Crossing the Coexistence Line of the Ising Model at Fixed Magnetization

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

?
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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?

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Ising model (or lattice gas)

• Magnetic transition
• Isomorphous with liquid-vapor transition
• Hamiltonian for s-sites and B-external field

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Finite size effects in Ising
Canonical (Lattice Gas)
Grand-canonical
?
finite lattice or finite drop?
A0
seek ye first the droplet and its
righteousness, and all things shall be added
unto you
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Clapeyron Equation for a finite drop

• Lowering of the isobaric transition temperature
  with decreasing droplet size

Clapeyron equation
Integrated
Correct for surface
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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?
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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 Fishers model but for a finite drop
  rather than the infinite bulk liquid

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Clue from Clapeyron
A0

• 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

Ei
Ef
A0-A
A
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Finite size effects Complement

• Infinite liquid

• Finite drop

• Generalization instead of ES(A0, A) use
  ELD(A0, A) which includes Coulomb, symmetry,
  etc.(tomorrows talk by L.G. Moretto)
• Specifically, for the Fisher expression

Fit the yields and infer Tc (NOTE this is the
finite size correction)
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Fisher fits with complement

• 2d lattice of side L40,fixed occupation r0.05,
  ground state drop A080
• Tc 2.26 - 0.02 to be compared with the
  theoretical value of 2.269
• Can we declare victory?

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Going from the drop to the bulk

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

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From Complement to Clapeyron

• In the limit of large A0gtgtA

Take the leading term (A1)
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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)

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Complement for Coulomb

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

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(Negative) Heat Capacities in Finite Systems

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

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Coulombs Quandary

• Solutions
• Easy
• Take the vapor at infinity!!
• Diverges for an infinite amount of vapor!!

• Coulomb and the drop
• Drop self energy
• Drop-vapor interaction energy
• Vapor self energy

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Generalization to nucleiheat capacity via
binding energy

• No negative heat capacities above A60

At constant pressure p,
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The problem of the drop-vapor interaction energy

• If each cluster is bound to the droplet (Qlt0),
  may be OK.
• If at least one cluster seriously unbound
  (QgtgtT), 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.

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Vapor self energy

• If Drop-vapor interaction energy is solved, then
  just take a small sample of vapor so that
  ECoul(self)/A ltlt 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!

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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

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A box for each cluster
s
s

s

• Saddle points Transition state theory guarantees
  in equilibrium with S

Isolate Coulomb from DF and divide away the
Boltzmann factor
Coulomb and all
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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)

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d2 Ising fixed magnetization (density)
calculations
? outside coexistence region ? inside coexistence
region
?, ? inside coexistence region
?, ? T gt Tc
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d2 Ising fixed magnetization M (d2 lattice gas
fixed average density ltrgt)

• Inside coexistence region
• yields scale via Fisher complement
• complement is liquid drop Amax(T)
• Surface tension g2
• Surface energy coefficient
• small clusters square-like
• Sc04g
• large clusters circular
• Lc02g?p
• Cluster yields from all L, M, r values collapse
  onto coexistence line
• Fisher scaling points to Tc

T 0
A0
Amax
Tgt0
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d3 Ising fixed magnetization M (d3 lattice gas
fixed average density ltrgt)

• Inside coexistence region
• yields scale via Fisher complement
• complement is liquid drop Amax(T)

T 0
A0

• Cluster yields collapse onto coexistence line
• Fisher scaling points to Tc

Amax
Tgt0
L
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Complement for excited nuclei

• 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

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Complement for excited nuclei

• Fisher scaling collapses data onto coexistence
  line

• Gives bulk
• Tc18.60.7 MeV

• Fisher ideal gas

• Fisher ideal gas

Fit parameters L(E), Tc, q0, Dsecondary
Fixed parameters t, s, liquid-drop coefficients

• pc 0.36 MeV/fm3
• Clausius-Clapyron fit DE 15.2 MeV

• rc 0.45 r0
• Full curve via Guggenheim

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Conclusions

• Surface is simplest correction for finite size
  effects (Rayleigh and Clapeyron)
• Complement accounts for finite size scaling of
  droplet
• For ground state droplets with A0ltltLd, 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 point extracted when complement
taken into account.