Content Heavy ion reactions started fragmenting nuclei in the 1980’s. Its study taught us that nuclear matter has liquid and gaseous phases, phase changes, critical behavior, and many rich phenomena. Here a summary of theoretical efforts leading to the understanding of the thermodynamics of nuclear matter will be presented, including recent ones that extend the phase diagram in a new direction: isospin. The nucleus is a liquid Experimental evidence Theoretical models Current efforts Conclusions

The nucleus is a liquid

1980: The nucleus is a liquid En early 80s the following was well known: The nuclear density The nuclear compressibility The nucleus was a Fermi system How does the nucleus behave at higher temperatures? The rest can be inferred by interpolation Behaves as a Fermi gas at low densities Compressibility: K=n(dp/dn)T (proportional to the slope) Stable nucleus: p = 0, density ~ 0.15 fm-3 What were the implication of these facts?

1980: The nucleus is a liquid Such behavior p – r is similar to that of liquids 1983-1985 Bertsch Siemens López . . . Could there be a change of phases líquid-gas in the nucleus? Was there any experimental evidence?

Experimental evidence

1980-2010: experimental evidence Fragmentation exists Gaseous phase: Separated particles Larger fragments decrease at higher energies  Evaporation Liquid phase: continuous medium

1980-2010: experimental Evidence Evidence of thermalization Kinetic energy and momentum of fragments follow Maxwell distributions  Thermalization

1980-2010: experimental Evidence Temperature measurements

1980-2010: experimental Evidence Temperature Caloric curve Temperature does not increase with an increase of energy  Change of phase

1980-2010: experimental Evidence Such evidence suggested this reaction mechanism: Heating compression Change of phaseExpansion

Theoretical models

1980-2000: theoretical models Statistical Models SMM Transition State Sequential Decay Dynamical Models Boltzmann, Vlasov, … Molecular Dynamics Quantum Classical

1985: SMM 1985-1990 Bondorf Schultz Barz Donangelo Sneppen Botvina Mishustin López … SMM “explores” the phase space during the “freeze-out” to determine the most probable breakups.

Data: peripheral collisions with Au 1985: SMM Theory: SMM Data: peripheral collisions with Au Successes: Reproduced mass distributions Very good PR Deficiencies: Did not use unique parameters No Kinetic information No de-excitation of fragments No inter-fragment interactions Unrealistic Freeze-out volume . . . Multics-NPA650 (1999) 329

1987: transition state treatment 1987 - 1990 Added inter-fragment interactions Added de-excitation of fragments But … it didn’t contain fragment dynamics and the freeze out volume was still a perfect sphere . . .

1988-1990: sequential decay Assumes a sequence of fissions Uses fission barriers Conserves energy and momentum De-excites fragments by evaporation But . . . Doesn’t explain non-sequential decays

> 1990: dynamical Models Kinetic Theory: Nordheim, Boltzmann, Vlasov 124Sn+124Sn, E/A=50 MeV Neck fragments b=7 fm multifragmentation b=0 fm Correct dynamics and geometry Uses Fermi energy distribution But . . .

>1995: Dynamical Models Kinetic Theory: Nordheim, Boltzmann, Vlasov Density evolution Momentum space - Use mean fields - Use “test particles” instead of real nucleons - Use Gaussian density distributions - Do not produce fragments - Fragments are identified by hand - Fragments do not de-excite by themselves - Use different parameters in different reactions

>1995: Dynamical Models Quantum Molecular Dynamics Pauli Potential Nuclear Force Coulomb Energy Kinetic Energy - Uses Gaussian distributions for p & n - Uses Mean Field Potentials - Obeys Fermi Statistics Uses correct dynamics & geometry But . . . .

>1995: Dynamical Models Quantum Molecular Dynamics - Does not produce fragments without external help - Does not de-excite fragments without external help Does not use a unique set of parameters But it is widely used (good PR)

>1995: Dynamical Models Classical Molecular Dynamics - Uses inter-nucleon potentials - Uses protons & neutrons Correct dynamics and geometry - Doesn’t use “test particles” - Does not use Gaussian distributions of density - Produces fragments without external help - De-excites fragments naturally Uses a unique set of parameters But . . . it uses classical mechanics and doesn’t obey Fermi statistics

>1995: Dynamical Models Classical Molecular Dynamics Potential Solve equation of motion (Verlet) Recognize clusters (MSE) Track evolutions in space-time

>1995: Dynamical Models Classical Molecular Dynamics

>1995: Dynamical Models Classical Molecular Dynamics Determine mass distributions It has been used to study: Critical phenomena Caloric curves Isoscaling

What do we know?

We now know much about the equation of state of nuclear matter What do we know? We now know much about the equation of state of nuclear matter

In a nutshell Nuclei in reactions are compressed, heated, and expand while going from a liquid phase to a liquid-gas mixture

Current efforts

Investigating neutron-rich nuclei Current efforts Investigating neutron-rich nuclei

The study of neutron-rich nuclei Current efforts The study of neutron-rich nuclei

Current efforts Isoscaling

Infinite nuclear matter Procedure to study phase diagram of nuclear matter Create an infinite system Select density r Select Temperature Equilibrate Measure Binding energy E(r,T) Pressure p(r,T) Compressibility K(r,T) Obtain equation of state Study other properties of system

System maintains crystalline structure at higher T Symmetric matter Binding energy of Pandha medium Higher T System maintains crystalline structure at higher T System produces pasta at r < 0.13 fm-3 at all T

Symmetric matter Low T structures T=1 MeV T=0.001 MeV

New phases in the nuclear phase diagram! Symmetric matter Significance? New phases in the nuclear phase diagram!

Pasta – Crystal: Latent heat needed 1st order phase transition? Symmetric matter Significance? Liquid-Gas - Pasta, a glass transition? Liquid-Gas - Crystal, freezing? Pasta – Crystal: Latent heat needed 1st order phase transition?

Extend what we know into the new dimension: Isospin Symmetric matter What’s next? Extend what we know into the new dimension: Isospin ?

Conclusions

Extra slides

There are more states available than nucleons Quantum caveats I There are more states available than nucleons Pauli blocking is not restrictive

Quantum caveats II Inter particle distance is larger than de Broglie wavelength for all cluster sizes