A Study of Atomic Displacements Produced in Cascades in Irradiated α-Zr by Using Molecular Dynamics Simulations
We study the cascades’ formation, development and annealing in pure zirconium crystals irradiated in different irradiation conditions. Statistical and geometric properties of cascades are studied in details by varying sample temperature, energy of primary knocked atoms, and direction of their motion...
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Інститут металофізики ім. Г.В. Курдюмова НАН України
2016
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| Cite this: | A Study of Atomic Displacements Produced in Cascades in Irradiated α-Zr by Using Molecular Dynamics Simulations / Yu.M. Ovcharenko, S.V. Kokhan, D.O. Kharchenko, X. Wu, B. Wen, L. Wu, and W. Zhang // Металлофизика и новейшие технологии. — 2016. — Т. 38, № 10. — С. 1303-1320. — Бібліогр.: 38 назв. — англ. |
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| author | Ovcharenko, Yu.M. Kokhan, S.V. Kharchenko, D.O. Wu, X. Wen, B. Wu, L. Zhang, W. |
| author_facet | Ovcharenko, Yu.M. Kokhan, S.V. Kharchenko, D.O. Wu, X. Wen, B. Wu, L. Zhang, W. |
| citation_txt | A Study of Atomic Displacements Produced in Cascades in Irradiated α-Zr by Using Molecular Dynamics Simulations / Yu.M. Ovcharenko, S.V. Kokhan, D.O. Kharchenko, X. Wu, B. Wen, L. Wu, and W. Zhang // Металлофизика и новейшие технологии. — 2016. — Т. 38, № 10. — С. 1303-1320. — Бібліогр.: 38 назв. — англ. |
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| description | We study the cascades’ formation, development and annealing in pure zirconium crystals irradiated in different irradiation conditions. Statistical and geometric properties of cascades are studied in details by varying sample temperature, energy of primary knocked atoms, and direction of their motion. A possibility of channelling at cascades development is shown; it results in formation of crowdions. A change in statistical properties of the crystal during cascades’ development and a relaxation time of cascades are studied. A possibility of formation of different-type defects after cascades’ annealing is discussed.
У даній роботі досліджуються процеси формування каскадів, їх проходження та відпал у чистих кристалах цирконію за різних умов опромінювання. Вивчаються статистичні та геометричні властивості каскадів за різних температур, енергій первинно вибитих атомів і напрямків їх направленого руху. Показано можливість каналювання при еволюції каскадів, що приводить до утворення краудіонів. Досліджено статистичні особливості проходження каскадів і характерний час відпалу. Проведено аналізу процесів формування структур точкових дефектів.
В данной работе исследуются процессы формирования каскадов, их прохождение и отжиг в чистых кристаллах циркония при различных условиях облучения. Изучаются статистические и геометрические свойства каскадов при различных температурах, энергиях первично выбитых атомов и направлениях их движения. Показана возможность каналирования при прохождении каскадов, что приводит к образованию краудионов. Исследованы статистические особенности прохождения каскадов и характерное время отжига. Проведён анализ процессов формирования структур точечных дефектов.
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1303
ВЗАИМОДЕЙСТВИЯ ИЗЛУЧЕНИЯ И ЧАСТИЦ
С КОНДЕНСИРОВАННЫМ ВЕЩЕСТВОМ
PACS numbers:31.15.xv, 34.20.-b,61.72.Bb,61.72.Cc,61.72.jd,61.72.jj,61.80.Az, 61.85.+p
A Study of Atomic Displacements Produced in Cascades
in Irradiated -Zr by Using Molecular Dynamics Simulations
Yu. M. Ovcharenko, S. V. Kokhan, D. O. Kharchenko, X. Wu*,
B. Wen*, L. Wu*, and W. Zhang*
Institute of Applied Physics, N.A.S. of Ukraine,
58 Petropavlivska Str.,
40000 Sumy, Ukraine
*The First Institute, Nuclear Power Institute of China,
328, the 1
st
Section, Changshundadao Road,
Shuangliu, Chengdu, China
We study the cascades’ formation, development and annealing in pure zirco-
nium crystals irradiated in different irradiation conditions. Statistical and
geometric properties of cascades are studied in details by varying sample
temperature, energy of primary knocked atoms, and direction of their mo-
tion. A possibility of channelling at cascades development is shown; it results
in formation of crowdions. A change in statistical properties of the crystal
during cascades’ development and a relaxation time of cascades are studied.
A possibility of formation of different-type defects after cascades’ annealing
is discussed.
Key words: molecular dynamics, embedded atom method, cascade, point de-
fects.
