Phase Coexistence and Phase Diagrams in Nuclei and Nuclear Matter Two avenues? Study the liquid (heat capacities) Study the vapor (vapor characterization) And a third? (Ising model, at your risk) Heat Capacities and finite size effects Clapeyron eq. and Lord Rayleigh Seek ye the drop and its righteousness… especially in Ising models Coulomb effects and heat capacities No negative heat capacities for A>60? Coulomb disasters and their resolution Back to the vapor Finite size effects Fisher generalized The complement does is it all! The way to infinite nuclear matter From Fisher to Clapeyron and back The data, finally! Phase Coexistence and Phase Diagrams in Nuclei and Nuclear Matter L. G. Moretto, J.B. Elliott, L. Phair

Motivation: nuclear phase diagram for a droplet? What happens when you build a phase diagram with “vapor” in coexistence with a (small) droplet? Tc? critical exponents? What can one really extract from the Fisher droplet model for very small systems?

Finite size effects in Ising Canonical (Lattice Gas) Grand-canonical ? finite lattice or finite drop? Density of occupied sites is fixed A0 … seek ye first the droplet and its righteousness, and all … things shall be added unto you…

(Negative) Heat Capacities in Finite Systems Inspiration from Ising To avoid pitfalls, look out for the ground state

Clapeyron Equation for a finite drop Lowering of the isobaric transition temperature with decreasiCng droplet size Clapeyron equation Integrated Lines are of constant radius. Correct for surface

Heat Capacity (boundary conditions) Evaporating droplet (Isobaric evaporation: p0 = p1 = p2) Open boundaries Periodic boundaries A0-1 p(A0-1) T(A0-1) A0 p0 T0 … p2 T2 0.5A0 p (0.5A0) T (0.5A0) … p (…) T (…) A0-A p1 T1

Example of vapor with drop The density has the same “correction” or expectation as the pressure Can we predict the bulk pressures and density by only measuring the droplet? Challenge: Can we describe p and r in terms of their bulk behavior?

Generalization to nuclei: heat capacity via binding energy No negative heat capacities above A≈60 At constant pressure p,

Coulomb’s Quandary Solutions: Coulomb and the drop Easy Take the vapor at infinity!! Diverges for an infinite amount of vapor!! Coulomb and the drop Drop self energy Drop-vapor interaction energy Vapor self energy

The problem of the drop-vapor interaction energy If each cluster is bound to the droplet (Q<0), may be OK. If at least one cluster seriously unbound (|Q|>>T), then trouble. Entropy problem. For a dilute phase at infinity, this spells disaster! At infinity, DE is very negative DS is very positive DF can never become 0.

Vapor self energy If Drop-vapor interaction energy is solved, then just take a small sample of vapor so that ECoul(self)/A << T However: with Coulomb, it is already difficult to define phases, not to mention phase transitions! Worse yet for finite systems Use a box? Results will depend on size (and shape!) of box God-given box is the only way out!

We need a “box” Artificial box is a bad idea Natural box is the perfect idea Saddle points, corrected for Coulomb (easy!), give the “perfect” system. Only surface binds the fragments. Transition state theory saddle points are in equilibrium with the “compound” system. For this system we can study the coexistence Fisher comes naturally

A box for each cluster • • • s s s Coulomb and all Saddle points: Transition state theory guarantees • in equilibrium with S Isolate Coulomb from DF and divide away the Boltzmann factor Coulomb and all

Solution: remove Coulomb This is the normal situation for a short range Van der Waals interaction Conclusion: from emission rates (with Coulomb) we can obtain equilibrium concentrations (and phase diagrams without Coulomb – just like in the nuclear matter problem)

Clusterization:cluster size distributions Fisher DF(A,T) parameterization Fisher’s formula: Clusterization in the vapor is described by associating surface free energy to clusters. This works well because nuclei are leptodermous (thin skinned) Fisher treats a non-ideal gas as an ideal gas of clusters. Surface energy Fisher’s model describes the aggregation of monomers into clusters in a vapor. That’s how the non-ideal nature of the gas is accounted for, by forming clusters. Specifically, there is a topological term in the front that describes the cluster yields at the critical point, there is a bulk term which measures how far away the system is from coexistence, and a boltzmann factor which has the surface energy of the cluster being emitted. Clusterization is described by associating surface energy to clusters and their yields. The Fisher model considers only bulk and surface terms. This approach should work well for nuclei because they are thin skinned. The resulting simplification is that one can take a non-ideal gas and treat it as an ideal gas of clusters. How shall we get calibrated with the Fisher model? (0.8minutes)

