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!

L. G. Moretto, L. G. Moretto,

J.B. Elliott, J.B. Elliott,

L. PhairL. 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?

Finite size effects in Ising

… seek ye first the droplet and its righteousness, and all … things

shall be added unto you…

?A0

Tc

Tcfinite lattice

or finite drop?

Grand-canonical Canonical (Lattice Gas)

(Negative) Heat Capacities in Finite Systems

• Inspiration from Ising– To avoid pitfalls, look out for the ground state

• Lowering of the isobaric transition temperature with decreasiCng droplet size

Clapeyron Equation for a finite drop

pp expc0

A1 3T

p exp

K

RT

dp

dTHm

TVm

Clapeyron equation

p p0 exp Hm

T

Integrated

Correct for surface

Hm Hm0 c0

A2 / 3

AHm

0 K

R

Heat Capacity (boundary conditions)

A0

p0

T0

A0-Ap1

T1

…p2

T2

Evaporating droplet (Isobaric evaporation: p0 = p1 = p2)

T A T A

11

A01/ 3 1 y 1/ 3

y A0 A

A0

Open boundaries

T A T A

1y2 / 3

A01/ 3 1 y

A0-1p(A0-1)T(A0-1)

0.5A0

p (0.5A0)T (0.5A0)

…p (…)T (…)

Periodic boundaries

Example of vapor with drop

• The density has the same “correction” or expectation as the pressure

pp expc0

A1 3T

p exp

K

RT

expc0

A1 3T

exp

K

RT

Challenge: Can we describe p and in terms of their bulk behavior?

Generalization to nuclei:heat capacity via binding energy

• No negative heat capacities above A≈60

dpp

A T

dA p

T A

dT 0

At constant pressure p,

p

A T

p

T

Hm

A T

p

dT A

pHm

T 2

T

A p

T

Hm

Hm

A T

Hm B(A)T

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!!

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,

E is very negativeS is very positive

F can never become 0.

FETS0

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

• Saddle points: Transition state theory guarantees • in equilibrium with S

•

•

s s

nS n0 exp F

T

Coulomb and all Isolate Coulomb from F and divide

away the Boltzmann factor

•

s

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)

• 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.

nA(T)q0A exp

AT

c0A

T

q0A exp

AT

c0A

Tc

c0A

T

Clusterization:cluster size distributions

Surface energy

Fisher F(A,T) parameterization

nA (T )exp F A,T T

F A,T A c0A T ln A

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

v AnAA

T vapor density

p T nAA

T vapor pressure

Finite size effects: Complement

• Infinite liquid • Finite drop

nA (T)C(A)exp ES (A)

T

nA (A0,T)C(A)C(A0 A)

C(A0)exp

ES (A0,A)

T

• 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:

nA (T)q0

A A0 A

A0 exp

c0 A (A0 A) A0

T

Fit the yields and infer Tc (NOTE: this is the finite size correction)

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 1 2 M = 0.9, = 0.05 M = 0.6, = 0.20

, inside coexistence region outside coexistence region inside coexistence region , T > Tc

• Inside coexistence region:– yields scale via Fisher

& complement– complement is liquid

drop Amax(T):

d=3 Ising fixed magnetization M (d=3 lattice gas fixed average density )

T = 0

T>0

Liquiddrop Vacuum Vapor

L

L

A0

Amax

Amax T A0 nA T AA1

AAmax

nA T exp F T F c0 A Amax T A Amax T

T lnA Amax T A

Amax T

• Cluster yields collapse onto coexistence line

• Fisher scaling points to Tc

c0(A+(Amax(T)-A)-Amax(T))/T

Fit: 1≤A ≤ 10, Amax(T=0)=100

nA(T

)/q

0(A

(Am

ax(T

)-A

) Am

ax(T

))-

Complement for excited nuclei

• Complement in energy– bulk, surface, Coulomb (self & interaction), symmetry, rotational

• Complement in surface entropy– Fsurface modified by

• No entropy contribution from Coulomb (self & interaction), symmetry, rotational– Fnon-surface= E, not modified by

nA T exp F T F F f Fi

E c0 A A0 A A0

T lnA A0 A

A0

A0-A A

Ff Ebind (A,Z) Tc0

Tc

A ln A

Ebind (A0 A,Z0 Z) Tc0

Tc

A0 A ln A0 A

E rot A0 A, A ECoul Z0 Z,Z;A0 A, A

A0

Fi Ebind (A0,Z0) E rot A0 Tc0

Tc

A0 ln A0

Complement for excited nuclei• Fisher scaling

collapses data onto coexistence line

• Gives bulk

Tc=18.6±0.7 MeV

• pc ≈ 0.36 MeV/fm3

• Clausius-Clapyron fit: E ≈ 15.2 MeV

• Fisher + ideal gas:

p

pc

T nA T

A

T nA Tc

A

• Fisher + ideal gas:

v

c

nA T A

A

nA Tc A

A

• c ≈ 0.45 0

• Full curve via Guggenheim

Fit parameters:L(E*), Tc, q0, Dsecondary

Fixed parameters:, , liquid-drop coefficients

ConclusionsNuclear dropletsIsing lattices

• 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 pointextracted whencomplement takeninto account.