У даній роботі досліджуються процеси формування каскадів, їх прохо-
дження та відпал у чистих кристалах цирконію за різних умов опроміню-
вання. Вивчаються статистичні та геометричні властивості каскадів за
різних температур, енергій первинно вибитих атомів і напрямків їх на-
правленого руху. Показано можливість каналювання при еволюції кас-
Corresponding author: Yuriy M. Ovcharenko
E-mail: dikh@ipfcentr.sumy.ua
Please cite this article as: Yu. M. Ovcharenko, S. V. Kokhan, D. O. Kharchenko,
X. Wu, B. Wen, L. Wu, and W. Zhang, A Study of Atomic Displacements Produced in
Cascades in Irradiated -Zr by Using Molecular Dynamics Simulations, Metallofiz.
Noveishie Tekhnol., 38, No. 10: 1303—1320 (2016), DOI: 10.15407/mfint.38.10.1303.
Металлофиз. новейшие технол. / Metallofiz. Noveishie Tekhnol.
2016, т. 38, № 10, сс. 1303—1320 / DOI: 10.15407/mfint.38.10.1303
Оттиски доступны непосредственно от издателя
Фотокопирование разрешено только
в соответствии с лицензией
2016 ИМФ (Институт металлофизики
им. Г. В. Курдюмова НАН Украины)
Напечатано в Украине.
1304 Yu. M. OVCHARENKO, S. V. KOKHAN, D. O. KHARCHENKO et al.
кадів, що приводить до утворення краудіонів. Досліджено статистичні
особливості проходження каскадів і характерний час відпалу. Проведено
аналізу процесів формування структур точкових дефектів.
Ключові слова: молекулярна динаміка, метода зануреного атому, каскад,
точкові дефекти.
В данной работе исследуются процессы формирования каскадов, их про-
хождение и отжиг в чистых кристаллах циркония при различных усло-
виях облучения. Изучаются статистические и геометрические свойства
каскадов при различных температурах, энергиях первично выбитых ато-
мов и направлениях их движения. Показана возможность каналирования
при прохождении каскадов, что приводит к образованию краудионов. Ис-
следованы статистические особенности прохождения каскадов и харак-
терное время отжига. Проведён анализ процессов формирования структур
точечных дефектов.
Ключевые слова: молекулярная динамика, метод погруженного атома,
каскад, точечные дефекты.
(Received May 25, 2016)
1. INTRODUCTION
From experimental and theoretical studies of pure metals and alloys
used in atomic energy facilities, it is well known that non-equilibrium
point defects produced by particle irradiation are able to manifest self-
organization, where clusters of point defects emerge due to an ensem-
ble of supersaturated non-equilibrium point defect rearrangement [1—
3]. Simplest (point defects) produced by irradiation influence can form
complexes of defects (bi- and three-vacancies) and complexes of inter-
stitials. During their diffusion, they can be captured by sinks, namely
dislocations, grain boundaries, voids, etc. Self-organization of ensem-
ble of defects results in formation of dissipative spatial or spatial-
temporal structures (clusters, voids) [4, 5]. These objects can form su-
perlattices repeating the lattice structure of a sample [4—6]. The for-
mation of such objects is possible in two cases: small clusters can
emerge if two or more defects of one sort (vacancies or interstitials) are
located at scales of several lattice constants, where their agglomera-
tion reduces the elastic energy of the crystal; objects at the nanometre
range (defect walls, voids) can be formed if the defect concentration
exceeds some critical value due to point defects supersaturation. An
observation of point defects’ self-organization into clusters, voids, and
formation of their superlattices were reported in important amount of
papers devoted to experimental and theoretical studies of irradiated
solids (see for example [1, 2, 7, 8] and citations therein).
During irradiation, most of non-equilibrium defects are produced,
A STUDY OF ATOMIC DISPLACEMENTS PRODUCED IN CASCADES IN -ZR 1305
resulting to degradation of physical and mechanical properties of the
corresponding constructions. Therefore, one of the main problems in
radiation material science belongs to study properties of defect micro-
structure formed during sustained irradiation. In general, one of low-
cost and perspective approach to study radiation damages in construc-
tion materials is the multiscale modelling [9], where processes of de-
fect formation, their dynamics, clustering and physical, and mechani-
cal properties change due to defects rearrangement can be analysed
self-consistently at different time and length scales. Among methods
used in multiscale modelling, one can issue ab-initio calculations, mo-
lecular dynamics simulations, phase field crystals modelling, model-
ling spatial dynamics of point defect concentrations within the frame-
work of the reaction rate theory, Monte Carlo simulations, and statis-
tical mechanics computations. Multiscale modelling can predict and
examine behaviour of already used structural materials by reducing
time and costs compared to corresponding experimental studies, and
design novel materials exploited in extreme non-equilibrium condi-
tions. The realization of such a multiscale modelling procedure is a
very ambitious problem requiring a huge database of structural ele-
ment parameters, the development of corresponding computation
codes cross-linked with different hierarchical levels of description,
and huge computational resources. Considering some narrow problems
related to study special aspects of above phenomena in most of cases,
one can use a combination of few approaches of this scheme. For exam-
ple, formation and dynamics of defects can be studied by combining at
least several methods, namely ab-initio computations and molecular
dynamics [10—12] or molecular dynamics and kinetic Monte Carlo sim-
ulations. Some hybrid methods such as phase-field crystal approach
[13—15] can be used to describe structural disorder formation, motion
of point and linear defects (dislocations), and transformations at mi-
cro- and mesoscopic scales [16—18]. As shown in Refs. [19—22], a com-
bination of phase field theory and elasticity theory allows one to study
a rearrangement of elastic stresses in the formation of defects in alloys
[23]. Self-consistent approaches combining phase field theory and dis-
location dynamics [24—26] was applied to explore the influence of mo-
bile dislocations onto phase decomposition processes [27]. To study the
formation of spatial-temporal structures on a surface of irradiated
samples, a combination of kinetic Monte Carlo and continual ap-
proaches gives results corresponding to the experimentally observed
picture of surface patterning [28, 29].
