Level 0: “Requirements”

1. Models of Fragment Production Chapter 1. Evaluation Criteria A. “Stationary (EQ) statistical Models”3. SMM or MMMC4. “Thermal Model”5. Fisher model6. Lattice gas and PercolationB. “Kinetic statistical decay models”1. Standard Statistical models (SSM) - GEMINI2. EESC. True dynamical models(classical to Quantal EoM)1. Classical LJ drop collisions2. BUU3. QMD4. AQMD…

2. EvaluationCriteria - for A and B Level 0: “Requirements” ETSB - Must acquire Energy of the “Transition-State” Barrier from “bath” CDM - Must account for all Competing Decay Modes EPFE - Must not rely on the presence of Extra-Physical Forces and Entities OtS- Must be open to scrutiny (e.g., published or available code) Level 1: “Issues” PRS - Plausible Reaction Scenario exists to establish initial conditions PTS - Plausible Time Scale separation between rxn dynamics and decay PPA - Plausible Physical Assumptions PMB- Plausible MB correlations and evolution thereof Pham - Plausible Hamiltonian that can yield this state counting Level 2: “Questions” CEO - Consistency with Experimental Observations

3. EvaluationCriteria – for C Dynamical Models Level 0: “Requirements” REV. - Is the model machinery REVersible? Level 1: “Issues” Pham - Plausible Hamiltonian PIBC – Plausible Initial and Boundary Conditions. GdSt. - Stable nuclear ground state with reasonable properties. PMB: - Plausible MB correlations (e.g., pairing, quarteting) and evolution thereof dealt with. Status of correlations in the initial state. Level 2: “Questions” ERG: To what extent are identifiable degrees of freedom equilibrated.

4. EvaluationCriteria – for C Dynamical Models Level 0: “Requirements” MF+NN - Mean field propagation and two-nucleon collisions Flct - Fluctuation/bifurcation/branching Pauli - Pauli principle should be satisfied. Level 1: “Issues” Pham - Plausible Hamiltonian PWF - Plausible approximation of quantum Wave Function PIBC -Plausible Initial and Boundary Conditions. GdSt - Nuclear ground state properties must be reasonable. PMB - Plausible MB correlations (e.g., pairing, quarteting) PEQ - Plausible description of ideal systems in EQuilibrium Level 2: “Questions” CEO - Consistency with experimental observations. FFM - When and how fragments are formed. ERG - To what extent are identifiable degrees of freedom equilibrated.

5. 1) “Standard” Statistical Models - SSM CDM – Does not provide for Spinoidal decomposition channel nor, until recently, fragment expansion. Multiparticle Transition states are not defined nor are the associated transient delays. In principle this could be done as it has been for fission. PTS –No TS separation between formation/EQ and decay likely for mid-rapidity emission. ONLY plausible for PLF or TLF decays or light-ion induced reactions. PMB– To some extent accounted for by level density: a(E*) but this is not done systematically. CEO- Massive fragment production << experimental observations. Requires upgraded concept of “compound nuclei”, where thermal expansion is accounted for and the role of the surface layer may be critical. Not trivial, because of the finite range of nuclear forces, momentum dependence of the interaction and collectivity. Large lower density regions may require coupling to cluster “preformation” logic. The future of this approach is the linking to DFT to get nuclear shape (density of states) and cluster properties.

6. 2) EESM ETSB - Fragment arrives at the saddle configuration NOT at the expense of thermal bath but by converting compressional energy into translational. This makes the emission non-statistical and therefore exo-statistical arguments must be invoked. OtS - Code unavailable. Published account is insufficient for replicating published predictions. PRS – No plausible explanation on how the system arrives at the initial state of an highly excited spherical system, at ground-state matter density and no radial velocity – model predictions rely critically on what the initial state is. Self-similar expansion is unlikely to be justified in most HI reactions (e.g., disproved by BUU calculations and simple Stokes-Navier hydrodynamics).

