As a result of irradiation with Xe with E = 250 MeV in InP at room temperature
[1] defects, similar to the "chain of pearls”, which are placed along the trajectory of the
ions at depths ranging from 35 to 100 nm and from 7 to 10 microns have been identified.
Such defects called tracks, and they can occur at different depths and have different
shapes. Tracks were examined like a chain of deal spherical regions; it was assumed that
each incident ion creates one such chain. In this model, we assume that the track is
formed randomly, but in that place of the ion path, where the energy value, which loses
each ion to the unity of the way, is above some threshold value.
As a result of irradiation the number of tracks will continue to grow, areas of the
single tracks modified substance continue to overlap, form of modificated matter
becomes more complicated, creating branched structure.
Percolation threshold, fraction of spanning cluster modified material for different
doses and different distributions of track areas in depth were evaluated. Based on the
scaling hypothesis large-scale curve were constructed, critical exponents for this
percolation model were established.
Calculated values of critical exponents were compared with the known values for
the model of continuous percolation. Parameters of the percolation and critical
exponents depend on the distribution of track areas in depth, which indicates the
difference in the order parameters of the track structure obtained for different
distributions in depth.
When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/2081

Demchyshyn, A.

Selyshchev, P.

SocioEconomic Challenges Journal (Sumy State University)

Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part II                  355 
 
CRITICAL EXPONENTS IN PERCOLATION MODEL  
OF TRACK REGIONS WITH DIFFERENT DEPTH  
DISTRIBUTION 
Demchyshyn Andriy, Selyshchev Pavlo  
Taras Shevchenko National University of Kyiv, Faculty of Physics , 03187, Ukraine, Kyiv 
 
ABSTRACT 
As a result of irradiation with Xe with E = 250 MeV in InP at room temperature 
[1] defects, similar to the "chain of pearls”, which are placed along the trajectory of the 
ions at depths ranging from 35 to 100 nm and from 7 to 10 microns have been identified. 
Such defects called tracks, and they can occur at different depths and have different 
shapes. Tracks were examined like a chain of deal spherical regions; it was assumed that 
each incident ion creates one such chain. In this model, we assume that the track is 
formed randomly, but in that place of the ion path, where the energy value, which loses 
each ion to the unity of the way, is above some threshold value. 
As a result of irradiation the number of tracks will continue to grow, areas of the 
single tracks modified substance continue to overlap, form of modificated matter 
becomes more complicated, creating branched structure. 
Percolation threshold, fraction of spanning cluster modified material for different 
doses  and  different  distributions  of  track  areas  in  depth  were  evaluated.  Based  on  the  
scaling hypothesis  large-scale curve were constructed, critical exponents for this 
percolation model were established.  
 Calculated values of critical exponents were compared with the known values for 
the model of continuous percolation. Parameters of the percolation and critical 
exponents depend on the distribution of track areas in depth, which indicates the 
difference in the order parameters of the track structure obtained for different 
distributions in depth.  
 
Key words: track, branched structures, swift heavy ions, the Monte Carlo method, 
critical exponents, percolation threshold, percolation. 
 
INTRODUCTION 
Under swift heavy ions (SHI) irradiation track of various shapes and 
lengths, filled with a modified (with respect to the material of the sample) ma-
terial, are formed in a solid. They are generated as a result of the strong relaxa-
tion of electronic excitations and formed along the trajectory of the ion. Tracks 
are beginning both from the irradiated surface and at some distance from it. 
Continuous and discontinuous cylindrical and spherical track field were found. 
Elongated defects, similar to the "chain of pearls" that are placed along the 
trajectory of the ions at depths ranging from 35 to 100 nm and from 7 to 10 
microns, were found after irradiation by xenon ions with E = 250 MeV in InP at 
room temperature [1]. Also in the iron garnet irradiated by ions with energies 
356                  Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part II           
 