A development of methods for multi-scale modelling allows one to
study specific materials used in real reactors, for example iron-based
or zirconium-based alloys. Considering zirconium as a base element,
one can note that zirconium as a fuel cladding material used for light
and heavy water reactors is established due to combination of its prop-
1306 Yu. M. OVCHARENKO, S. V. KOKHAN, D. O. KHARCHENKO et al.
erties. It has good corrosion resistance to high temperature, high-
pressure water and steam, it manifests good mechanical ductility and
stress, and it has low thermal neutron cross section. At the same time,
it is exploited for reactor core internals such as fuel element supports,
pressure tubes, core support members, and, different instrumentation
assemblies. It has the following useful properties which are superior
comparing to stainless steels to such as: corrosion resistance, strength
under neutron bombardment, neutron economy, its use as fuel clad-
ding elements allows one to exclude water chemistry problems, it man-
ifests suitability for primary steam systems, it is widely used in non-
nuclear applications (for example, in chemistry processing plants as
chemical reactor vessels). At the same time, zirconium cannot be weld-
ed to iron, copper, or nickel based alloys due to formation of brittle in-
termetallics. Zirconium is characterized by low solubility of impurities
like phosphorus, silicon, carbon.
As far as this material and alloys based on pure zirconium are used
as a construction material in atomic energy facilities, in real situa-
tions, it usually affected by neutron and ion irradiation. Therefore, a
study of cascades development in such materials is allows one to under-
stand reason leading to physical and mechanical properties degrada-
tion under particle irradiation. From practical viewpoint, main mate-
rials are binary and ternary alloys with small concentration of alloying
elements (up to two percent). Hence, the main problem in their study
lies in description of defect microstructure behaviour. From the other
hand, one should know properties of defect microstructure evolution
in pure materials in order to compare dynamics of defects in alloys and
make some recommendations for using these materials in real reactors,
to extend an exploiting time of such materials.
Therefore, in this work, we focus our attention in studying defect
microstructure behaviour in pure zirconium as a based element for
cladding materials. The aim of the work is to study dynamics of cas-
cade development in -Zr crystal irradiated at different temperatures
and direction of primary knocked atom with different direction of
their motion. In our study, we use molecular dynamics simulation
technique, where embedded atom methodology is used to describe be-
haviour of defect microstructure in real metallic system, namely, pure
-Zr crystal. We consider not only number of defects in cascades and a
change of geometrical properties of cascade at different irradiation
conditions; here, attention is also paid to study statistical properties
change in irradiated crystals. We analyse defects remaining after cas-
cade annealing.
The work is organized in the following manner. In Section 2, we dis-
cuss methodology of our simulations. Section 3 is devoted to study
simulation results. Here, we initially discuss methods used to correct
identification of atoms belonging to cascades. Next, we consider sta-
A STUDY OF ATOMIC DISPLACEMENTS PRODUCED IN CASCADES IN -ZR 1307
tistical properties of cascades, lifetime for cascades and types of de-
fects realized after cascade annealing. Finally, we conclude in Sec-
tion 4.
2. METHODOLOGY OF SIMULATION PROCEDURE
To study dynamics of atomic displacements realized in cascades in sol-
ids subjected to an irradiation influence, we will use the molecular dy-
namics (MD) methods, realized in programming programs package
LAMMPS [30]. We perform MD simulation for pure -Zr by using the
interatomic interaction potential [31], obtained with the help of em-
bedded atom method [32]. All calculations were done for 3D Zirconium
crystal containing 1.8105
atoms by using GPU computing, which al-
lows us the computations accelerated by 20 times comparing to the
standard CPU one.
A simulation procedure corresponds to numerical solutions of clas-
sical Newton equations,
2
ext
2
i i
i i
i
d U
m
ddt
r
F
r
,
for each of 1.8105
atoms. Here,
ext
iF is the vector of all equilibrated
forces related to potential energy of i-th atom defined by the radius
vector ri and the mass mi.
