3, the joint m ult iplicit y dist ribut ion P( m, m ) observed for the 09 Bi+ 136 Xe reaction at Lab = 6 MeV is shown as a scatter plot of neutron m ultiplicity vs. LCP m ultiplicity. The dotted curves m ark the positions of the ridge lines of the respective m ost probable m ult iplicit ies for t hree bombarding energies, corrected for differences in detection efficiencies [ Gaw05]. The overlap of t hese lines dem onstrates independence of bom barding energy and a dom i- nantly statistical origin of the particles. By sorting data corresponding t o j oint m ult iplicit y bins defined by segments (Fig. 5) cut perpendicular to the ridge line, one n LCP

5 4 Figure 4 : Joint multiplicity distributions (filtered) predicted by various models for the 09 Bi+ 136 Xe reaction at Lab = 40 MeV [Gaw05]. obtains an approxim ate rendering of the data with respect to dissipated energy or impact parameter. Prim ary PLF reconstruction [Gaw05] from the m easured rem nants and associated neutrons and light charged particles in the sam e 09 Bi+ 136 Xe reactions provides inform ation on the m ean energy sharing between PLF and TLF as a function of total energy loss. I t turns out that, like at low energies [ Sch84], so also for the intermediate bombarding energy range, the excitation energy division evolves gradually with decreasing im pact param eter, from E * E * toward the * * equal-tem perature lim it ( PLF TLF ). This behavior is another hallmark of the dissipative reaction mechanism. Very different model predictions for the overall joint m n /m LCP m ultiplicity distribution are illustrated in Fig. 4. Prim ary fragm ent s predict ed by t he NEM [ Sch87] have been assum ed in the sim ulation shown in the top two panels. Their sequential decay has been m odeled using the statistical code GEMINI (top) or the m ulti-fragm entation code SMM [ Bon95] (middle), respectively. The bottom panel of Fig. 4 represents results obtained for QMD calculations [ Luk93], using again GEMINI for modeling of fragment decay. The comparatively best agreem ent with the experimental data is obtained with the NEM/ GEMI NI calculations (Fig. 4, top), even though these calculations over predict both neutron and LCP m ult iplicit ies for high dissipat ed energies. This discrepancy can be traced back to an under estim ation of the em ission of relatively stable, neutron rich clusters. Both SMM and QMD fragm ent at ion calculat ions yield charact erist ic m ult i- plicit y pat t erns at variance wit h experim ent al data. For example, a prom inent peak in the QMD calculations at low joint m ultiplicities not seen in the data suggests that QMD fragments are largely too cold. Similar com parisons of other experim ental and sim ulation data dem onstrate that presently no m odel perform s satisfactorily and in a consistent fashion for the entire experimental data set. NEM calculations (CLAT, [ Sch87] ) describe rather well the dissipative PLF dynam ics but do not treat direct or sequential PLF or TLF emission of intermediate-mass clusters (IMF, defined by 3 Z IMF 15). Such observations em phasize the lim ited utility of com parisons of theoretical predictions to only subsets of reaction data. Furthermore, measurements [Dje01, Gaw05] of the m arginal j oint m ult iplicit y dist ribut ions for event s wit h different clust er m ult iplicit ies have indicated an unexpected and strong dom inance of clusters in the statistical com petition of nuclear decay channels, for m edium t o high excit at ions [ Tok05]. Failures t o explain t he large clust er/ LCP branching rat ios wit hin t radit ional st at ist ical t heory suggest new st at ist ical decay phenomena. While no quantitative reaction m odel is available, the experim ental cluster data are similar to the em ission patterns of nucleons and LCPs (p, d, t, He) and support a generally dissipative react ion scenario wit h an additional source of non- st at ist ical part icles. The Galilei- invariant velocity plots of Fig. 5 show that, with increasing beam projectile velocity (left vs. right panel), the I MF velocity distributions stretch with increasing projectile velocity. Sim ulations of detector response reveal I MF cluster sources tracing kinem atically with reconstructed PLF and TLF velocity vectors. These com ponents are naturally assigned to sequential cluster evaporation from the respective hot PLF and TLF primaries. Their presence does not contradict a basically dissipative reaction environment. However, a purely sequential two-emitter m odel is unable to describe experim ental I MF (or LCP) em ission patterns. The bulging contour lines seen in Fig. 5 between the PLF and TLF cluster components represent an additional com ponent of clusters associated with the kinem atical center of an interm ediate velocity source (IVS), reported first [ Lot9] in very peripheral 09 Bi+ 136 Xe collisions associated with relatively cold PLFs and TLFs. This observation is taken as an indication of a non-statistical origin of I VS clusters, resulting possibly from fragm entation of a neck form ed between em erging PLF and TLF. Corresponding observations for other system s PLF TLF

