146
I. Gudyma and A. Maksymov
The simplest fluctuations are the uncorrelated ones and can be described as a white
Gaussian stochastic process. We started our analysis with this type of fluctuations.
In this approach, each site fluctuates separately and is insensitive to the influence of
any other sites from the system. Such type of fluctuations considered for separate
SCO compounds can be characterized statistically in the following manner [15–17]:
ξ(t) = 0, ξ(t)ξ(t
) = 2ε
2
δ(t − t
),
(4)
where the fluctuation strength ε characterizes the spectral density of stochastic variable ξ i (t), and t and t
are distinct times. The type of fluctuations introduced here is
more common for classical macroscopic systems; however, they can be successfully
used in microscopic models as well.
The completely uncorrelated outcomes of white fluctuations serve as a convenient
mathematical model of the real noise, but they also bear some physical inconsistencies, such as the infinite white noise energy. In reality, the autocorrelation time of
noise may be very small but different from zero. Thus, the system is under colored
noise action, which may be described in terms of the Ornstein–Uhlenbeck (OU)
process with an exponential correlation function
ξ(t)ξ(t
) =
ε
2
τ
exp
|t − t
|
τ
,
(5)
where τ is the autocorrelation time which is also related to the cutoff frequency
characteristic to the Lorentzian power spectrum of OU noise.
With decreasing the system size of spin-crossover crystals, the surface effects
become crucial for determining their properties. The main difference that drastically
impacts the magnetic properties of spin-crossover nanocrystal is the different couplings for molecules inside the lattice and on the surface. For obtaining the model
for such spin-crossover lattice, we split the coupling term in (1) into three separate couplings: the one describing the interaction of bulk molecules, the second
one describing the coupling of molecules on the surface, and additional one that is
responsible for linking the bulk of nanocrystal with its surface. The Hamiltonian of
the model is following [18–20]:
H = −
i h i s i −
b
J
b
i j s i s j
−
s
J
s
i j s i s j −
bs
J
b−s
i j s i s j .
(6)
Here, J
α
i j is short-range coupling between nearest-neighbor sites, given in energy
units, and is considered as a parameter of the theory. The superscript index α =
s, bs, b describes three different interaction cases displayed on the simplified illustration of the model shown in Fig. 1. The three kinds of bonds shown in Fig. 1 arise
from surrounding the bulk sites with a surface of one site thick and corresponds to (i)
the coupling of molecules on the surface J
s (blue); (ii) the coupling of entire surface
with bulk part of nanocrystal J
bs (green) which is realized through some intermedi-
I. Gudyma and A. Maksymov
The simplest fluctuations are the uncorrelated ones and can be described as a white
Gaussian stochastic process. We started our analysis with this type of fluctuations.
In this approach, each site fluctuates separately and is insensitive to the influence of
any other sites from the system. Such type of fluctuations considered for separate
SCO compounds can be characterized statistically in the following manner [15–17]:
ξ(t) = 0, ξ(t)ξ(t
) = 2ε
2
δ(t − t
),
(4)
where the fluctuation strength ε characterizes the spectral density of stochastic variable ξ i (t), and t and t
are distinct times. The type of fluctuations introduced here is
more common for classical macroscopic systems; however, they can be successfully
used in microscopic models as well.
The completely uncorrelated outcomes of white fluctuations serve as a convenient
mathematical model of the real noise, but they also bear some physical inconsistencies, such as the infinite white noise energy. In reality, the autocorrelation time of
noise may be very small but different from zero. Thus, the system is under colored
noise action, which may be described in terms of the Ornstein–Uhlenbeck (OU)
process with an exponential correlation function
ξ(t)ξ(t
) =
ε
2
τ
exp
|t − t
|
τ
,
(5)
where τ is the autocorrelation time which is also related to the cutoff frequency
characteristic to the Lorentzian power spectrum of OU noise.
With decreasing the system size of spin-crossover crystals, the surface effects
become crucial for determining their properties. The main difference that drastically
impacts the magnetic properties of spin-crossover nanocrystal is the different couplings for molecules inside the lattice and on the surface. For obtaining the model
for such spin-crossover lattice, we split the coupling term in (1) into three separate couplings: the one describing the interaction of bulk molecules, the second
one describing the coupling of molecules on the surface, and additional one that is
responsible for linking the bulk of nanocrystal with its surface. The Hamiltonian of
the model is following [18–20]:
H = −
i h i s i −
b
J
b
i j s i s j
−
s
J
s
i j s i s j −
bs
J
b−s
i j s i s j .
(6)
Here, J
α
i j is short-range coupling between nearest-neighbor sites, given in energy
units, and is considered as a parameter of the theory. The superscript index α =
s, bs, b describes three different interaction cases displayed on the simplified illustration of the model shown in Fig. 1. The three kinds of bonds shown in Fig. 1 arise
from surrounding the bulk sites with a surface of one site thick and corresponds to (i)
the coupling of molecules on the surface J
s (blue); (ii) the coupling of entire surface
with bulk part of nanocrystal J
bs (green) which is realized through some intermedi-