crossing the coexistence line of the ising model at fixed magnetization l. phair, j. b. elliott, l....
TRANSCRIPT
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.