6 5 have been made wit h LCPs and neut rons [ Que93], which also suggest I VS sources of highenergy particles. The curved lines in Fig. 5 trisecting the velocity distributions of clusters (3 Z IMF 15) for the 09 Bi+ 136 Xe reaction [ Gaw05] attem pt to separate a transversal component of IVS clusters from t hose em it t ed sequent ially from PLF ( forward sect or) or TLF ( backward sector). Differential sorting of data according t o dissipat ed energy illustrates the evolution of the different identified I MF sources of clusters wit h im pact parameter. Unlike PLF and TLF cluster components, IVS clusters are present essentially at all im pact param e- ters. Mean I VS clust er m ult iplicities increase with decreasing im pact Figure 5 : Galilei-invariant velocity plots of IMF clusters from the param eter m ore 09 Bi+ 136 Xe reaction, vs. velocity components parallel and transversal to strongly than the the beam. The curved lines provide an arbitrary separation of PLF and TLF form er com ponents sources from an intermediate-velocity (IVS) source of clusters. [Gaw05] such t hat I VS cluster em ission is emphasized am ong events selected according to high overall cluster m ultiplicities. IVS cluster particles also have a harder energy spectrum, in the respective em itter c.m. rest fram es, than those evaporated from PLF or TLF. Typical logarithm ic slope param eters for the I VS cluster energy spectra range from T IVS = 15 to T IVS = 5 MeV, for events where sequentially emitted LCPs suggest nuclear tem peratures of only T PLF = 5-8 MeV. The fact that I VS cluster m ultiplicity, atomic number and energy distributions all depend on bombarding energy points to a dynamical production m echanism. Even so, with decreasing im pact param eter the distinction between different em it t ers becom es m ore difficult and am biguous. Higher resolut ion and st at ist ical accuracy are required to perform a more specific analysis of the different cluster components. While the above (hypothetical) I VS sources of neutrons, LCPs, and I MF clusters represent prom pt non-equilibrium em ission m echanism(s), it is not known at which stage of reaction these particles are emitted and whether light- particle and cluster mechanisms are identical. Preequilibrium transport theory applicable to intermediate-energy heavy-ion, or light-ion spallation, reactions has not advanced significantly beyond model parameterizations [You94, Bou04]. However, im portant in-m edium NN interactions render bulk em ission unlikely. Consequently, such em ission processes can be used to probe surface structure and dynam ics, e.g., the presence of nucleonic correlations in the nuclear surface. It turns out that surface modes are largely responsible also for equilibrium evaporation of clusters from hot, expanded nuclei. Equilibrium and non-statistical cluster processes are cause and/ or effect of a transform ation of the dissipative reaction m echanism at interm ediate energies, which otherwise describes the overall reaction scenario. They are im portant in an arsenal of tools available to study the physics of the nuclear surface. Examples of nuclear surface studies will be discussed next. III. Surface interactions and phenomena The nuclear surface plays an im portant role in a large range of nuclear phenom ena, from elastic scattering to fusion reactions, as well as in compound nuclear decay. The specific density profile of the nuclear surface is a consequence, and therefore a m easure, of the effective forces and m ean field interactions [ Sey61]. The surface is also the dom ain where m ulti-nucleonic correlations ( clust ering) and scat t ering occur, influencing st ruct ure and nucleus- nucleus int eractions. The neutron skin [ Mye90] is an exam ple of statistical multi-nucleon correlations related

7 6 to the nuclear sym m etry energy and the iso-eos. Few-nucleon correlations in the surface are im portant [ Jon04] for the stability of exotic (halo) nuclei. There is also reason to anticipate important new insights into nuclear dynam ics from a study of surface phenom ena of hot nuclei and their transm utation. For exam ple, the long-standing puzzle of nuclear m ulti-fragmentation may well be explained by the role played by the surface of a hot nucleus in its disintegration. I n order t o develop a consist ent pict ure, establishing a plausible, holist ic pict ure and scenario of nuclear react ions of int erest here is prerequisit e to a deduct ion of specific physical param eters such as those of the iso-eos. I n com parisons to theoretical calculations, good reproduction of observed gross trends in projectile/ target reaction dynam ics with respect to changes in bom barding energy and dissipat ed energy or ot her m easures of im pact parameter are requisite for m ore detailed com parisons. Reaction codes available so far for such com parisons are based on essentially classical BUU-type, MD-type and INC cascade models. 1. Conservative surface interactions I n peripheral collisions and during the approach phase, nuclei interact dom inantly via their surfaces. Det ailed surface profiles for nuclear charge and m ass densit y dist ribut ions are predicted by m icroscopic Thom as-ferm i, Hartree-Fock and relativistic m ean field calculations, based on specific sets of effective in-m edium (contact or finite-range) forces [ Sey61, Blo78, Mye91, Chr95, Ala96, Sch84]. Calculations predict a neutron skin, a neutron-rich surface layer that has so far eluded detailed experim ental study. They also estimate the effective m ean field, fusion and fission barriers and model a host of dissipative proximity effects [Sch84]. The conservative nucleus-nucleus interaction can be tested in more detail in elastic and inelastic scattering, transfer, knock-out