1. W. Bauer, Michigan State University Prologue 1985XXIII“Multi-Fragmentation Reactions Within the Nuclear Lattice Model” 1993XXXI“Pion Correlations in Proton-Induced Reactions …” 1999XXXVII“Sub-Poissonian Fluctuations …” 2007XLV“Fragment Size Rankings …” 2008XLVI“Double Beta Decay” 2010XLVIII“The Physics and Astrophysics of FRIB” 2012L… anything I want …

2. W. Bauer, Michigan State University The nuclear fragmentation problem and Bormio's contributions to its solution Wolfgang Bauer

3. W. Bauer, Michigan State University Taking Things Apart – A Universal Problem Initial state “Yard Sale”

4. W. Bauer, Michigan State University Wikipedia U. Post D. Dean U. Mosel WB Bormio 1985

5. W. Bauer, Michigan State University Nuclear Fragmentation Data Proton-induced multifragmentation Here: Tevatron data (blue circles) Minimum bias data – Impact parameter integrated – Mixing of different event classes

6. W. Bauer, Michigan State University Equation of State / Phase Transition? State variables: pressure, temperature, density (internal energy, chemical potential, strangeness, …) Equation of state: relationship between state variables,. – Thermodynamic equation describing state of matter under given physical conditions – Example: Ideal gas: pV = nRT – Example: Ultra-relativistic fluid: – More realistic equations of state need to contain phase transitions, coexistence regions, critical points, …

7. W. Bauer, Michigan State University EoS for C 2 H 6 (Ethane) (a)ideal gas EoS (b)Redlich Kwong EoS, p(T,V) (c)Maxwell construction (d)First-order phase transition, terminating in critical point

8. W. Bauer, Michigan State University Nuclear EoS Can be computed, if you know nuclear force – Short-range repulsive – Medium-range attractive – Long-range repulsive (Coulomb) Here: Skyrme – Note: Coexistence region, critical point. Mean-field approximation ( Sauer, Chandra, Mosel, NPA 264 (1976) )

9. W. Bauer, Michigan State University Nuclear Matter Phase Diagram o Two phase transitions in nuclear matter:  “Liquid Gas” ç Hadron gas / QGP / chiral restoration o Problems / Opportunities: ç Finite size effects (finite size scaling!  ) ç Is there equilibrium? (  )  Measurement of state variables ( , T, S, p, …  ) ç Migration of nuclear system through phase diagram (non-equilibrium processes) (  ) ç Near critical point(s): Critical slowing down! Not sufficient time for equilibrium phase transition! (  ) NUCLEAR SCIENCE, A Teacher ’ s Guide to the Nuclear Science Wall Chart, Figure 9-2

10. W. Bauer, Michigan State University Astrophysics Connection o Exploration of the drip lines below charge ~40 via projectile fragmentation reactions o Determination of the isospin degree of freedom in the nuclear equation of state o Astrophysical relevance (origin of heavy elements!) o Review: Li, Ko, WB, Int. J. Mod. Phys. E 7

11. W. Bauer, Michigan State University Percolation o Short-range NN force: nucleons in contact with nearest neighbors o Expansion (thermal, compression driven, dynamical, …) o Bonds between nucleons rupture o Remaining bonds bind nucleons into fragments o One control parameter: bond breaking probability o Can be done on large lattices with “exact” infinite size limits o Not necessarily an equilibrium phase transition theory! o Also applies to non-equilibrium situations WB et al., PLB 150, 53 (1985) WB et al., NPA 452, 699 (1986) X.Campi, JPA 19, L917 (1986) T. Biro et al., NPA 459, 692 (1986) J. Nemeth et al., ZPA 325, 347 (1986) …

12. W. Bauer, Michigan State University Breaking Probability Determined by the excitation energy deposited Infinite simple cubic lattice: – 3 bonds/nucleon – It takes 5.25 MeV to break a bond Proton-induced: Glauber theory – p break proportional to path length through matter General relation between p break and T: G = generalized incomplete gamma function, B = binding energy per nucleon T. Li et al., PRL 70 (generalization of Coniglio-Klein for Fermi systems) Or … obtain E* or T from other model or directly from experiment

13. W. Bauer, Michigan State University Event-by-Event Analysis Near critical point, information on fluctuations is essential; averaging destroys it Promising candidates: E-by-E moment analyses e = event, n e (i) = # of times i is contained in e E-by-E for different observables can generate N -dimensional scatter plots Big question: How to sort events into classes? Natural choice: If you know control parameter, use it! (easy for theory, impossible for experiment) Closest choice: observable that is ~linear in control parameter. Attempt: use charged particle multiplicity, m.