All calculations were performed according to the potential No. 3 for
pure -Zr taken from the work [31], where embedded atom method was
exploited [32]. A choice for the interaction potential No. 3 is stipulated
by following criteria, which are made to study cascades’ dynamics. It
gives correct results comparing to experimental study of crystal lattice
behaviour of -Zr at elevated temperatures and corresponds good to
define defects formation energies at irradiation influence. According
to the embedded atom method, for the interatomic potential energy,
one has ( ( )) (1/2) ( ),i ij iji j i j
U F
r r where rij is a radius-
vector between two i-th and j-th atoms, is an interaction function of
pairwise distance, is a contribution related to electronic charge den-
sity acting from j-th atom onto i-th atom, F is an ‘embedding’ func-
tion related to energy needed to put i-th atom into the electronic cloud.
In MD simulation procedure, there are several limitations, approx-
imations and parameters, which define accuracy of obtained results.
Among them, the most important ones are interatomic interaction po-
tential, simulation time-step, total simulation time, initial conditions,
etc. Hence, the first important task in MD simulations is to prepare a
model system for further study.
In Table 1, we collect the main parameters used for simulations of
1308 Yu. M. OVCHARENKO, S. V. KOKHAN, D. O. KHARCHENKO et al.
atomic displacements in cascades and to study defect structure evolu-
tion in -Zr crystal subjected to irradiation influence.
In cascades study, it is important that the maximal size of cascade
should be less than the size of the studied crystals. Hence, the linear
size of the model can be varied with a change in the energy of bombard-
ing particle, affected the maximal size of cascade. The most important
parameters for our MD simulations are the thermostat parameter
0.2 ps and the simulation time-step t (0.1—1) fs. The thermostat
parameter is the mean time when energy exchanges between the sys-
tem and the thermostat are realized. As far as energy in the crystal
passed from atom to atom, therefore, the mean time for an energy ex-
change between two atoms or between atom and thermostat is defined
by the mean period of atom oscillations. Hence, the quantity is the
mean period of atom oscillations in a crystal. The simulation time-step
t should be much more less than the quantity for detailed study of
atoms’ positions.
In our computing, we use two values for simulation time step. The
minimal one was chosen as /2000 , that gives t 0.1 fs. This value
was used at that time, when cascade of atomic displacements occurs
and large number of atoms attains large values of kinetic energy com-
paring to atoms, which do not take part in cascade. The maximal value
for the simulation time-step was fixed as t 1 fs ( /200 ). It was used
for simulations before cascade appearance and after its relaxation.
The value of can be obtained in the framework of harmonic ap-
proximation in the following way. It is known that atoms of the crystal
oscillate with a frequency ranging from zero up to the Debye fre-
quency D. The corresponding distribution function g() over oscillat-
ing frequencies obeys the relation [33]:
2 3
D
( ) 9 / .g N (1)
TABLE 1. The main parameters for cascades modelling in -Zr crystal.
Parameter Value
Simulation box size, nm 16.3116.9515.98
Number of atoms N 1.8105
Crystal temperature T, K 300, 400, 500
Primary knock-on atom (PKA) energy EPKA, keV 2, 6, 10
PKA moving direction <0001>, 0110
Thermostat parameter , ps 0.2
Time step t, fs 0.1—1
Total time of simulation t, ps 610
A STUDY OF ATOMIC DISPLACEMENTS PRODUCED IN CASCADES IN -ZR 1309
Rewriting Eq. (1), one can obtain the distribution function over oscil-
lation period as follows:
3
3 4
D
72
( ) .
N
g T (2)
By taking into account that the total number of elastic waves generat-
ed by atoms in the crystal is equal to 3N, one can write a formula for
the mean period of atomic oscillations:
D
3
3 3
DD2 /
24 3
.
d
(3)
By using the experimental data for the Debye temperature, TD 291 K,
for pure -Zr and the relation
D D
/ ,kT where , k are the Boltz-
mann and Dirac constants, respectively, one finds the mean period of
atomic oscillations, 0.2 ps .
3. RESULTS AND DISCUSSIONS
3.1. Cascade Domains Identification Procedure
During cascades formation and their evolution, domains of structural
disorder are formed, where crystalline lattice is broken totally. Here,
most of atoms attain kinetic or potential energy, which differs essen-
tially from that out of cascade. It allows one to distinguish cascade do-
main in a crystal according to values of kinetic and potential energies
of atoms. However, this procedure of cascade identification cannot
give the total information about the size of the cascade domains and
the number of atoms in cascades with good accuracy. Indeed, a finite
number of atoms belonging to the structural disorder configuration
due to their chaotic location can have both kinetic and potential ener-
gies of the same values that outside cascade atoms have. Therefore,
this procedure should generalize by considering structural configura-
tions of atoms in the crystal during the cascade development.
