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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.