7. EESM: High saddle Nonthermalized compressional energy (large!!) “0” “70” “70” “5” /  o = 0.3(1.7) (MeV) Subtracted from separation energy (Q)!

8. 3) SMM and MMMC CDM - Most of the energetically allowed phase space benefiting competition is excluded by disallowing thermal expansion of nuclear matter. The models are in fact not microcanonical in an essential manner. EPFC - Reliance on the presence of an active wall at the boundary of the freezeout volume. This active wall helps populating parts of the phase space that would otherwise be “out of bounds” for fragmentation. Most (!) of the configurations considered are in fact spurious – feature particles entering the freezout volume from outer space with substantial velocities (these are the particles reflected from the non-physical boundary). PRS- “Fuzzy” narrative is used to describe, how the system arrives at a freezeout configuration – cannot be expressed in meaningful cartoons.

9. MMMC/SMM: Non-physical fragment, reflected from a non-exis-tent wall – the boundary of the freezeout volume; most of the considered configu-rations contain such offen-ding fragments! Most likely configurations of thermally expanded liquid, surrounded by gas (that is free to go) are disregarded.

10. 4)``Thermal Model” PRS- Reliance on the presence of freezeout volume. PHam- No interaction between clusters. Interaction required to explore PS. It is unclear that this PS population could actually be generated by a real Hamiltonian. While the formal issue might not in practice turn our to be relevant – it should be kept in mind. Comparison to dynamical models is required.

11. 5) “Fisher’s” Model ES - Considers forming fragments of liquid within a volume filled with gas and not releasing them to surrounding vacuum (to allow detection). CDM - Gaseous phase is in fact free to go. EPFE- A container is implicit in considering gaseous phase close to the condensation point. OtS- The derivation of essential relationships not published. Link of the published model to nuclear physics scenario not clear. PRS- No “depictable” scenario of arriving at “Fisher’s point”. PPA - Origin and justification of the assumption that the L-G surface free energy is a linear function of temperature is missing. PHAM- No interaction between clusters. Interaction required to explore PS. It is unclear that this PS population could actually be generated by a real Hamiltonian. While the formal issue might not in practice turn our to be relevant – it should be kept in mind. Comparison to dynamical models is required to justify.

12. Fisher’s Model Transiently formed fragment; Formation (not an observable) governed by surface (between liquid and gas) free energy - not trivial to calculate and not provided by the model, but critical for what is being called Fisher’s scaling. Gas close to the condensation point

13. 6) lattice gas and percolation models ETSB - Fragment formation is being equated to fragment emission EPF- Reliance on the presence of a containment “vessel” (lattice boundary). PPS- Breaking of bonds is by far insufficient for the fragment being emitted – fragments have typically near-zero binding energy (separation energy), yet they are not free to go because of the Coulomb barrier. PRS- Not clear why it is relevant for the behavior of non-contained nuclear matter.

14. Mean Field Models with Fluctuation Pauli OK in principle, but ? in practice. Flct Various implementations: BOB, Df=f(1-f), Small Ntest, ... Not clear how the total energy is conserved. Not clear why bifurcation is absent before MF instability. PWF No constraint on f(r,p). Semiclassical aproximation. PIBC Applicable from t = -infinity to [1-3]00 fm/c. Some ambiguity in identifying fragments. Statistical decay is not well described (?) PEQSystems in a box. Caloric curve extracted? Is it quantal? PMB No MB. Not reliable for n, p, d, t, alpha... CEO Too small yeild of light IMF's. FFMSpinodal decompositoin. Equal-size fragmentation. ERG(N-Z)/A distribution is very sharp.

15. QMD Pauli Only in Pauli-blocking and initialization. Pauli potential is artificial and has side-effects. Flct By NN collisions and initialization. PWF Wave packet, but DxDp < hbar/2. Usually Dp=0. This is another practical source ofFlct. PIBC Applicable from t = 0 to [0-3]00 fm/c. Ground state is not very stable. Fragments often need to be recognized at an early stage. PEQ Statistics is expected to be classical. PMBSome classical MB? But not good for n, p, d, t, alpha... CEO IMF yeild is not enough in many cases. FFMFragments can be identified at an early stage.

