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A. V. Dvornichenko and O. M. Shchokotova
of the growing surface is universal, invariant to a change in system parameters and
satisfies the Zipf law p(s) ∝ s
−1 . According to the relation p(s)ds = f (R)d R with
s ∝ R
2 for the distribution f (R) one gets the same law f (R) ∝ R
−1 .
It should be noted that the obtained distribution in a Zipf law form could not give
any information about the mean and most probable size of the adsorbate islands.
In order to get the appropriate distribution characterizing such properties, next, we
will calculate the area occupied by adsorbate on the half-height of each pyramidal
island. We assume that each pyramidal island is characterized by a form of circle
on the half-height. Next, for the fixed simulation time instant initially we define
the height difference of the highest pyramidal island, calculate the corresponding
half-height of it as h = (φ max − φ min )/2 and area of the cross section of the highest
pyramid s max . Next, we calculate the inclination angle of the pyramidal structure
in the standard manner: tan(γ ) = h/
√
s max /π , by assuming that at fixed system
parameters all pyramidal structures are characterized by the same angle γ . Finally, by
calculating half-heights of all pyramidal structures {h i } (assuming that φ min remains
constant at fixed time instant for all structures) and by using the value tan(γ ), we
calculate the distribution of radii of pyramidal islands on their own half-heights [51].
Obtained results for the distributions f (R/ are shown in Fig. 11a, b, at fixed
system parameters and different time instants; and at fixed time instant and different
system parameters, respectively. Here numerical data are shown by symbols; solid
lines correspond to the approximation of numerical data by log-normal distribution
for the quantity x ≡ R/ in the following form f (x; μ, σ ) =
1
xσ
√
2π
e
−
(ln x−μ) 2
2σ 2
,
where μ and σ correspond to the shape and log-scale, accordingly. It follows that
this distribution is universal (does not depend on simulation time) and is invariant to
a change in the main system parameters.
Fig. 10 Snapshots of the cross section of pyramidal structures on the mean height level of the
growing surface at F = 4, ε = 4, ν = 0.1 at different time instants (a). Dependencies of the total
area of all islands area on the mean height level at ν = 0.1 (b). Dependencies of the mean number of
islands on the mean island area at different time instants at ν = 0.1 (c). The distribution of adsorbate
islands over sizes f (s) at ε = 4, F = 4, ν = 0.1 and t = 50 is shown in the insertion in plot c
A. V. Dvornichenko and O. M. Shchokotova
of the growing surface is universal, invariant to a change in system parameters and
satisfies the Zipf law p(s) ∝ s
−1 . According to the relation p(s)ds = f (R)d R with
s ∝ R
2 for the distribution f (R) one gets the same law f (R) ∝ R
−1 .
It should be noted that the obtained distribution in a Zipf law form could not give
any information about the mean and most probable size of the adsorbate islands.
In order to get the appropriate distribution characterizing such properties, next, we
will calculate the area occupied by adsorbate on the half-height of each pyramidal
island. We assume that each pyramidal island is characterized by a form of circle
on the half-height. Next, for the fixed simulation time instant initially we define
the height difference of the highest pyramidal island, calculate the corresponding
half-height of it as h = (φ max − φ min )/2 and area of the cross section of the highest
pyramid s max . Next, we calculate the inclination angle of the pyramidal structure
in the standard manner: tan(γ ) = h/
√
s max /π , by assuming that at fixed system
parameters all pyramidal structures are characterized by the same angle γ . Finally, by
calculating half-heights of all pyramidal structures {h i } (assuming that φ min remains
constant at fixed time instant for all structures) and by using the value tan(γ ), we
calculate the distribution of radii of pyramidal islands on their own half-heights [51].
Obtained results for the distributions f (R/ are shown in Fig. 11a, b, at fixed
system parameters and different time instants; and at fixed time instant and different
system parameters, respectively. Here numerical data are shown by symbols; solid
lines correspond to the approximation of numerical data by log-normal distribution
for the quantity x ≡ R/ in the following form f (x; μ, σ ) =
1
xσ
√
2π
e
−
(ln x−μ) 2
2σ 2
,
where μ and σ correspond to the shape and log-scale, accordingly. It follows that
this distribution is universal (does not depend on simulation time) and is invariant to
a change in the main system parameters.
Fig. 10 Snapshots of the cross section of pyramidal structures on the mean height level of the
growing surface at F = 4, ε = 4, ν = 0.1 at different time instants (a). Dependencies of the total
area of all islands area on the mean height level at ν = 0.1 (b). Dependencies of the mean number of
islands on the mean island area at different time instants at ν = 0.1 (c). The distribution of adsorbate
islands over sizes f (s) at ε = 4, F = 4, ν = 0.1 and t = 50 is shown in the insertion in plot c