and, for light systems, also in fusion [Sch84, Ala96]. These methods are sensitive to strong-absorption and critical fusion radii, R SA and R Fus, respectively, providing m easures of m atter density radii of exotic nuclei near the neutron and proton drip lines [ She01, Tan95, Han03] and even their surface density profile. I t is obvious that elastic scattering and direct reactions will continue to be used in the explorations of new dom ains of exotic nuclei far off stability. Strong energy and isospin dependencies of the NN cross sections ( nn, np ) in the intermediate energy range induce sim ilar behavior of nucleus-nucleus surface interactions. I n addition to the norm al, iso-scalar interaction, the proximity potential acquires an iso-vector com ponent. The potential can be cast [Pom80] into a fam iliar separable proxim ity form, R R V ( s) I I ( s) 1 T1 T 1 T1 T R 1 R (3) where the term s denote the surface separation (s), the radii (R i ), the average (bulk) neutron excesses ( Ii ( Ni Zi ) / A i ) and isospins T i of the interacting nuclei, respectively. T ( ) 1T s is the form factor. Even though the form factor in Equ. (3) has significant m agnitude, the iso-vector potential is expected to contribute only app. 5% to the total interaction [Chr95], if neutron and proton surface density profiles are similar as for most beta-stable nuclei. System atics of reaction cross sections [ Kox87] feature bulk and surface contributions to the effective strong-absorption distance. Due to the decrease of the NN cross sections ( nn, np ) the energy dependent surface t ransparency term should increase in m agnit ude for bombarding energies approaching 100 MeV. To est ablish experimentally t he surface t ransparency is important because of its relation to the optical potential and its effect on for elastic and inelastic cross sections, in particular for transfer and knock-out processes [Fuc94, Per99, Han03, Jon04]. On the other hand, matter density distributions featuring halos or other structure associated with the presence of surface clusters should be m ore readily detectable in elastic scattering. As an example, the experimental (solid squares) angular distribution for elastic scattering of 6 He on 1 C [ AlK96] is com pared in Fig. 6 to theoretical predictions based on few-body eikonal (solid curve) and folding (dashed) models. Such comparisons have been used to determine the 3-body 6 He wave function. The st udies m ent ioned above can serve as illust rat ions of a broad dom ain of int erplay between structure and m echanistic understanding required in investigations of exotic light nuclei. The m ain goals of reaction studies in this dom ain are a determ ination of param eters and trends in effective nucleus-nucleus interactions, more specifically of the optical model potential and the

8 7 corresponding transm ission coefficients for particles and for fusion. There are obvious connect ions between the forces at work in the dom ains of elastic scat t ering and direct react ions and t hose responsible for substantially inelastic and strongly dam ped processes. Figure 6 : Experim ent al ( squares) elastic-scattering angular distribution for 6 He on 1 C and several m odel predictions. [AlK96]. Prom pt particle em ission in dissipative surface interactions Nucleon- nucleon ( NN) int eract ions in t he surface are responsible for t he attract ive nucleusnucleus potentials, as well as a host of dissipative phenomena. The latter are observable in inelastic nucleus-nucleus scattering and via the prom pt em ission of non-statistical particles. I n a one-body (mean- field) NEM pict ure [ Blo78, Ran87], dissipative forces are generated through the exchange of independent nucleons bet ween proj ect ile and t arget, by collisions of nucleons with m oving walls of the m ean field (U), or through inelastic excitations. I t is always due to randomizing in-medium NN collisions that irreversibility arises. In a short mean-free path, two-body scenario, dissipation arises entirely from collisions between individual nucleons [Alb77, Ayi87]. The scenario assumed in the, essent ially classical, BUU and BNV- t ype t ransport calculat ions invokes bot h aspect s. I n t hese approaches, the Boltzmann transport equation d f f d f v r f ru pf dt t dt coll (4) for the phase space (Wigner) distribution functions f r, p is solved for each nucleon. The mean field U(, p, ) is generally density ( ), m om entum (p), and isospin ( ) dependent. It is often derived independently from an energy density functional based on effective zero-range Skyrm e interactions, e.g. [Nor00], U( n, p) UV ( n, p) UW ( n, p) U Coul ( p) (5) Even though the Skyrm e force has zero range, the effective m ean field has an inhomogeneous Weizsäcker term U W, which also induces a specific surface layer interaction that is different from the m ean interaction in the bulk. The field (Equ. (5)) can also be expressed as a sum of an isoscalar term, U 0 ( ), and an isovector term depending on the n/ p asym m etry energy approximately as. The r.h.s. of Equ. ( 4) describes independent NN collisions (1, U : ( ) ( ) n p n p 1, ) in the approximation df dt ' ' 1 dpdp1dp 6 coll ' ' ' ' ' ' 1v1 f1ff1 f f1fff p1 p p1 p (6) They depend mainly on in-medium cross section 1 and relative NN velocity v 1. Pauli blocking factors for the scattering are abbreviated as ff. The in-medium cross section 1 is itself a function of the relative velocity, v 1, and hence of the bombarding energy. Quite characteristically, the cross section decreases with increasing bombarding energy and increasing medium density. As an exam ple, the density contours in Fig. 7 show a snapshot of a mid-peripheral 48 Ca on 11 Sn collision modeled in BUU [code: LiK98] with an isospin dependent mean field equivalent to a soft equation of state (K 00 MeV). Seen in Fig. 7 is a projectile-like fragment (PLF) trailing a neck connecting it to the target-like fragm ent (TLF). The neck m atter reflects substantial m ixing of projectile and target nucleon fluids earlier in the collision. Neck and surfaces of both PLF

