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 nA(T) of size A in a vapor at temperature T

• Cluster concentration nA(T ) + ideal gas law PV = T

€

ρv = AnAA

∑ T( )

vapor density

€

p=T nAA

∑ T( )

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?

€

H =− J ijsisjij

∑ −B sii=1

N

∑

€

J ij =J, i and j neighboring sites

0, otherwise

⎧ ⎨ ⎩

€

s=1

-1

⎧ ⎨ ⎪

⎩ ⎪ ⇒s+12

=1, occupied

0, empty

⎧ ⎨ ⎪

⎩ ⎪

• Magnetic transition

• Isomorphous with liquid-vapor transition

• Hamiltonian for s-sites and B-external field

Ising model (or lattice gas)

Finite size effects in Ising

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

shall be added unto you…

?A0

€

Tc

Tc∞finite lattice

or finite drop?

Grand-canonical Canonical (Lattice Gas)

• Lowering of the isobaric transition temperature with decreasing droplet size

Clapeyron Equation for a finite drop

€

p = p∞ expc0

A1 3T

⎡ ⎣ ⎢

⎤ ⎦ ⎥= p∞ exp

K

RT

⎡ ⎣ ⎢

⎤ ⎦ ⎥

€

dp

dT=

ΔHm

TΔVm

Clapeyron equation

€

⇒ p ≈ p0 exp −ΔHm

T

⎛

⎝ ⎜

⎞

⎠ ⎟Integrated

Correct for surface

€

ΔHm = ΔHm0 + c0

A2 / 3

A= ΔHm

0 +K

R

Example of vapor with drop

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

€

p = p∞ expc0

A1 3T

⎡ ⎣ ⎢

⎤ ⎦ ⎥= p∞ exp

K

RT

⎡ ⎣ ⎢

⎤ ⎦ ⎥

€

ρ =ρ∞ expc0

A1 3T

⎡ ⎣ ⎢

⎤ ⎦ ⎥= ρ∞ exp

K

RT

⎡ ⎣ ⎢

⎤ ⎦ ⎥

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

Clue from the multiplicity distributions

• Empirical observation: Ising multiplicity distributions are Poisson

€

P ma( ) =ma

ma e− ma

ma!

– 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

A0

A0-A A

Ei

Ef€

ΔHm → ΔHm0 +

c0

A1/ 3

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−τ A0 − A( )−τ

A0−τ

exp −c0ε Aσ + (A0 − A)σ − A0

σ( )

T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

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

€

nA (T) = q0A−τ exp −c0εAσ

T

⎡

⎣ ⎢

⎤

⎦ ⎥

Fisher fits with complement

• 2d lattice of side L=40,fixed occupation ρ=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?

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 A0>>A

€

nA (T) = q0

A−τ A0 − A( )−τ

A0−τ

exp −c0ε Aσ + (A0 − A)σ − A0

σ( )

T

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

€

≈q0A−τ 1+ τA

A0

⎛

⎝ ⎜

⎞

⎠ ⎟exp −

c0εAσ

T

⎡

⎣ ⎢

⎤

⎦ ⎥exp

c0εσA

TA01−σ

⎡

⎣ ⎢

⎤

⎦ ⎥

Take the leading term (A=1)

€

⇒ nA (T) ≈ nAFisher (T) 1+

τ

A0

⎛

⎝ ⎜

⎞

⎠ ⎟exp

c0εσ

A01−σ T

⎡

⎣ ⎢

⎤

⎦ ⎥

€

≈nAFisher(T)exp

K

RT

⎡ ⎣ ⎢

⎤ ⎦ ⎥

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 • 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

€

dp =∂p

∂A T

dA +∂p

∂T A

dT = 0

At constant pressure p,

€

€

∂p

∂A T

≈ −p

T

∂ΔHm

∂A T

€

∂p

dT A

≈ pΔHm

T 2

€

⇒∂T

∂A p

=T

ΔHm

∂ΔHm

∂A T

€

ΔHm ≈ −B(A) + T

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,

ΔE is very negativeΔS is very positive

ΔF can never become 0.

€

ΔF=ΔE−TΔS=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

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

•

•

s s

€

nS = n0 exp −ΔF

T

⎛

⎝ ⎜

⎞

⎠ ⎟ Coulomb and all Isolate Coulomb from ΔF 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)

d=2 Ising fixed magnetization (density) calculations

€

M =1− 2ρ M = 0.9, ρ = 0.05 M = 0.6, ρ = 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):

• Surface tension =2• Surface energy coefficient:

– small clusters square-like:

•Sc0=4

– large clusters circular:•

Lc0=2• Cluster yields from all L,

M, ρ values collapse onto coexistence line

• Fisher scaling points to Tc

d=2 Ising fixed magnetization M (d=2 lattice gas fixed average density ρ)

T = 0

T>0

Liquiddrop Vacuum Vapor

L

L

A0

Amax

€

Amax T( ) = A0 − nA T( )AA=1

A<Amax

∑

€

nA T( )∝ exp −ΔF T( )

ΔF = S c0Aσ +Lc0 Amax T( ) − A( )σ

−L c0Amax T( )σ

( )ε

+ Tτ lnA Amax T( ) − A( )

Amax T( )

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

• 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 ρ)

T = 0

T>0

Liquiddrop Vacuum Vapor

L

L

A0

Amax

€

Amax T( ) = A0 − nA T( )AA=1

A<Amax

∑

€

nA T( )∝ exp −ΔF T( )

ΔF = c0 Aσ + Amax T( ) − A( )σ

− Amax T( )σ

( )ε

+ Tτ 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– ΔFsurface modified by

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

€

nA T( )∝ exp −ΔF T( )

ΔF =F f −Fi

= ΔE + c0 Aσ + A0 − A( )σ

− A0σ

( )ε

+ Tτ lnA A0 − A( )

A0

⎛

⎝ ⎜

⎞

⎠ ⎟

A0-A A

€

Ff = Ebind (A,Z) − Tc0

Tc

Aσ + τ ln A ⎛

⎝ ⎜

⎞

⎠ ⎟

+ Ebind (A0 − A,Z0 − Z) − Tc0

Tc

A0 − A( )σ

+ τ 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

A0σ + τ 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: ΔE ≈ 15.2 MeV

• Fisher + ideal gas:

€

p

pc

=

T nA T( )A

∑

T nA Tc( )A

∑

• Fisher + ideal gas:

€

ρv

ρ c

=

nA T( )AA

∑

nA Tc( )AA

∑

• ρc ≈ 0.45 ρ0

• 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.