27 Cooperative Phenomena in Spin-Crossover Molecular Crystals
429
27.2 Ising-Like Model of Spin-Crossover Nanocrystal
Interacting with the Surroundings
With the achievement of nanoscopic sizes of spin-crossover nanocrystals, the
dynamics of surface molecules becomes relevant and can influence the phase
stability and transformation kinetics of entire solids. Despite the variety of the
models that are able to describe the physical processes at microscopic space and
time scales, there is a lack of descriptions which account for linkage between the
surface of the crystal and its volume. Since the sizes of solid nanocrystals decrease,
the increase in surface area-to-volume ratio makes the surface-environment effects
become more pronounced. Therefore the importance of theoretical model that could
explain the role of surface of separate spin-crossover nanocrystal is evident. Phenomenologically, the interactions between any spin-crossover nanoparticles located
into the lattice can be modeled in a simplest way by the Ising-like Hamiltonian
H = −
b
J
b
ij s i s j −
s
J
s
ij s i s j −
s−b
J
s−b
ij s i s j −h 0
i
s i −
s
i
h i (t)s i . (27.1)
Here s i is a fictitious spin (pseudospin) operator which has two eigenvalues ±1,
corresponding to the HS and LS states of respective i-s molecule, and < ij >
denotes the summation over all nearest neighboring spin pairs. The Ising-like
Hamiltonian describes the elastic interaction between spin states via the nearneighbor coupling of two-level units. We based our study on 3D crystal lattice
with open boundary conditions. In this case it is obvious that depending whether
the particle is located on the surface or inside the crystal, the number of bonds
associated to it is different. In framework of mean field approximation, it means
that the interacting force between the sites on the surface and inside the volume is
not the same. The intersite short-range coupling constants J α
ij , measured in energy
units, are parameters of the theory, where an index α = b, s, b − s corresponds
to occupied pairs of bulk sites, surface sites, and bulk surface sites. The schematic
illustration of the model composed of a bulk sites which are surrounded by a surface
of one site thick is shown in Fig. 27.1. In this figure are displayed only two kinds
of bonds: one for surface sites and another one for bulk sites since the interaction
between the surface sites and bulk sites is considered the same as for the sites inside
the lattice, i.e., the b − s = b which leads to the decomposition of the Hamiltonian
into bulk and surface parts [6].
The effective external field describing the result of surrounding action of crystal
on single molecular magnet is
h 0 = −
1
2
(Δ − k B T ln g),
(27.2)
where Δ stands for the energy difference between HS and LS states (the enthalpy
change associated with the LS → HS conversion) for an individual spin-crossover
429
27.2 Ising-Like Model of Spin-Crossover Nanocrystal
Interacting with the Surroundings
With the achievement of nanoscopic sizes of spin-crossover nanocrystals, the
dynamics of surface molecules becomes relevant and can influence the phase
stability and transformation kinetics of entire solids. Despite the variety of the
models that are able to describe the physical processes at microscopic space and
time scales, there is a lack of descriptions which account for linkage between the
surface of the crystal and its volume. Since the sizes of solid nanocrystals decrease,
the increase in surface area-to-volume ratio makes the surface-environment effects
become more pronounced. Therefore the importance of theoretical model that could
explain the role of surface of separate spin-crossover nanocrystal is evident. Phenomenologically, the interactions between any spin-crossover nanoparticles located
into the lattice can be modeled in a simplest way by the Ising-like Hamiltonian
H = −
b
J
b
ij s i s j −
s
J
s
ij s i s j −
s−b
J
s−b
ij s i s j −h 0
i
s i −
s
i
h i (t)s i . (27.1)
Here s i is a fictitious spin (pseudospin) operator which has two eigenvalues ±1,
corresponding to the HS and LS states of respective i-s molecule, and < ij >
denotes the summation over all nearest neighboring spin pairs. The Ising-like
Hamiltonian describes the elastic interaction between spin states via the nearneighbor coupling of two-level units. We based our study on 3D crystal lattice
with open boundary conditions. In this case it is obvious that depending whether
the particle is located on the surface or inside the crystal, the number of bonds
associated to it is different. In framework of mean field approximation, it means
that the interacting force between the sites on the surface and inside the volume is
not the same. The intersite short-range coupling constants J α
ij , measured in energy
units, are parameters of the theory, where an index α = b, s, b − s corresponds
to occupied pairs of bulk sites, surface sites, and bulk surface sites. The schematic
illustration of the model composed of a bulk sites which are surrounded by a surface
of one site thick is shown in Fig. 27.1. In this figure are displayed only two kinds
of bonds: one for surface sites and another one for bulk sites since the interaction
between the surface sites and bulk sites is considered the same as for the sites inside
the lattice, i.e., the b − s = b which leads to the decomposition of the Hamiltonian
into bulk and surface parts [6].
The effective external field describing the result of surrounding action of crystal
on single molecular magnet is
h 0 = −
1
2
(Δ − k B T ln g),
(27.2)
where Δ stands for the energy difference between HS and LS states (the enthalpy
change associated with the LS → HS conversion) for an individual spin-crossover