9 8 and TLF are rather diffuse. At the tim e (t= 105 fm / c) of the snapshot, the neck is seen to be on the verge of disintegrating, breaking up into two or three small fragments (clusters) and a num ber of individual nucleons. Depending on duration of attachm ent of neck clusters to either PLF or TLF and t heir t ransversal m ot ion, correlations of the final cluster velocities with those of PLF or TLF are expected [ Col03, Hud05, DeF05]. In BUU approxim ation, extent and cohesion of neck matter are affected only by isospin dependent mean-field and NN collisions. Resulting correlations between the n/ p (N/ Z) ratios of prom pt nucleons and clusters can be tested experimentally. The em ergence of other nucleons seen in Fig. 7 during the reaction justifies the qualification prom pt, even though they m ay have been em itted only after the approach phase of the collision. Because of the absence of NN correlations from the BUU model true nucleation into clusters does not take place, clusters form tem porarily due to density fluctuations, but particles can easily rearrange subsequently. Specific surface interactions and the restricted quantal level density in the neck [ Tok0] are neglected. The basic concepts of classical m ean field dynamics and two-body NN collisions are ubiquitous, underlying also QMD, I NC, and hydro dynam ical Figure 7: BUU simulation of the exit channel (t=105 fm/c) of 11 Sn+ 48 Ca collision at E/A=50 MeV at b=6.5 fm. Figure 8: Illustration of the disintegration of 197 Au after annihilation of an antiproton (impact from right). Snapshot of a hydro- dynamical calculation. [Nix88] models, which differ m ostly in technical detail. The extreme two-body scenario form s the basis of intra- nuclear cascade (INC) m odels for nucleoninduced spallat ion react ions [ Cug97]. I t also underlies eikonal, m ult iple ( Glauber) scat t ering m odels of heavy-ion fragm entation reactions [ Ber88, Shu03] with weak absorption. These concepts have also been em ployed in exciton m odels [ Bla75, Kal77, You94] of pre-equilibrium cascades and associated emission of pre-equilibrium nucleons. Figure 8 renders results of a hydro-dynam ical calculation [Nix88] showing a rarefaction wave and the beginning (non-equilibrium ) decay of a 196 Pt nucleus following absorption and annihilation of an antiproton by a 197 Au target nucleus. Close to the im pact zone (on the r.h.s.), the surface of the expanding nucleus has frayed already and is ready to eject several pieces of matter of different sizes, nucleons and nuclear clusters, while the bulk remains basically intact. This picture is plausible also for reactions induced by m assive projectiles, as suggested by (the few) existing pertinent experim ents. For exam ple, in a study [ Hil88] of 3 A-MeV 3 S induced fusion reactions with 144 Sm and 154 Sm targets, non-equilibrium (NE) neutrons and protons are observed with relatively hard energy spectra and em ission patterns sim ilar to those associated with the I VS source introduced in Sect. I I (cf. Fig.5). Their m ultiplicities are not consistent with average bulk abundances and contradict one- body m odel [ Bon80, Ran87] predict ions of prom pt part i- cle emission. Non-equilibrium, prom pt em ission of energetic nucleons has been st udied experimentally [Sch01] for t he pair of reactions 11 Sn+ 48 Ca and 11 Sn+ 40 Ca at E = 35 A MeV. Here, the transfer of neutrons and protons between projectile and target nuclei appears to follow initially the gradient of the underlying isospin driving pot ent ial. Gradually wit h decreasing im pact param eter, the isospin-dependent nucleon diffusion m echanism leads to a net transfer of protons from 40 Ca projectile to the 11 Sn target, while the PLF and TLF in the 11 Sn+ 48 Ca reaction preserve the initial N/ Z asym m etry. The response of the diffusion m echanism t o t he N/ Z asym m et ry of t he react ion partners is underst ood t o result mainly from differences in avail-

10 9 Figure 7 : Ratio of m ultiplicities of non-statistical neutrons and protons em itted in the two reactions 11Sn+ 48Ca and 11Sn+ 40Ca at e = 35 MeV vs. total excitation energy. [Agn97, Sch01] n able, Pauli-restricted phase space, which should diminish with increasing bombarding energy. In contrast, an impressive gradual process of m ixing of the nucleon fluids of projectile and target is observed in t he flow of high- energy, non- st at ist ical nucleons from t he I VS source introduced earlier. Figure 9 illustrates the experim ental m ultiplicity ratio M n / M p for such nucleons as a function of total excitation or decreasing impact parameter. For 11 Sn+ 48 Ca, this ratio starts from very large values M n / M p values, approaching gradually the bulk value. For the neutron poorer system 11 Sn+ 40 Ca, the ratio rem ains at M n / M p 1.6 for all m easured excitation