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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.