16. AMD Pauli No way to violate it. But some approximation is introduced at several places. Flct By NN collisions and wave packet splitting. PWF Wave packet with DxDp = hbar/2. Change of wave packet shape is considered byFlct. Not clear why Flct should occur to form wave packets. PIBC Applicable from t = -infinity to +infinity, in principle. Ground state is uniquely determined and completely stable. Description of statistical decay has not been tested well. PEQ Consistent with quantum statistics and phase transition. PMBSome classical MB? But not very good for n, p, d, t. CEO IMF yeild is good if a reasonable Flct is chosen. ERG(N-Z)/A distribution is similar to statistical prediction.

17. FMD Pauli No way to violate it. Flct Not yet. (Only in b and the orientation of initial nuclei.) Deterministic dynamics is not suitable for fragmentation. PWF Slater det of Wave packets with variable width. Pham Based on realistic nuclear force. Short range correlations are considered. GdSt Good for nuclear structure problems (with projections). PIBC Applicable from t = -infinity to +infinity, in principle. Ground state is uniquely determined and completely stable. Description of statistical decay has not been tested well. PEQ Consistent with quantum statistics and phase transition. Does it depend on the definition of temperature? PMBAlpha clustering appears in structure calc. No paring?

18. CoMD Pauli Respected by introducing a stochastic process. Not derived or justified form a first principle. Flct NN collisions and the “collisions” for Pauli. PWF Wave packet with DxDp < hbar/2. Dp is small. This is another practical source of Flct. PIBC Applicable from t = 0 to [1-3]00 fm/c. Ground state is more stable than QMD. What about statistical decay? PEQ Has not been studied. Will be quantal and fermionic? PMBBetter than QMD for alpha. Still too many nucleons. CEO Some multifragmentation data are well reproduced.

19. Classical MD MF+NN, Pauli, Flct, Pham, PWF Deterministic Newtonian dynamics with two-body force. Not justified for nuclear systems. But some general features of fragmentation can be studied. PEQ Liquid-gas phase transition exists. But the statistics is classical by definition. FFMFragments can be recognized in an early stage. EGO Chaos is often discussed. How does it apply to quantum systems?

20. Importance of Surface Effects • Fragment formation always requires an increase in surface area. • Both, statistical and dynamical models must be sensitive to the cost of such an increase, expressed in terms of surface free energy (SFE), manifesting itself as surface tension (ST). • SFE decreases with increasing excitation energy. Eventually, va-nishes (at a temperature different from the critical temperature). • SFE depends on the surface profile (in matter density), which in turn depends on the (finite) range of nuclear interactions. • Surface profile changes with excitation energy (max. of entropy). • No liquid-gas (L-G) separation in small systems interacting via finite range forces (truth neglected by all (?) L-G propositions). • Vanishing of SFE (surface tension) should result in power-law distributions for fragment masses.

21. Compound Nucleus at High Excitation Weisskopf:

22. Surface Entropy

23. The Benefit of Surface Entropy; Only Massive Fragments Benefit • Known from fission studies -> asaddle/an Increased SSurface Any single saddle micro-configuration still “extremely” unlikely because of high barrier (standard problem). However, there is an “extremely” large number of such micro-configurations -> high probability for one of them being populated.

24. Promises and Challenges of Surface Physics • Surface is characterized by matter density profile, isospin profile, effective nucleon masses, surface level density parameter, surface energy, surface entropy – all of which affect decay modes and composition of fragments and all of which depend on excitation energy. • Surface free energy vanishes at -> power-law distributions. • Surface is a perfect multi-body correlator – particles have there near zero radial velocity, which allows them in concerted fashion to deform the surface, occasionally to a saddle configuration.