1. Decay of Metastable States
in Spin-Crossover Solids
Ranjit Chacko 1,2
Harvey Gould 1,2 W. Klein 2
1. Clark University 2. Boston University

2. h temperatures because it where g = gHS / gLS denotes the degeneracy ratio between the
ition between the LS and HS and LS states.
ges of the pressure, mag- The other important characteristic is the intermolecular
Spin-Crossover Solids
7
When interactions be- interaction. For the cooperative property in the SC transition,
raction changes smoothly
the interactions become RH
erative phenomena.8 The Vintra(R) RL
s sharper with increasing
the interaction exceeds a
discontinuous. In order to
perties of SC compounds,
stable nature of such mo-
the spin-crossover transi-
acteristics of the system.
he intramolecule Hamil-
an energy difference be-
and different degenera-
the HS state and the LS D
RL
pin state at the ith site by R
or HS. The intramolecule RH
y
FIG. 1. Color online Schematic picture of the energy structure
si . 1
of a molecule. The left right minimum corresponds to the LS HS
state. In the inset, schematic pictures of a lattice of LS molecules
of the degeneracy as a left , and the distortion caused by a HS molecule in a lattice of LS
• Molecules have low spin(LS) and high spin(HS) states.
n use an effective Hamil- molecules right , are illustrated.
• HS state has larger radius, higher energy, and higher degeneracy.
014105-1 ©2008 The American Physical Society
• Effective long range interaction because of lattice distortion.

3. Wajnflasz-Pick Model
1
H = −J σi σj + (D − kb T ln g)σi
<i,j>
2 i

4. Wajnflasz-Pick Model
NN Ising interaction
1
H = −J σi σj + (D − kb T ln g)σi
<i,j>
2 i

5. Wajnflasz-Pick Model
NN Ising interaction
effective field
1
H = −J σi σj + (D − kb T ln g)σi
<i,j>
2 i

6. Wajnflasz-Pick Model
NN Ising interaction
effective field
1
H = −J σi σj + (D − kb T ln g)σi
<i,j>
2 i
energy difference between HS
and LS states
D>0

7. Wajnflasz-Pick Model
NN Ising interaction
effective field
1
H = −J σi σj + (D − kb T ln g)σi
<i,j>
2 i
energy difference between HS
and LS states g is the ratio of degeneracy of HS
D>0 to LS states

8. the ratio of the radii to be RHS =RLS ˆ 1:1. H nnn expresses on the details of the c
that the data given here
Modeling Elastic Interactions
elastic interaction for next-nearest-neighbor pairs (hhi; jii).
In this study, we set the ratio of the spring constants, k1 =k2 , we perform simulation
to be 10 [35]. MCSs at several point
For the simulation, we adopt the NPT-MC method [36] First, we study how t
Konishi et al., PRL, 2008
for the isothermal-isobaric ensemble with the number of the HS fraction fHS
of molecules N, the pressure of the system P, and the k1 …ˆ 10k2 †. In Fig. 2 w
Miyashita et al, Phys. Rev. B, 2008
temperature T. The thermodynamic potential for the of k1 with g…ˆ gHS =
isothermal-isobaric ensemble is the enthalpy, H ˆ k1 ˆ 10, 20, 30,40, an
• Replace NN Ising interaction with elastic interaction. of
E ‡ PV, where E is the energy and V is the volume 10, the transition is g
• Model exhibits mean fieldstates of the system are specified by 4N ‡
the system. The behavior. transition becomes sha
• mean field 1 variables (n1 ; ; nN ; r1 ; ; rN ; V). In the NPT-MC
critical exponents
method, we have the following detailed balance condition 1
• spinodal nucleation
HS molecule 0.8
LS molecule
HS fraction
0.6
0.4
Rl
Ri
. .
0.2
Spring constant k1
Spring constant k2 0
0 0.2
Rj . . Rk
FIG. 2 (color online).
fraction fHS …T† with g
FIG. 1 (color online). Schematic illustration of the present (red squares), 20 (green
model. HS/LS molecule consists of Fe atom (red/blue circle) (blue triangles), and 50 (

9. Classical vs. Spinodal Nucleation
Classical nucleation Spinodal Nucleation
Geometry Compact Ramified
Small difference from
Density Same as stable phase
metastable phase
Growth Mode Grows at surface Fills in

10. Long Range Ising SC Model
• Model ordering behavior of SC solids with
Ising interaction as in WP model.
• But, use weak long range interaction
between spins to model the effective long
range interaction due to elastic forces in SC
solids.