energies. High-sensitivity studies of the neutron-richness in non-statistical nucleon em ission are highly desirable for a num ber of cross bom bardm ents involving system s with different N/ Z asymmetries. Tentatively, the above large neutron enhancem ent observed for non-statistical nucleon em ission is taken as evidence for a surface emission mechanism cont radict ing Ferm i j et m echanisms [ Bon76, Ran87] requiring a long m ean free path. Such enhancem ent seem s to be incom patible also wit h a rapid m ixing of proj ect ile and t arget nucleon fluids favored more by the BUU than the QMD approach. One is hence encouraged to search for other, presum ably surface em ission m echanism s. Fortunately, in peripheral collisions t he com plicat ing effects of different effective m asses for neutrons and * * protons [ LiC04], m ( ) m ( ) induced by the (quadratic) m om entum dependence of the m ean field are minimized since the effective m asses approach their free values at low matter densities. Several conceptually different m echanism s can contribute to prom pt, non-statistical em ission of fast nucleons. I n a one-body diabatic [ Nor00] process, the m oving walls of the m ean field U of a fast-approaching nucleus can transfer m om entum to individual nucleons and cause t heir ej ect ion. The spect rum of such nonstatistical nucleons should depend on the geometrical overlap of proj ect ile and t arget nuclei and show noticeable im pact param et er dependence. Their kinem a- t ical cent er should coincide wit h t he overall cent er- ofmass, and their m ean kinetic energies should increase approxim at ely in proport ion t o t he relative nucleusnucleus velocity. I ncreased ejection efficiency (particle multiplicity) of this process is expected at increased bombarding energies, in contrast to ejection induced in NN collisions. Because of a relat ively weak isospin dependence expected for the m ean field in peripheral collisions, neutrons and protons should be em itted with m ultiplicities reflecting the respective local abundances and show sim ilar energy spectra. Com bining t he t wo feat ures m ent ioned for diabat ic nucleon em ission, one should observe an anti-correlation between the neutron/ proton m ultiplicity ratio and the average kinetic energy of these particles, for a given bom barding energy. No detailed theoretical calculations are available as yet to suggest a magnitude for the effect. The second process contributing to the prom pt, em ission of fast NE nucleons is due to NN scattering in the overlap region of projectile and target m ass density distributions (cf. Fig. 7) modeled by Boltzm ann collision term s in Equ. (6). The particles em itted in this process should display kinematical symmetry with respect to the NN center-of-mass. The dependence of NN scattering on relative energy and local matter density should induce a corresponding behavior of t he average m ult iplicit ies for NE nucleons. At high nuclear t em perat ures, Pauli blocking becomes less effective, increasing slightly the NN scattering probability. This cancellation should therefore be less important in peripheral collisions. The relative im portance of NN scattering for NE nucleon ejection can be judged from the m agnitude of the (two-body) surface viscosity with which the NE nucleon m ultiplicity should be directly correlated. A distinctive signature of this correlation is the strong iso-spin dependence of the NN scattering cross section and hence of viscosity. Calculations performed [ Col98] in the p

11 10 framework of isospin dependent BNV transport have already demonstrated a dependence of relat ive probabilit ies for dissipat ive and fusion- t ype collisions on average neut ron/ prot on densities, but attributed the effect to the iso-spin dependence of the m ean field. NN collisions also contribute t o t he phenom enon, as shown [ LiC98] for Sn+ Sn react ions wit h large proj ectile/target isospin asymmetries. Studies of NE surface cluster emission could help solve the puzzle. Very little is known about such processes in com plex nuclear reactions at interm ediate energies. In peripheral heavy-ion reactions, neutron-rich NE clusters have been observed [ Lot9, Sch94, Mon94, Tok95, Sos01, DeF05] with em ission patterns characteristic of an I VS source. These clusters could com e from the decay of a neck zone [ Tok95] but could com e from an earlier breakup phase. Correspondingly, the IVS cluster N and Z distributions should reflect either the mixed projectile/target matter in the neck region or indicate an angle-dependent strong m em ory of projectile and target N/Z ratios. I f NE nucleons and clusters are produced contem poraneously in sim ilar m echanism s, their production yields should exhibit characteristic correlations (or anticorrelations) in m ultiplicity and energy spect ra. I n a sim ple classical random- coalescence approach, t he m om ent um dist ribut ion of a clust er ( N, Z) produced by a single effect ive source is est im at ed by simple expressions, e.g. A 1 N Z (, ) S N, Z, E, s A N Z 4 p0 V 3 n p p 3 3 n pp pa dp N! Z! eff 3 A dpn dpp n p d P N Z d P d P (7) 3 3 in term s of the probabilities for single nucleons, d P dp. Here, the volum e V, the N and Z values, as well as the neutron and proton densities, n and p, are those of the effective source (e.g., the neck zone), and p 0 is an adjustable param eter denoting a coalescence radius in mom entum space. The kinetic factor S