Fisher Droplet Model (FDM) FDM developed to describe formation of drops in macroscopic fluids FDM allows to approximate a real gas by an ideal gas of monomers, dimers, trimers, ... ”A-mers” (clusters) The FDM provides a general formula for the concentration of clusters nA(T) of size A in a vapor at temperature T Cluster concentration nA(T ) + ideal gas law PV = T vapor density vapor pressure 

Ising model (or lattice gas) Magnetic transition Isomorphous with liquid-vapor transition Hamiltonian for s-sites and B-external field We chose the Ising model. The Ising model describes a magnetic phase transition. It describes the behavior of a lattice of spins that have an attractive interaction when neighboring spins are parallel. And there is an exact mapping of the magnetic phase transition to the liquid-gas phase transition. In this mapping we chose up spins as occupied sites and down spins as unoccupied. Then a clustering algorithm provides abundances of clusters in the gas phase in coexistence with the percolating “liquid” cluster. So we took a 3 dimensional Ising model… (0.5minutes)

Finite size effects: Complement Infinite liquid Finite drop Generalization: instead of ES(A0, A) use ELD(A0, A) which includes Coulomb, symmetry, etc.(tomorrow’s talk by L.G. Moretto) Specifically, for the Fisher expression: Simplify Fisher for a moment to see what needs to be done Fit the yields and infer Tc (NOTE: this is the finite size correction)

Clue from the multiplicity distributions Empirical observation: Ising multiplicity distributions are Poisson Meaning: Each fragment behaves grand canonically – independent of each other. As if each fragments’ component were an independent ideal gas in equilibrium with each other and with the drop (which must produce them). This is Fisher’s model but for a finite drop rather than the infinite bulk liquid Link complement and Clapeyron

Clue from Clapeyron Ei Ef A0 A0-A A Rayleigh corrected the molar enthalpy using a surface correction for the droplet Extend this idea, you really want the “separation energy” Leads naturally to a liquid drop expression Ei Ef A0-A A

Fisher fits with complement 2d lattice of side L=40, fixed occupation r=0.05, ground state drop A0=80 Tc = 2.26 +- 0.02 to be compared with the theoretical value of 2.269 Can we declare victory? Fits are good. Not really unexpected…

From Fisher to Clapeyron

Going from the drop to the bulk We can successfully infer the bulk vapor density based on our knowledge of the drop.

d=2 Ising fixed magnetization (density) calculations  M = 0.9, r = 0.05  M = 0.6, r = 0.20  outside coexistence region  inside coexistence region ,  inside coexistence region ,  T > Tc

d=2 Ising fixed magnetization M (d=2 lattice gas fixed average density <r>) Inside coexistence region: yields scale via Fisher & complement complement is liquid drop Amax(T): Surface tension g=2 Surface energy coefficient: small clusters square-like: Sc0=4g large clusters circular: Lc0=2gp Cluster yields from all L, M, r values collapse onto coexistence line Fisher scaling points to Tc T = 0 A0 Liquid drop Vacuum Vapor Amax T>0 L

d=3 Ising fixed magnetization M (d=3 lattice gas fixed average density <r>) Inside coexistence region: yields scale via Fisher & complement complement is liquid drop Amax(T): T = 0 A0 Cluster yields collapse onto coexistence line Fisher scaling points to Tc Liquid drop Vacuum Vapor c0(As+(Amax(T)-A)s-Amax(T)s)e/T Fit: 1≤A ≤ 10, Amax(T=0)=100 nA(T)/q0(A(Amax(T)-A)/Amax(T))-t Amax T>0 L

Complement for excited nuclei A0 A0-A A Complement in energy bulk, surface, Coulomb (self & interaction), symmetry, rotational Complement in surface entropy DFsurface modified by e No entropy contribution from Coulomb (self & interaction), symmetry, rotational DFnon-surface= DE, not modified by e

Complement for excited nuclei Fisher scaling collapses data onto coexistence line Gives bulk Tc=18.6±0.7 MeV Fisher + ideal gas: Fisher + ideal gas: Fit parameters: L(E*), Tc, q0, Dsecondary Fixed parameters: t, s, liquid-drop coefficients pc ≈ 0.36 MeV/fm3 Clausius-Clapyron fit: DE ≈ 15.2 MeV rc ≈ 0.45 r0 Full curve via Guggenheim

Conclusions Ising lattices Nuclear droplets Bulk critical point Surface is simplest correction for finite size effects (Rayleigh and Clapeyron) Complement accounts for finite size scaling of droplet For ground state droplets with A0<<Ld, finite size effects due to lattice size are minimal. Surface is simplest correction for finite size effects(Rayleigh and Clapeyron) Complement accounts for finite size scaling of droplet In Coulomb endowed systems, only by looking at transition state and removing Coulomb can one speak of traditional phase transitions Bulk critical point extracted when complement taken into account.