11. Modeling Long-Range Interactions
1
H = −J σi σj + (D − kb T ln g)σi
2 i
[i,j]
F
2
1
-1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1
m
-1
-2
-3

12. Ising Model and Percolation
• Mapping Ising model to percolation model gives us a
geometric way of thinking about critical behavior.
• Coniglio-Klein bond probability gives same critical point and
same critical exponents as Ising model.
• Clusters are statistically independent.
• Basis of cluster flipping algorithms like Wolff, Swendsen-Wang.
p = 1 − e−2βJ
A. Coniglio and W. Klein, J. Phys. A 1980

13. Ising Model and Percolation
• Mapping Ising model to percolation model gives us a
geometric way of thinking about critical behavior.
• Coniglio-Klein bond probability gives same critical point and
same critical exponents as Ising model.
• Clusters are statistically independent.
• Basis of cluster flipping algorithms like Wolff, Swendsen-Wang.
p = 1 − e−2βJ
A. Coniglio and W. Klein, J. Phys. A 1980

14. Ising Model and Percolation
• Mapping Ising model to percolation model gives us a
geometric way of thinking about critical behavior.
• Coniglio-Klein bond probability gives same critical point and
same critical exponents as Ising model.
• Clusters are statistically independent.
• Basis of cluster flipping algorithms like Wolff, Swendsen-Wang.
p = 1 − e−2βJ
A. Coniglio and W. Klein, J. Phys. A 1980

15. Ising Model and Percolation
• Mapping Ising model to percolation model gives us a
geometric way of thinking about critical behavior.
• Coniglio-Klein bond probability gives same critical point and
same critical exponents as Ising model.
• Clusters are statistically independent.
• Basis of cluster flipping algorithms like Wolff, Swendsen-Wang.
p = 1 − e−2βJ
A. Coniglio and W. Klein, J. Phys. A 1980

16. Ising Model and Percolation
• Mapping Ising model to percolation model gives us a
geometric way of thinking about critical behavior.
• Coniglio-Klein bond probability gives same critical point and
same critical exponents as Ising model.
• Clusters are statistically independent.
• Basis of cluster flipping algorithms like Wolff, Swendsen-Wang.
p = 1 − e−2βJ
A. Coniglio and W. Klein, J. Phys. A 1980

17. Ising Model and Percolation
• Mapping Ising model to percolation model gives us a
geometric way of thinking about critical behavior.
• Coniglio-Klein bond probability gives same critical point and
same critical exponents as Ising model.
• Clusters are statistically independent.
• Basis of cluster flipping algorithms like Wolff, Swendsen-Wang.
p = 1 − e−2βJ
A. Coniglio and W. Klein, J. Phys. A 1980

18. Ising Model and Percolation
• Mapping Ising model to percolation model gives us a
geometric way of thinking about critical behavior.
• Coniglio-Klein bond probability gives same critical point and
same critical exponents as Ising model.
• Clusters are statistically independent.
• Basis of cluster flipping algorithms like Wolff, Swendsen-Wang.
p = 1 − e−2βJ
A. Coniglio and W. Klein, J. Phys. A 1980

19. Ising Model and Percolation
• Mapping Ising model to percolation model gives us a
geometric way of thinking about critical behavior.
• Coniglio-Klein bond probability gives same critical point and
same critical exponents as Ising model.
• Clusters are statistically independent.
• Basis of cluster flipping algorithms like Wolff, Swendsen-Wang.
p = 1 − e−2βJ
A. Coniglio and W. Klein, J. Phys. A 1980

20. Ising Model and Percolation
• Mapping Ising model to percolation model gives us a
geometric way of thinking about critical behavior.
• Coniglio-Klein bond probability gives same critical point and
same critical exponents as Ising model.
• Clusters are statistically independent.
• Basis of cluster flipping algorithms like Wolff, Swendsen-Wang.
p = 1 − e−2βJ
A. Coniglio and W. Klein, J. Phys. A 1980

21. Ising Model and Percolation
• Mapping Ising model to percolation model gives us a
geometric way of thinking about critical behavior.
• Coniglio-Klein bond probability gives same critical point and
same critical exponents as Ising model.
• Clusters are statistically independent.
• Basis of cluster flipping algorithms like Wolff, Swendsen-Wang.
p = 1 − e−2βJ
A. Coniglio and W. Klein, J. Phys. A 1980

22. Percolation at the Spinodal
Unger, Klein, Phys. Rev. B, 1994
p = 1 − e−2βJ(1−ρs )
• For Ising model with long range interactions
spinodal line can also be mapped to
percolation transition.
• Bond probability now depends on density of
stable phase spins.

23. Entropic Barrier to Nucleation
Klein et al., Phys. Rev. E 2007
• Typically many clusters of connected spins will
span one correlation volume.
• Coalescence of clusters occurs at nucleation.
• Reduction in number of independent clusters
means loss of entropy.

24. Percolation in Infinite Range
Ising Model
• Bonds can be placed between any two aligned spins.
• Equivalent to Erdos-Renyi graph theory.
• In Erdos-Renyi graphs a “giant component” appears
at a critical value of the probability.
• This critical value corresponds to the Unger-Klein
probability at the spinodal density.
• Coalescence of clusters begins before nucleation.

25. Spinodal Nucleation in SC Solids
• Investigate alternative derivation of bond
probability through branching process
argument.
• More detailed study of spinodal nucleation in
elastic model.
• identify precursors to nucleation?
• Investigate compressible Ising model as model
for phase transition kinetics in SC solids.