represents sym bolically the successive pick-up/attachment probabilit ies of t he involved nucleons. I n principle, S should be a quant al t ransm ission/ attachm ent factor depending on Coulom b barrier and the internal cluster structure, in particular on the num ber of accessible states, but also on relative kinetic energies (E), Q values, and channel spins (s). For cluster form ation in the diffuse nuclear surface, such term s should be crucial. Nevertheless, this im portant term is consistently ignored in coalescence analyses where the factor S is set to unity. There have been few measurements [e.g., Hag00] of com plex part icle em ission interpreted in t erm s of the simple one- source coalescence scenario. The considerably simpler p-induced spallation reaction m ay be m ore appropriate for applicat ions of coalescence concept. However, systemat ic st udies [ Her06] of com plex part icle em ission in p- induced reactions at E p = 1-.5 GeV on a variety of targets have dem onstrated an inadequacy of the coalescence model to account consistently for com plex particle yields and spectra. Model fits require readjustm ent for each particle and each target. This deficiency exhibited by the coalescence m odel in relatively sim ple situations casts doubt on its viability for an interpretation of cluster em ission in heavy-ion reactions. The im port ance of t he dissipative capture and at t achm ent dynam ics for t he cluster em ission probability is illustrated schematically in Fig. 10. Here the summed frequency of a cluster m ass is plotted vs. m ass num ber A for a given num ber of reaction events in 1 C elastic (top) and inelastic Figure 10: Cluster size distributions for p-induced breakup of 1 C in a schematic model with low (top) and high (bottom) viscosity [Qui05] (bottom) breakup induced by energet ic prot ons. The process has been sim ulated in a schem atic linear reaction model [ Qui05] without (top) and with (bottom ) dissipation. The results suggests that at zero surface viscosity there is

12 11 little propensity for 1 C to break up. The random breakup m ass distribution is peaked at symm etric splits. I n contrast, a high 1 C surface viscosity enables the projectile to drag out m ass from the target nucleus, producing a U-shaped cluster m ass distribution (bottom panel). I n this schematic model, viscosity simulates the kinetics of nucleon capture and attachment to a cluster formed at the end of an intranuclear cascade. I n surface emission modes probed in peripheral collisions, viscosity effects should be even m ore im portant. Here they are related to the internal structure (density of states) of the clusters, which have to be able to absorb collision energy. In this fashion, cluster form ation depends on therm odynam ic cluster properties as well as on the kinetics of the aggregation process. The above qualitative discussion attributes to the internal structure of clusters an im portant role in fast aggregation and nucleation processes. Methods of identifying clusters in theoretical m odel sim ulations, e.g., in deriving BUU or QMD based predictions for cluster production, should consider t he int ernal ( quant al) st ruct ure of these light nuclei. Random proxim it y of nucleons in phase space may be a necessary but insufficient condition for cluster formation. The m obility of nucleons exchanged between interaction partners trivially influences the m akeup of clusters prom ptly em itted from the projectile/ target overlap zone, e.g., from the neck. This m obility can be m easured in isospin diffusion experim ents m onitoring the consistencies of PLF and TLF in peripheral collisions. Early experim ents [Sch01] quoted above (cf. Fig. 9) suggest t hat int eract ion t im es in peripheral react ions vary sufficiently st rongly wit h im pact parameter to allow one to study the tim e dependent m ixing process of projectile and target nucleons. This process should be observable in the changing A/ Z distributions of clusters form ed dynam ically in surface interactions, suggesting a series of prom pt cluster/ PLF (TLF) coincidence experiments. The NE part icle and clust er dist ribut ions obt ained in exclusive coincidence experim ents should provide im portant inform ation on the neck dynam ics in heavy-ion collisions. Quantal effects m ay play an im portant role in the form ation and decay of the neck zone, which should receive attention. 3. Role of the surface in equilibrium nuclear decay I n addit ion t o prom pt clust er em ission in peripheral ( and probably cent ral) collisions, sequent ial clust er em ission has been observed experimentally [Lot9, Dj e01, Sch03] in coincidence with projectile-like fragments. For exam ple, Fig. 5 illustrates the kinem atical tracking of cluster em ission patterns with PLF velocity not anticipated by conventional statistical m odels, or by m odels developed for statistical m ulti-fragmentation. Traditional estim ates of reaction and relaxation tim es (cf. Fig. 1) assert that reaction prim aries are so unstable that they vaporize before significant accelerat ion in t he Coulom b field could occur. This cont radict ion can be resolved by generalizing the view of the traditional com pound nucleus (CN) to include surface structure, in particular its density of states [ Tok81]. I ncreased overall CN stability results from expansion cooling, and unusual clust er/ part icle branching is produced by effect s of surface entropy. This has been