27 Cooperative Phenomena in Spin-Crossover Molecular Crystals
431
was used by Tissot et al. [15] to explain the experiments of control of the thermal
hysteresis of the prototypal spin transition in F e I I (phen) 2 (NCS) 2 compound via
the microcrystallite environment. Muraoka et al. [16] considered the edge molecules
as inactive HS ones, whereas all inside molecules are active in the framework of the
Ising-like model. We proceed somewhat differently. Since a Gaussian stochastic
process is archetypical for contact with thermal surrounding system, we use the
statistical characteristic of the fluctuating external field as follows:
h i (t) = 0,
(27.3)
h i (t)h j (t
)
= ε
2 δ(t − t
).
(27.4)
Here < . . . > represents an ensemble average over all realizations of the fluctuating
field. Besides ε is a measure of the strength of random surface fields, and δ ij is the
Kronecker delta symbol. The basic idea behind Langevin approach is to assume that
the fluctuations on each atomic site can be represented by a Gaussian white noise
term. Stronger fluctuations lead to increasing the width of the Gaussian distribution.
In the usual consideration, each molecule on surface is assumed to be instantaneous coupled to a thermal reservoir, and heat currents may arise between
the reservoirs in the stationary state scenario. The surface itself provides a new
intrinsically low-symmetry environment that can trigger structural distortions and
modify the ligand field. Thus the order parameter is sensitive to the stochastic
environment process with respond to the system, acting as a disorder force. In
fact, the surrounding environment is the main factor governing the spin transition
behavior of spin-crossover materials. Inasmuch as spin crossover is an inherently
ligand field problem that this coupling is through random variations of the field.
27.3 Monte Carlo Simulation of Magnetic Behavior in
Spin-Crossover Nanocrystals
For the spin-crossover system described by the Hamiltonian (27.1), we have
provided a series of three-dimensional (3D) Monte Carlo (MC) simulations based
on the Metropolis algorithm. The MC Metropolis algorithm provides a natural
way to simulate temperature effects where dynamics are not required due to the
rapid convergence to equilibrium and relatively simple implementation. In order to
reproduce the experimental situations, we have used for simulations 3D cubic lattice
with size L 3 , where L is the length of each side, and open boundary conditions.
For simulations the following parameters of the system were fixed: The intersite
interaction of inside molecules is chosen at J b = 85, the degenerating ratio is
g = 150, and the energy gap is Δ = 900. In this way we characterize the behavior
of temperature transition curves which are the main characteristic for this type of
Ising-like model. The numerical solutions of Hamiltonian (27.1) were obtained for
5000 MC steps assuming that the coupling of bulk sites with the surface ones is the
same like for the bulk sites, i.e., J bs
ij = J b
ij .
431
was used by Tissot et al. [15] to explain the experiments of control of the thermal
hysteresis of the prototypal spin transition in F e I I (phen) 2 (NCS) 2 compound via
the microcrystallite environment. Muraoka et al. [16] considered the edge molecules
as inactive HS ones, whereas all inside molecules are active in the framework of the
Ising-like model. We proceed somewhat differently. Since a Gaussian stochastic
process is archetypical for contact with thermal surrounding system, we use the
statistical characteristic of the fluctuating external field as follows:
h i (t) = 0,
(27.3)
h i (t)h j (t
)
= ε
2 δ(t − t
).
(27.4)
Here < . . . > represents an ensemble average over all realizations of the fluctuating
field. Besides ε is a measure of the strength of random surface fields, and δ ij is the
Kronecker delta symbol. The basic idea behind Langevin approach is to assume that
the fluctuations on each atomic site can be represented by a Gaussian white noise
term. Stronger fluctuations lead to increasing the width of the Gaussian distribution.
In the usual consideration, each molecule on surface is assumed to be instantaneous coupled to a thermal reservoir, and heat currents may arise between
the reservoirs in the stationary state scenario. The surface itself provides a new
intrinsically low-symmetry environment that can trigger structural distortions and
modify the ligand field. Thus the order parameter is sensitive to the stochastic
environment process with respond to the system, acting as a disorder force. In
fact, the surrounding environment is the main factor governing the spin transition
behavior of spin-crossover materials. Inasmuch as spin crossover is an inherently
ligand field problem that this coupling is through random variations of the field.
27.3 Monte Carlo Simulation of Magnetic Behavior in
Spin-Crossover Nanocrystals
For the spin-crossover system described by the Hamiltonian (27.1), we have
provided a series of three-dimensional (3D) Monte Carlo (MC) simulations based
on the Metropolis algorithm. The MC Metropolis algorithm provides a natural
way to simulate temperature effects where dynamics are not required due to the
rapid convergence to equilibrium and relatively simple implementation. In order to
reproduce the experimental situations, we have used for simulations 3D cubic lattice
with size L 3 , where L is the length of each side, and open boundary conditions.
For simulations the following parameters of the system were fixed: The intersite
interaction of inside molecules is chosen at J b = 85, the degenerating ratio is
g = 150, and the energy gap is Δ = 900. In this way we characterize the behavior
of temperature transition curves which are the main characteristic for this type of
Ising-like model. The numerical solutions of Hamiltonian (27.1) were obtained for
5000 MC steps assuming that the coupling of bulk sites with the surface ones is the
same like for the bulk sites, i.e., J bs
ij = J b
ij .