around 12 MeV in a mode of high speed [2] with the energy loss in the range of 
4.5 -7 KeV·nm^(-1) the appearance of spherical domains of the modified mate-
rial were observed. Discrete tracks point-shaped and oblong dark spots with a 
diameter equal to an average of about 3 - 10 nm along the trajectories of incom-
ing ions in the form of a "chain of pearls" with a number of "pearls" in the track 
of two to five pieces were observed on the bright-field pictures in the paper [3]. 
Physical mechanisms of track formation, probability distribution of their 
formation along the trajectory and in the chain have not been yet fully clarified. 
The distribution of the tracks in the depth of the sample did not correlate with 
either the distribution of implanted ions, or the distribution of point defects 
formed by the ions. The experimental results do not uniquely determine the 
distribution of tracks. 
However, it is widely recognized that the track is formed as a result of re-
distribution of the energy transmitted by one ion in the electron subsystem. 
Also it is known that the track is formed randomly in that place of the ion path, 
where  the  value  of  missing  energy  per  unit  of  ion  path  is  above  a  certain  
threshold, in a place where the rush of energy release is presented. It was noted 
that the simulation results of energy spectra using the SRIM 2008, shows that 
the maximum energy during the passage of high-energy ions are also located in 
an average of 25-40 nm. From the experimental data it is known that the tracks 
in the form of a chain of spheres are created at different depths (between 0 and 
50 nm ,between 0 and 500 nm) for different materials and conditions of expo-
sure. For example, for InP irradiated by xenon ions, the average distance be-
tween tracks spherical averages of 25-40 nm, length of the chain is in the range 
from 50 nm to 150 nm. 
Study of changes in material properties as a result of electronic excitation 
upon irradiation by fast heavy ions and the creation of latent tracks in these 
materials is an exciting and promising field for new research and development. 
However, despite the relevance of the topic and practical significance of the 
expected results, the formation of branched structures of the individual tracks 
has been insufficiently studied. 
FORMULATION OF THE PROBLEM 
We simulated a sample in the form of plane-parallel plate with a thickness 
of 50 to 300 nm. For the calculations was chosen fragment of the plate in the 
shape of the box size from 50 to 300 nm. The case where the ion energy is suffi-
cient to create a chain of spheres modified substances, energy is 20-40 MeV / 
amu around was modeled. Each incident ion creates a track in a sequence of a 
certain number of spherical regions. The trajectories of all ions are parallel to 
each other. In this model, we simulate different distribution areas in depth, such 
as: a) a sphere in the chain b) a sphere in a chain or two areas in the chain) is a 
sphere or two areas or three areas. And so on up to six spheres in the chain. 
Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part II                  357 
 
The first spherical region is equally likely to appear anywhere in the seg-
ment of the trajectory of an ion from its point of penetration into the sample to the 
point of 2ȡ, where ȡ - the average distance between the spheres. In the calcula-
tions ȡ is equal to 40 nm. The distance from the first sphere to the second is cho-
sen randomly from range of values between 0 and 2ȡ, the form of distribution 
doesn’t depend on the «history» (it is the same as for the preliminary scope, only 
shifted along the ion path in the appearance point of the previous sphere). Some 
specific distribution of tracking areas in depth was obtained as a result, distribu-
tion for the case when there are maximum three spheres in the chain is shown in 
Fig.1. 
Consecutively two 
surfaces were exposed 
to radiation. 
With the irradia-
tion modified substance 
areas of individual 
tracks start to overlap, 
as a result modified 
form of matter becomes 
more complicated, 
forming a branched structure. Some part of this branched structure is bordered on 
one surface of the sample, while others lie on the boundary with the opposite 
surface. When these parts connect one to another, they will create so-called 
"spanning cluster," which percolates from one surface to another. 
RESULTS AND DISCUSSION 
Computed de-
pendence of the 
branched structure 
surface area on the 
dose and from different 
angles has previously 
been reported (Fig.2).  
This nontrivial 
dependence is related 
with the distribution of 
track regions in depth, 
namely, the correlation 
between the modified 
regions of different 
tracks as a "chain of 
spheres." Scaling hy-
 
Fig.1.- Distribution for a chain with one, two or three 
spheres. 
 
Fig.2. - Dependence of the branched structure surface 
area on the dose and from different angles 
358                  Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part II           
 
pothesis has been used to separate influence of this distribution on the angular 
and dose dependence of structural parameters. 
Percolation threshold, fraction of spanning cluster modified material for 
different doses and different distributions of track areas in depth for different 
scales of the sample were evaluated. Based on the scaling hypothesis large-
scale curve were constructed, critical exponents for this percolation model was 
established.  
Scale was changed in the following way: the sample size was increased in 
two times, irradiation dose – in four times, the number of spheres in the chain 
in two times (to make modified structure density and percolation effects didn’t 
change). 
For each scale dependence of the appearance frequency of first spanning 
cluster (percolation) on the number of spheres was obtained, in each case deal 
number of areas, where percolation is most frequent, was defined. This value 
corresponds to a value of modified material share, which is the percolation 
threshold for corresponding finite size of sample and will approach the percola-
tion threshold of the infinite cluster. Using the percolation threshold depend-
ence on the number of track regions in the sample the percolation threshold of 
the infinite cluster was determine. 
Differences between the critical exponents of this model and the well-
known model of uniform distributed spherical domains in volume (continuous 
percolation model) were received. 
CONCLUSIONS 
As a result, it was found that the difference critical exponents of this mod-
el and the continuous percolation model indicates that the dependence of the 
modified structure area on the dose and the angle related with the distribution 
of track regions in the depth of the sample. Also, correlation of individual track 
regions result in higher ratio of the critical exponents than in a continuous per-
colation model, that talks about the higher connectivity of track regions struc-
ture in this model. Also it causes fact that percolation threshold in this case is 
below the percolation threshold of continuous percolation. Created model based 
on experiments and has vivid scaling properties, that allows us to predict differ-
ent parameters of the modified structure at different scales of the sample. 
REFERENCES 
[1] Herre O., Wesch W., Wendler E. Phys. Rev. B —1998.—vol. 58 —p.4832. 
[2] Meftah Ⱥ., Brisard F., Costantini J. M. et al. Physical Review B.- 1993. — Vol. 48. 
— P. 920 – 925. 
[3] Gaiduk P.I.,  Nylandsted Larsen A. et  al.  Phys.  Rev. B.,  — 2002. — Vol. 66.— P. 
045316-2 - 045316-3. 
 