In our simulations, we initially prepare the sample of -Zr at fixed
temperature with ideal structure configuration. To distinguish atoms
belonging to cascade and atoms of a bulk -phase, the Common Neigh-
bour Analysis (CNA) was exploited [34—36]. According to this ap-
proach, two nearest neighbour atoms assumed to be bounded if the dis-
tance between them does not exceed the cutting radius rcut. Its value
should lie in the interval between two peaks in a radial distribution
function of atomic positions [37]. Obviously, a suitable choice of values
of rcut is the mean value between distance between selected atom to it
nearest neighbours and a distance between selected atom and next
1310 Yu. M. OVCHARENKO, S. V. KOKHAN, D. O. KHARCHENKO et al.
nearest neighbours. According to this scheme, one can choose a value
for the cutting radius for different type of crystalline lattices.
In Figure 1, we plot graphical explanation of cascade identification
by using values of kinetic and potential energies and the CNA method.
The PKA energy was taken EPKA 10 keV at temperature T 300 K;
PKA direction is <0001>; the time for the cascade development is
t 5.4 ps.
By comparing energy difference of all atoms (see Fig. 1, a), one finds
that only NC 8314 number of atoms of the crystal can be distin-
guished as atoms belonging to the cascade. The cascade surface and
volume of the cascade are SC 4311.88 nm2
and VC 23282.1 nm3. By
using the CNA method, we can mark atoms by phase compound differ-
ence, which belong to the cascade with h.c.p. symmetry breaking (see
Fig. 1, b). Indeed, it becomes possible as far as at initial stage (before
cascade formation) all atoms obey h.c.p. symmetry. In this case, one
gets the following information: the number of atoms having h.c.p.
symmetry breaking is NC 15292, whereas the cascade surface and its
volume take values SC 4808.40 nm2
and VC 35378.2 nm3, respective-
ly. Here, we need to stress that the CNA method cannot give total in-
formation about cascade properties, if it is used separately. Indeed,
due to chaotic atomic mixing in cascades, some atoms can randomly
have configuration of nearest neighbours allowing one to identify
them as atoms with h.c.p. symmetry. In such a case, the CAN method
fails, and these atoms with randomly formed h.c.p. configuration can
be identified as atoms out of cascades. In other time step, microstruc-
ture of cascade is changed, some other atoms in cascade can be charac-
terized randomly by the h.c.p. symmetry, and as a result, the CAN
a b c
Fig. 1. The cascade identification in -Zr crystal with N 180000 atoms by
analysing energy of atoms and atom neighbour positions (phase compound) at
T 300 K, EPKA 10 keV, PKA direction is <0001> at time t 5.4 ps (cross-
section of the crystal with a cascade): identification by energy only (a); identi-
fication by phase compound only (b); identification by energy and phase com-
pound (c).
A STUDY OF ATOMIC DISPLACEMENTS PRODUCED IN CASCADES IN -ZR 1311
method can fail in this way. To avoid such ambiguity, we propose to
combine above two methods for correct identification of atoms in cas-
cades. Therefore, next, we consider phase compound of atoms by using
the CAN method and analyse difference in kinetic and potential ener-
gies of all atoms in the system. The corresponding illustration of such
combined approach application is shown in Fig. 1, c. Here, we get the
correct number of atoms in cascade NC 15763, surface and volume of
excited ensemble of atoms SC 4792.53 nm2, VC 36407.8 nm3, consid-
ered below as real cascade.
The corresponding analysis of volume and surface of all studied cas-
cades of their evolution allows us to conclude that all atoms inside the
disturbed ensemble are identified as atoms belonging to cascades.
Therefore, the proposed combined approach can be used well to study
properties of cascades in h.c.p. crystals. It can be developed to study
cascades formation and their evolution in metals and alloys with other
symmetries.
3.2. Statistical Characteristics of Cascades
By using above generalized approach for cascades identification, we
studied dynamics of the number of atoms in cascades, cascade surface
and volume, respectively, by changing energy of PKA and the temper-
ature of the crystalline system. In our study, we have considered
changes in of internal energy, volume and density of irradiated sam-
ples.
In Figure 2, snapshots are shown for different stages of one cascade
formation, its evolution and annealing in pure -Zr, where the temper-
ature of unirradiated sample was chosen as T 300 K, the energy for
PKA we set EPKA 10 keV, PKA moves in the direction <0001> (see
Fig. 2, a). According to obtained results, one finds that, during time
interval 0.5 pst , the cascade attains it maximal size (cf. Figs. 2, b,
c) related to a thermal spike. After this stage, the annealing of cascade
is realized.