recognized in recent work [ Tok05] m odeling, largely analytically, t he behavior of hot nuclei in term s of a harm onic interaction Ferm i gas (HI FG) m odel. The decay patterns of such nuclei resemble those of a phase transition. I n statistical theory, the self-organization of an isolated system is m odeled m aximizing entropy S. Considering an excited nucleus as an interacting Ferm i gas, this m axim ization is achieved through an expansion to equilibrium density eq, which can be significantly lower than the ground state m atter density 0. For nuclei, the quantum features of the diffuse surface play an im portant role, since here the num ber ( ) of available states is com paratively large, m aking the surface entropy (S = ln ) a decisive factor in the quantal expansion process. Defining the nuclear temperature T of the microcanonical nucleus in the usual fashion, the entropy s (in units of the Boltzmann constant k B ) of a nucleon is given by s( T, ) m * kf ( ) T (8) Here, the density dependence of the Ferm i m om entum k F ( ) (3 ) is responsible for the enhanced entropy per particle in diffuse density regions such as the nuclear surface, and the quantity m* is the effective nucleonic m ass, accounting for momentum-dependent interactions. This relation leads to the free energy per nucleon [Sto79], 1 3

13 1 * m f ( T, ) (0, ) T s( T, ) (0, ) T kf( ) The quantity (0, ) is the total energy ( EOS ) per nucleon (kinetic plus potential) in T=0 matter of density. The thermal excitation energy per nucleon is given by the difference (9) * * ( T ) T, ( T ) 0, (0) a( ( T ), m ) T (10) Equations (8)-(10) describe the essence of the expansion of a hot nucleus and its dependence on t he isoeos. Finally, t he int roduct ion of t he level densit y parameter a est ablishes a conceptual connection between the nuclear EOS, the param eters of nuclear expansion, and nuclear decay. Obviously, statistical branching in nuclear decay occurs according to entropy, i.e., the level densities of different daughter nuclei. Approximately, one can determine the contribution of the intrinsic, thermal energy E to the total excitation energy, E*, by subtracting a m ean com pression energy, E E (1 ), * comp B equ o which is approxim ated here by a quadratic form involving the binding energy E B. The equilibrium sit uat ion, i.e., t he act ual densit y of t he nucleus, is defined by m axim izat ion of t he entropy, * th S T a E a E E (11) * * * (, ) ( ) th( ) ( )( comp( )) of the expanded nucleus by varying the density for the fixed total excitation energy. For simplicit y, a radial densit y profile corresponding t o self- sim ilar expansion of t he ground st at e density distribution is em ployed. As suggested by experim ental system atics [ Tok81], the level density parameter has density dependent volume and surface terms. The relative importance of the surface term to the total level density increases with the surface area and is higher for lighter and expanded nuclei ( < 0). Since the expansion against the EOS restoring force requires energy, the temperature T of the expanded nucleus must decrease to (1) * 1 Eth equ 1 3 T ( ) E E (1 ) a a o o * equ B o The schematic m odel caloric curve (Equ.(1)) graphed in Fig. 11 dem onstrates a st riking nonm onotonic behavior. The theoretical relation between tem perature and excitation energy depends also on t he effect ive nucleon m asses, a dependence t hat flattens [ Sob04] the caloric curve som ewhat. The general dependence of the caloric curve in the HI FG m odel asserts that there is a m axim um tem perature that a nucleus can sustain. This is a unique model prediction that follows wit hout reliance on hypothetical constructs of ext ernal gas pressures. Equation ( 1) further suggests that it is possible in principle to study Figure 11: Equilibrium temperature the isoeos by m easuring the caloric curve T(E*) of vs. excitation energy for an A=00 finite nuclei. I t is encouraging that the im portant feature of the caloric curve showing a plateau of constant nucleus. tem perature for a range of excitations is borne out by experimental caloric curve systematics [Nat0]. The decrease in therm al energy due to expansion cooling increases the overall lifetim e of the nucleus against particle decay [ Tok05], explaining qualitatively why it has been possible to observe sequential (PLF) cluster decay following heavy-ion reactions. Surface entropy also changes the statistical competition between evaporation of clusters and light particles [Dje01]. The fact that a hot nucleus can generate a dinuclear system with significantly increased surface area produces a new, prom pt fission instability. I t turns out that the several tens of MeV high barriers (V int ) hindering the em ission of m assive clusters (3 Z cl 10) from heavy nuclei do not change substantially when expansion is taken into account. However, at total excitation

