Crossing the Coexistence Line of the Ising Model

Crossing the Coexistence Line of the Ising Model at Fixed Magnetization L. Phair, J. B. Elliott, L. G. Moretto

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 n. A(T) of size A in a vapor at temperature T • Cluster concentration n. A(T ) + ideal gas law PV = T vapor density vapor pressure

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?

Ising model (or lattice gas) • Magnetic transition • Isomorphous with liquidvapor transition • Hamiltonian for s-sites and B-external field

Finite size effects in Ising Canonical (Lattice Gas) Grand-canonical finite lattice or finite drop? ? A 0 … seek ye first the droplet and its righteousness, and all … things shall be added unto you…

Clapeyron Equation for a finite drop • Lowering of the isobaric transition temperature with decreasing droplet size Clapeyron equation Integrated Correct for surface

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

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

Clue from Clapeyron • 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 A 0 Ei Ef A 0 -A A

Finite size effects: Complement • Infinite liquid • Finite drop • Generalization: instead of ES(A 0, A) use ELD(A 0, A) which includes Coulomb, symmetry, etc. (tomorrow’s talk by L. G. Moretto) • Specifically, for the Fisher expression: Fit the yields and infer Tc (NOTE: this is the finite size correction)

Fisher fits with complement • 2 d lattice of side L=40, fixed occupation r=0. 05, ground state drop A 0=80 • Tc = 2. 26 +- 0. 02 to be compared with theoretical value of 2. 269 • Can we declare victory?

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

From Complement to Clapeyron • In the limit of large A 0>>A Take the leading term (A=1)

Summary • Understand the finite size effects in the Ising model at fixed magnetization in terms of a droplet (rather than the lattice size) – Natural and physical explanation in terms of a liquid drop model (surface effects) – Natural nuclear physics viewpoint, but novel for the Ising community • Obvious application to fragmentation data (use the liquid drop model to account for the full separation energy “cost” in Fisher)

Complement for Coulomb • NO e • Data lead to Tc for bulk nuclear matter

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

Coulomb’s Quandary Coulomb and the drop 1) Drop self energy 2) Drop-vapor interaction energy 3) Vapor self energy Solutions: 1) Easy 2) Take the vapor at infinity!! 3) Diverges for an infinite amount of vapor!!

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

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 • • Saddle points: Transition state theory guarantees • in equilibrium with S Coulomb and all Isolate Coulomb from DF and divide away the Boltzmann factor

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)

d=2 Ising fixed magnetization (density) calculations = 0. 9, r = 0. 05 M = 0. 6, r = 0. 20 M inside coexistence region , outside coexistence region inside coexistence region T > Tc ,

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

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

Complement for excited nuclei A 0 -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 Me. V • Fisher + ideal gas: Fit parameters: L(E*), Tc, q 0, Dsecondary Fixed parameters: t, s, liquid-drop coefficients • pc ≈ 0. 36 Me. V/fm 3 • Clausius-Clapyron fit: DE ≈ 15. 2 Me. V • rc ≈ 0. 45 r 0 • Full curve via Guggenheim

Ising lattices Conclusions Nuclear droplets • Surface is simplest correction for finite size effects (Rayleigh and Clapeyron) • Complement accounts for finite size scaling of droplet • For ground state droplets with A 0<