Let us consider stages of cascade development in detail. At initial
stages of cascade formation when energy of knocked atoms takes ex-
tremely large values comparing to other atoms of the crystal, channel-
ling processes can be realized. It is well seen in Fig. 2, a (thin part of
the cascade). In order to explain possibility of channelling, we have
considered trajectories of particles responsible for cascade develop-
ment. The corresponding atomic configuration and trajectories of at-
oms at initial stages of cascade formation are shown in Figs. 3, a, b. It
is well seen that, along indicated direction of motion of PKA, it moves
along the line. The channelling distance of PKA is C 11 nm. In time
instants when cascade attains maximal size (thermal spike), one can
observe one-dimensional thickening of atoms meaning formation of
1312 Y
crowdion
processes
straight-
In Fig
fragmen
It is well
large dom
domains
(Fig. 2, d
To ma
late time
the samp
c, e). Tim
number o
symmetr
vided at t
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were cho-
each tem-
vely same
Zr contain-
V: t 0.1 ps
.
A STUDY OF ATOMIC DISPLACEMENTS PRODUCED IN CASCADES IN -ZR 1313
at fixed PKA energy and temperature independently on the direction
of PKA atom motion. Next, we consider their quantitative differences
in details.
We summarize calculated values for maximal cascade volume
(max)
C
V ,
cascade surface
(max)
C
S and number of atoms in cascades
(max)
C
N in -Zr
crystals at different vales of PKA energy and different direction of
PKA motions at 300, 400, 500 K in the Table 2.
According to our analysis of obtained data, we have found that at
temperatures of the sample 400 K and 500 K all cascades with PKA di-
rection <0001> are characterized by larger sizes comparing to cas-
cades simulated at the same other parameters with PKA direction
0110 . We have found that cascade sizes are increases with an in-
crease in the irradiation sample temperature independently on the
PKA directions. It means that, at elevated temperatures of irradiated
a b
c d
Fig. 3. Channelling and crowdion formation in crystal -Zr during cascade
development: perturbations of atoms at channelling (a), trajectories of atoms
in cascade at channelling (b), perturbation of atoms at crowdions formation
(c), trajectories of atoms at crowdions formation (d).
1314 Yu. M. OVCHARENKO, S. V. KOKHAN, D. O. KHARCHENKO et al.
sample, irradiation damages will grow. Illustrations for cascades at
their maximal sizes allow one to see crowdions formation as was shown
in Fig. 3, c.
a b
c d
e f
Fig. 4. Time dependences of statistical parameters of cascades at tempera-
tures 300 K and 500 K; PKA direction is <0001>. Panels a—f correspond to
dynamics of internal energy, cascade volume, volume of the sample, cascade
surface, density of the sample, and number of atoms in cascade, respectively.
A STUDY OF ATOMIC DISPLACEMENTS PRODUCED IN CASCADES IN -ZR 1315
3.3. Determination of Cascades Lifetime
One of the important quantities in studying cascade dynamics is the
relaxation time of cascades C (lifetime of cascade). To define this
quantity, one should consider dynamics of the kinetic and potential
energy of the system during cascade evolution. Indeed, as far as, even
at large values of the energy of PKA, the cascade will have macroscopic
size, one can expect that, in a massive sample, a temperature of cascade
will tend to the temperature of the sample very fast, whereas crystal
will have unrelaxed (unannealed) defect structure characterized by
large amount of metastable non-equilibrium defects. By assuming the
temperature as a quantity related to kinetic energy of all atoms of the
crystal, one can consider dynamics of the corresponding kinetic and
potential energies to define cascade lifetime C.
According to protocols for kinetic and potential energies shown in
Figs. 5, a, b, one finds that, even at time interval of 10 ps, the tempera-
ture (kinetic energy) of the crystal takes values of that before atomic
TABLE 2. Maximal values of cascade volume
(max)
C
V , its surface
(max)
C
S , and
number of atoms in cascade
(max)
C
N in -Zr sample at different values of PKA
energy, direction of PKA motion and temperature of the sample.
EPKA, keV 2 6 10
Direction <0001> 0110 <0001> 0110 <0001> 0110
T
3
0
0
K
(max) 3
C
,nmV 8212.26 9104.83 20878.50 21363.80 32161.60 33418.70
(max) 2
C
,nmS 1324.24 1730.95 2709.99 3235.76 4318.11 4374.50
(max)
C
N 3629 4126 9090 9436 13897 14422
Cascade
form
T
4
0
0
K
(max) 3
C
,nmV 9259.53 9195.95 24741.00 24449.10 37117.80 32880.60
(max) 2
C
,nmS 1557,20 1470.88 3626.45 3821.14 5230.81 4181.34
(max)
C
N 4163 4022 10700 10759 16216 14367
Cascade
form
T
5
0
0
K
(max) 3
C
,nmV 9402.16 8659.34 25781.10 25651.60 41374.00 37973.80
(max) 2
C
,nmS 1743.58 1276.91 3809.81 3503.99 5175.19 4685.57
(max)
C
N 4255 3838 11122 10984 17551 16179
Cascade
form
1316 Yu. M. OVCHARENKO, S. V. KOKHAN, D. O. KHARCHENKO et al.
collisions (around initial temperature). As far as crystal after this time
interval is characterized by large amount of non-equilibrium defects,
which are able to diffuse along the crystal, the potential energy will
change slowly comparing to kinetic energy. Therefore, to calculate
cascade lifetime, one can study dynamics of potential energy as most
appropriate quantity to measure lifetime for non-equilibrium (meta-
stable) defect structure in the sample.