14 13 energies of the order of E*/A = (4-5) MeV, the di-nuclear saddle-point configuration of a cluster (A cl, Z cl ) and the residue (A res =A-A cl, Z res -Z cl ) attains a higher entropy, S din, than that (S m ) of the parent mono-nucleus (A, Z), E * Vint E * Vint Sdin( Acl, Zcl, Ares, Z, E*) S( Acl, Zcl, ( )) S( Ares, Zres, ( )) (13) A A Therefore, the fission-like process can overcome unfavorable energetics (V int» T), provided that there is a sufficiently large entropy gain. I n a canonical description, this is due to the relation between the corresponding free energies F: cl res F E a T F E V a T (14) * * m m m m din din int din din Here, a m and a din are the corresponding level density param eters. The entropy gain in cluster form ation is the larger, the lower the average density and the higher the surface diffuseness of the cluster are. Entropy associated with internal structure has to be added to that of the final state, amplifying the im portance of the entropy contribution to the free energy (Equ.(14)). Formation of clusters with a higher level density is favored relative to nuclei with only a few excited levels and can dominate even the release of nucleons or alpha particles. With increasing excitation the surface of a hot nucleus softens, with an HIFG surface tension vanishing as E * S const 1 4 r 0 a T (15) Here, is the surface area and a is the surface level density param eter, is given relative to the liquid-drop surface energy E, E E A F A r A and F is the ratio of this area to the surface of the corresponding spherical nucleus [Tok81]. Equation (15) describes gradual softening of the surface with increasing temperature, where the surface tension vanishes at the critical tem perature T. Using liquid-drop m odel surface energy constants of b s = MeV, a radius param eter of r 0 =1.fm, and a = 0.74 MeV -1 for the nuclear ground state, one can derive an upper limit for this critical temperature, (16) T a 8MeV (17) For an expanded nucleus, the specific surface energy should be sm aller, and the level density param eter larger, than for its ground state density. A m ore precise value for T is presently not available, but the critical tem perature predicted by the HI FG m odel is m uch sm aller than the approximately 18-0 MeV predicted by standard matter calculations [Ban90]. A better theoretical determ ination of this critical tem perature in a m ore detailed and rigorous t heory is highly desirable. A body of t heoret ical work [ Col94, LiK98, Nor00, Cho05] discusses critical temperatures for mechanical and chemical instabilities of nuclei or nuclear matter to develop. I nstabilities of hom ogeneous nuclear droplets with sharp, structure-less surfaces have been studied [ Nor00] based on effective (Skyrm e-type) in-m edium forces. Neglecting surface entropy, isentropic expansion m odes of such nuclei are calculated. With these restrictions [Nor00], one predicts a temperature and isospin dependent surface tension of the form as s 1 a 1 T (18) 4 r 0 q Here, the constant a and the central nuclear density ( q ) are determined by the assumed Skyrm e force. The parameter defines a soft ( = MeV - ) or stiff ( = MeV - ) EOS. The isospin term ( ) in Equ. (18) ensures that the surface tension is sm aller for more iso-asymmetric systems. According to this equation, the surface energy of a sharp-surface nucleus increases with temperature, opposite to what is expected for real nuclei. Assuming the sam e density dependence of the Skyrm e forces for the entire range of densities and excitation

15 14 Figure 1: Surface tension as a function of total excitation energy (Es), as predicted by SkM* Skyrme forces for an expanded nucleus (A= energies of interest, one simulates the neglected surface entropy effect by inserting into Equ. (18) the HIFG equilibrium density and tem perature functions. The result depicted in Fig. 1 indicates that, for an A= 00 nucleus, t he surface t ension should vanish at E* 1.7 GeV (T < 8 MeV), in agreem ent with the critical temperature of Equ. ( 17). At t his t emperature the surface m elts initiating com plex nuclear disintegration, a process that has hallmarks of a phase transition. Together with the EOS defining the bulk com pressibility and the restoring force for bulk norm al m odes, the surface t ension plays a crucial role in t he developm ent of nuclear inst abilit ies. The liquiddrop m odel predict s [ Boh69] for t he st iffness of a surface m ode of m ultipolarity in a nucleus (A, Z), in the usual notation for liquid-drop m odel parameters b s and b 3e 5r, C 0 bs 3 5 ( 1) Z C ( 1)( ) A bc (19) 4 ( 1) A 1 3 The energy of the mode is related to stiffness C and deformation parameter as E = C. The stiffness parameter C is mainly determined by the first (surface) term in Equ.(19), which is proportional to the surface energy. For Ferm i m atter, surface and Coulomb parameters have temperature dependencies of the form [Kol04] 5 4 cr 4 C cr T T T bs ( T ) 17.5 MeV b ( T ) 0.7MeV T T MeV Figure 13: Liquid- drop model stiffness parameters C for =, 4, 6, 8 (bottom to top) and T c = 8 MeV. (0) Taking for the critical tem perature the value quoted in Equ. (17), i.e., T c = T = 8 MeV, one obtains results shown in Fig. 13. A fission-like ( = ) instability arises for a 11 Sn nucleus already at approxim ately T = 4.3 MeV. Higher tem peratures are predicted for the limiting temperatures for higher m odes, However, as gathered from Fig. 13, the lim iting tem perature is non-linear in the m ultipolarity. As the m ultipolarity increases, t he lim it ing t em perature T approaches the series lim it, lim T T. For temperatures approaching this limit, a large number of nearly degenerat e surface m odes can be excit ed, providing for a range of regular and irregular nuclear (surface) shapes. The above considerations can be made more accurately within the HI FG m odel, which gives a consistent account of expansion cooling and surface m elting effects. Here surface oscillations are driven by the corresponding free energy, c