  


Critical exponents in percolation model of track regions with different depth distribution

Electronic Sumy State University Institutional Repository

Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part II                  355  CRITICAL EXPONENTS IN PERCOLATION MODEL  OF TRACK REGIONS WITH DIFFERENT DEPTH  DISTRIBUTION Demchyshyn Andriy, Selyshchev Pavlo  Taras Shevchenko National University of Kyiv, Faculty of Physics , 03187, Ukraine, Kyiv  ABSTRACT As a result of irradiation with Xe with E = 250 MeV in InP at room temperature [1] defects, similar to the "chain of pearls”, which are placed along the trajectory of the ions at depths ranging from 35 to 100 nm and from 7 to 10 microns have been identified. Such defects called tracks, and they can occur at different depths and have different shapes. Tracks were examined like a chain of deal spherical regions; it was assumed that each incident ion creates one such chain. In this model, we assume that the track is formed randomly, but in that place of the ion path, where the energy value, which loses each ion to the unity of the way, is above some threshold value. As a result of irradiation the number of tracks will continue to grow, areas of the single tracks modified substance continue to overlap, form of modificated matter becomes more complicated, creating branched structure. Percolation threshold, fraction of spanning cluster modified material for different doses  and  different  distributions  of  track  areas  in  depth  were  evaluated.  Based  on  the  scaling hypothesis  large-scale curve were constructed, critical exponents for this percolation model were established.   Calculated values of critical exponents were compared with the known values for the model of continuous percolation. Parameters of the percolation and critical exponents depend on the distribution of track areas in depth, which indicates the difference in the order parameters of the track structure obtained for different distributions in depth.   Key words: track, branched structures, swift heavy ions, the Monte Carlo method, critical exponents, percolation threshold, percolation.  INTRODUCTION Under swift heavy ions (SHI) irradiation track of various shapes and lengths, filled with a modified (with respect to the material of the sample) ma-terial, are formed in a solid. They are generated as a result of the strong relaxa-tion of electronic excitations and formed along the trajectory of the ion. Tracks are beginning both from the irradiated surface and at some distance from it. Continuous and discontinuous cylindrical and spherical track field were found. Elongated defects, similar to the "chain of pearls" that are placed along the trajectory of the ions at depths ranging from 35 to 100 nm and from 7 to 10 microns, were found after irradiation by xenon ions with E = 250 MeV in InP at room temperature [1]. Also in the iron garnet irradiated by ions with energies 356                  Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part II            around 12 MeV in a mode of high speed [2] with the energy loss in the range of 4.5 -7 KeV·nm^(-1) the appearance of spherical domains of the modified mate-rial were observed. Discrete tracks point-shaped and oblong dark spots with a diameter equal to an average of about 3 - 10 nm along the trajectories of incom-ing ions in the form of a "chain of pearls" with a number of "pearls" in the track of two to five pieces were observed on the bright-field pictures in the paper [3]. Physical mechanisms of track formation, probability distribution of their formation along the trajectory and in the chain have not been yet fully clarified. The distribution of the tracks in the depth of the sample did not correlate with either the distribution of implanted ions, or the distribution of point defects formed by the ions. The experimental results do not uniquely determine the distribution of tracks. However, it is widely recognized that the track is formed as a result of re-distribution of the energy transmitted by one ion in the electron subsystem. Also it is known that the track is formed randomly in that place of the ion path, where  the  value  of  missing  energy  per  unit  of  ion  path  is  above  a  certain  threshold, in a place where the rush of energy release is presented. It was noted that the simulation results of energy spectra using the SRIM 2008, shows that the maximum energy during the passage of high-energy ions are also located in an average of 25-40 nm. From the experimental data it is known that the tracks in the form of a chain of spheres are created at different depths (between 0 and 50 nm ,between 0 and 500 nm) for different materials and conditions of expo-sure. For example, for InP irradiated by xenon ions, the average distance be-tween tracks spherical averages of 25-40 nm, length of the chain is in the range from 50 nm to 150 nm. Study of changes in material properties as a result of electronic excitation upon irradiation by fast heavy ions and the creation of latent tracks in these materials is an exciting and promising field for new research and development. However, despite the relevance of the topic and practical significance of the expected results, the formation of branched structures of the individual tracks has been insufficiently studied. FORMULATION OF THE PROBLEM We simulated a sample in the form of plane-parallel plate with a thickness of 50 to 300 nm. For the calculations was chosen fragment of the plate in the shape of the box size from 50 to 300 nm. The case where the ion energy is suffi-cient to create a chain of spheres modified substances, energy is 20-40 MeV / amu around was modeled. Each incident ion creates a track in a sequence of a certain number of spherical regions. The trajectories of all ions are parallel to each other. In this model, we simulate different distribution areas in depth, such as: a) a sphere in the chain b) a sphere in a chain or two areas in the chain) is a sphere or two areas or three areas. And so on up to six spheres in the chain. Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part II                  357  The first spherical region is equally likely to appear anywhere in the seg-ment of the trajectory of an ion from its point of penetration into the sample to the point of 2ȡ, where ȡ - the average distance between the spheres. In the calcula-tions ȡ is equal to 40 nm. The distance from the first sphere to the second is cho-sen randomly from range of values between 0 and 2ȡ, the form of distribution doesn’t depend on the «history» (it is the same as for the preliminary scope, only shifted along the ion path in the appearance point of the previous sphere). Some specific distribution of tracking areas in depth was obtained as a result, distribu-tion for the case when there are maximum three spheres in the chain is shown in Fig.1. Consecutively two surfaces were exposed to radiation. With the irradia-tion modified substance areas of individual tracks start to overlap, as a result modified form of matter becomes more complicated, forming a branched structure. Some part of this branched structure is bordered on one surface of the sample, while others lie on the boundary with the opposite surface. When these parts connect one to another, they will create so-called "spanning cluster," which percolates from one surface to another. RESULTS AND DISCUSSION Computed de-pendence of the branched structure surface area on the dose and from different angles has previously been reported (Fig.2).  This nontrivial dependence is related with the distribution of track regions in depth, namely, the correlation between the modified regions of different tracks as a "chain of spheres." Scaling hy- Fig.1.- Distribution for a chain with one, two or three spheres.  Fig.2. - Dependence of the branched structure surface area on the dose and from different angles 358                  Nanomaterials: Applications and Properties (NAP-2011). Vol. 2, Part II            pothesis has been used to separate influence of this distribution on the angular and dose dependence of structural parameters. Percolation threshold, fraction of spanning cluster modified material for different doses and different distributions of track areas in depth for different scales of the sample were evaluated. Based on the scaling hypothesis large-scale curve were constructed, critical exponents for this percolation model was established.  Scale was changed in the following way: the sample size was increased in two times, irradiation dose – in four times, the number of spheres in the chain in two times (to make modified structure density and percolation effects didn’t change). For each scale dependence of the appearance frequency of first spanning cluster (percolation) on the number of spheres was obtained, in each case deal number of areas, where percolation is most frequent, was defined. This value corresponds to a value of modified material share, which is the percolation threshold for corresponding finite size of sample and will approach the percola-tion threshold of the infinite cluster. Using the percolation threshold depend-ence on the number of track regions in the sample the percolation threshold of the infinite cluster was determine. Differences between the critical exponents of this model and the well-known model of uniform distributed spherical domains in volume (continuous percolation model) were received. CONCLUSIONS As a result, it was found that the difference critical exponents of this mod-el and the continuous percolation model indicates that the dependence of the modified structure area on the dose and the angle related with the distribution of track regions in the depth of the sample. Also, correlation of individual track regions result in higher ratio of the critical exponents than in a continuous per-colation model, that talks about the higher connectivity of track regions struc-ture in this model. Also it causes fact that percolation threshold in this case is below the percolation threshold of continuous percolation. Created model based on experiments and has vivid scaling properties, that allows us to predict differ-ent parameters of the modified structure at different scales of the sample. REFERENCES [1] Herre O., Wesch W., Wendler E. Phys. Rev. B —1998.—vol. 58 —p.4832. [2] Meftah Ⱥ., Brisard F., Costantini J. M. et al. Physical Review B.- 1993. — Vol. 48. — P. 920 – 925. [3] Gaiduk P.I.,  Nylandsted Larsen A. et  al.  Phys.  Rev. B.,  — 2002. — Vol. 66.— P. 045316-2 - 045316-3.    

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Critical exponents in percolation model of track regions with different depth distribution

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