To determine cascade lifetime C, one can use protocols for the num-
ber of defects in cascade from one hand and a relaxation time for po-
tential energy. Here, we have restriction related to determination of
the number of atoms in cascades. The number of defects in cascade is
determined by very complicated procedure by taking into account two
criteria related to energy difference of atoms in the sample and the
CNA method. Therefore, one can use the simplest way to define cas-
cade lifetime as a time when the potential energy Ep of the sample de-
creases in 2.71e times. Here, we do not need to take care about con-
figurations of nearest neighbours of selected atom, all needed infor-
mation about ensemble of non-equilibrium defects can be found in pro-
tocols for potential energy of the crystal. To approximate obtained da-
ta in protocols for Ep, we use standard exponential decaying fitting,
p p0 pd C
exp( / ),E E E t where Ep0 is the potential energy before cas-
cade formation; Epd is the energy contribution responsible for nonequi-
librium defects formation.
In Table 3, calculated values of cascade lifetime C depending on
PKA energy EPKA and PKA direction are shown at different tempera-
tures.
It is logically explained that cascade lifetime C will increase with
growth in PKA energy EPKA at fixed temperature. Indeed, the larger
energy of PKA the larger will be cascade size. Therefore, the formed
crowdions, which will move from the centre of the cascade, will be sep-
a b
Fig. 5. Protocols for temperature (kinetic energy) T and potential energy Ep of
the sample.
A STUDY OF ATOMIC DISPLACEMENTS PRODUCED IN CASCADES IN -ZR 1317
arated at elevated distances. As usual, crowdions are related to inter-
stitials. Hence, after very short time interval of cascade annealing, one
can observe separated interstitials in the crystal. As far as initially,
the studied sample had ideal crystalline structure, therefore, the num-
ber of formed interstitials after cascade thermal spike should corre-
spond to the number of vacancies. If we increase the energy of PKA,
then, the number of such point defects (Frenkel pairs) will grow and
domain size of cascade development will increase. It means that the
crystalline system should have much more time for relaxation related
to recombination of vacancies and interstitials for large cascades.
Comparing data related to different irradiation temperatures, one
finds that initial temperature of the sample does not affect onto cas-
cade life-time at low PKA energies (EPKA 2 keV, 6 keV); whereas, at
elevated PKA energy (EPKA 10 keV), the life time increases with the
sample temperature increase. Comparing data for C for different PKA
directions at the same values of PKA energy and temperature, one
finds that the life time is lower if the 0110 PKA direction is cho-
sen comparing to PKA direction <0001> at low temperatures of the
sample.
3.4. Type of Defects after Cascade Relaxation
During cascades annealing, a large amount of point defects remains in
a crystal comparing to their equilibrium values. These defects lead to
an emergence of uncompensated stresses in a crystal that increases to-
tal energy of the crystalline system. To compensate such stresses and
to reduce the energy of the system, point defects are arranged into
clusters and, therefore, from energetic viewpoint, a cluster of defects
is more energetically favourable than dissolution of separated non-
equilibrium defects in a bulk. In our simulations in -Zr crystals, it
was found that special type of defects can emerge, where some of them
can be considered as stable and some are metastable ones (see Fig. 6).
Here, one can observe separated interstitials (se Fig. 6, a) and intersti-
tials in a form of a dumbbell (two nearest interstitials are located be-
TABLE 3. Relaxation time for cascades in -Zr crystal.
EPKA, keV 2 6 10
PKA
direction
<0001> 0110 <0001> 0110 <0001> 0110
(300 K)
C
,ps 1.63 1.19 2.13 2.09 2.33 2.22
(400 K)
C
,ps 1.43 1.39 1.96 1.85 2.35 2.34
(500 K)
C
,ps 1.35 2.32 2.19 2.44 2.65 2.85
1318 Yu. M. OVCHARENKO, S. V. KOKHAN, D. O. KHARCHENKO et al.
tween nearest atomic planes) [38], clusters of interstitials (three near-
est interstitials shown in Fig. 6, b), single vacancy, bi-vacancy clus-
ters, and three-vacancy clusters (see Figs 6, c, d, e, f), and vacancy
clusters with more than three vacancies (see Figs. 6, g, h).