14. W. Bauer, Michigan State University Continuous Phase Transition Near critical point, we expect scaling behavior: all physical quantities have power-law dependencies on the control parameter No characteristic scales in observables Critical exponents of power-laws are main quantities of interest – , Cluster size: – , Order parameter: – , Divergence of s: Hyper-scaling assumption (Determine 2 critical exponents sufficient)

15. W. Bauer, Michigan State University Finite Size Scaling Phase transitions strictly only defined for (almost) infinite systems Lattice calculations work on finite lattices and extrapolate to infinite lattices (hardest part!) Finite size scaling exponent, – Modify control parameter by – Modify order parameter by Opportunity for nuclear physics: Learn about extreme finite size scaling in real systems M. Thorpe, MSU

16. W. Bauer, Michigan State University EOS-TPC: Gilkes et al., PRL 73 Complete reconstruction of events: all charges recovered Assume charged-particle multiplicity is proportional to control parameter Find critical value, m c ; extract critical exponents Assuming validity of hyper-scaling: universality class of transition is completely determined Interesting data; incorrect interpretation Determining Critical Exponents

17. W. Bauer, Michigan State University mcmc M 2 ~|m-m c | - o 1 A GeV Au + C o Data are integrated over all residue sizes and excitation energies o Complete detection of all charges o Data: red circles o Assumption: total multiplicity is proportional to control parameter o Percolation model: green histograms Strong evidence for continuous phase transition  Percolation model contains critical events  Strong evidence for continuous phase transition o But: WB & A.Botvina, PRC 52 WB & A.Botvina, PRC 55

18. W. Bauer, Michigan State University ISiS BNL Experiment 10.8 GeV p or  + Au Indiana Silicon Strip Array Experiment performed at AGS accelerator of Brookhaven National Laboratory Viola, Beauliau, et al.

19. W. Bauer, Michigan State University Energy Deposition Reaction: pi,  +Au @AGS Very good statistics (~10 6 complete events) Philosophy: Don’t deal with energy deposition models, but take this information from experiment! Detector acceptance effects crucial – filtered calculations, instead of corrected data Parameter-free calculations

20. W. Bauer, Michigan State University Experiment / Theory Comparison Filter very important – Sequential decay corrections huge Good agreement

21. W. Bauer, Michigan State University Scaling Idea (Elliott et al.): If data follow scaling function with f(0) = 1 (think “exponential”), then we can use scaling plot to see if data cross the point [0,1] -> critical events

22. W. Bauer, Michigan State University Scaling of ISIS Data Most important: critical region and explosive events probed in experiment Possibility to narrow window of critical parameters –  : vertical dispersion –  : horizontal dispersion – T c : horizontal shift  Analysis to find critical exponents and temperature M. Kleine Berkenbusch et al., PRL 88 Scaled control parameter Result: = 0.5 +- 0.1  = 2.35 +- 0.05 T c = 8.3 +- 0.2 MeV

23. W. Bauer, Michigan State University Freeze-Out Density Percolation model only depends on breaking probability, which can be mapped into a temperature. Q: How to map a 2-dimensional phase diagram? A: Density related to fragment energy spectra; Coulomb many-body expansion of pre-fragments WB, Nucl.Phys. A787, 595c (2007)

24. W. Bauer, Michigan State University 625 MeV Xe 35+ Cheng et al., PRA 54 Binding energy of C 60 : 420 eV Buckyball Fragmentation

25. W. Bauer, Michigan State University Cross-Disciplinary Comparison o Left: Nuclear Multifragmentation o Right: Buckyball Fragmentation o Histograms: Percolation Models o Similarities: ç U - shape (b-integration)  Power-law for imf’s (1.3 vs. 2.6) ç Binding energy effects provide fine structure Data: Bujak et al., PRC 32 LeBrun et al., PRL 72 Calc.: WB, PRC 38 Cheng et al., PRA 54

26. W. Bauer, Michigan State University

28. Colleagues Funding Students Larry Phair Holger Harreis Marko Kleine Berkenbusch Brandon Alleman Collaborators (Theory) Scott Pratt Kerstin Paech Ulrich Mosel Ulrich Post Collaborators (Experiment) Vic Viola (and Indiana group) Konrad Gelbke (and MSU group) Don Gemmel (and ANL group) US National Science Foundation A.v.Humboldt Forschungspreis