Phase Coexistence and Phase Diagrams in Nuclei and Nuclear Matter Two avenues? –Study the liquid (heat capacities) –Study the vapor (vapor characterization)

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.