We identify positions of such defects by using potential energy val-
ues of atoms and centrosymmetry parameter (CSP) defined as follows:
V /2
2
/2
1
,
N
i i N
i
CSP r r (3)
where ri and riN/2 are radius-vectors for spatial coordinates computed
from a marked central atom toward pair of diametrically opposite
neighbour atoms with respect to central one, NV is a coordination num-
ber for the crystal (for pure -Zr one has NV 12) [36].
To identify defects and their position, we took into account the fol-
lowing criteria. If there are not defects near the marked atom, then
CSP takes values about zero. However, in the case of symmetrical loca-
tion of pair of defects (see, for example, Fig. 6, g), CSP takes around
f 2.63eV f 5.97eV
f 2.32eV
f 4.35eV
a b c d
f 4.35eV metastable metastable
metastable
e f g h
Fig. 6. Types of defects, their formation energy f, and defects’ life-time
emergent after cascade annealing at short time scales: single interstitial (a),
three-interstitial cluster (b), single vacancy (c), bi-vacancy cluster (d), bi-
vacancy cluster with vacancies located in nearest atomic planes (e), three-
vacancy cluster (two vacancies are in the same plane and one vacancy in near-
est plane) (f), three-vacancy with one interstitial (metastable object) (g), va-
cancy cluster containing six vacancies and two interstitials (metastable ob-
ject) (h).
A STUDY OF ATOMIC DISPLACEMENTS PRODUCED IN CASCADES IN -ZR 1319
zero values. In such a case, one should analyse potential energy values
(values of the potential energy of the atom in defect free zone and zone
containing defects are different always). It was found that clusters of
defects in Figs. 6, b, d, e are stable; they can change their position and
diffuse, whereas clusters in in Figs. 6, f, g, h, are metastable; they can
transformed in more stable configurations.
4. CONCLUSIONS
Molecular dynamics simulations in pure -Zr crystals irradiated at dif-
ferent irradiation regimes allows us to find following regularities in
defect microstructure behaviour.
At initial stages of cascades formation when kinetic energy of
knocked atoms takes extremely large values comparing to that in a
bulk, channelling processes are realized. These processes lead to atomic
displacements far from the centre of cascades. It was shown that, at
elevated energies of primary knocked atoms up to 10 keV, the channel-
ling distance takes value around 11 nm. A time interval before thermal
spike is around 0.4 ps; it does not depend on the temperature of the
crystal, energy and direction of motion of primary knocked atom. It
was shown that at time instants related to thermal spike emergence
crowdions are formed. These objects move from the centre of cascade
toward undisturbed zone of the bulk leading to formation of intersti-
tials. Considering change of geometric properties of cascades, we have
shown that the size of studied cascades (volume, number of atoms in
cascades) increases with the temperature growth. By studying defect
microstructure evolution at cascade annealing, it was found that the
relaxation time of cascades could play a role of an effective parameter
describing radiation damages at molecular dynamics simulations.
This work was financially supported by the Sichuan Province Inter-
national Science and Technology Cooperation and Exchange Research
Program (2016HH0014) and the China Postdoctoral Science Founda-
tion (2015M582575).
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/HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0 i kasnijim verzijama.)
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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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| id | nasplib_isofts_kiev_ua-123456789-112628 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1024-1809 |
| language | English |
| last_indexed | 2025-12-07T18:14:11Z |
| publishDate | 2016 |
| publisher | Інститут металофізики ім. Г.В. Курдюмова НАН України |
| record_format | dspace |
| spelling | Ovcharenko, Yu.M. Kokhan, S.V. Kharchenko, D.O. Wu, X. Wen, B. Wu, L. Zhang, W. 2017-01-24T12:43:02Z 2017-01-24T12:43:02Z 2016 A Study of Atomic Displacements Produced in Cascades in Irradiated α-Zr by Using Molecular Dynamics Simulations / Yu.M. Ovcharenko, S.V. Kokhan, D.O. Kharchenko, X. Wu, B. Wen, L. Wu, and W. Zhang // Металлофизика и новейшие технологии. — 2016. — Т. 38, № 10. — С. 1303-1320. — Бібліогр.: 38 назв. — англ. 1024-1809 DOI: 10.15407/mfint.38.10.1303 PACS: 31.15.xv, 34.20.-b, 61.72.Bb, 61.72.Cc, 61.72.jd, 61.72.jj, 61.80.Az, 61.85.+p https://nasplib.isofts.kiev.ua/handle/123456789/112628 We study the cascades’ formation, development and annealing in pure zirconium crystals irradiated in different irradiation conditions. Statistical and geometric properties of cascades are studied in details by varying sample temperature, energy of primary knocked atoms, and direction of their motion. A possibility of channelling at cascades development is shown; it results in formation of crowdions. A change in statistical properties of the crystal during cascades’ development and a relaxation time of cascades are studied. A possibility of formation of different-type defects after cascades’ annealing is discussed. У даній роботі досліджуються процеси формування каскадів, їх проходження та відпал у чистих кристалах цирконію за різних умов опромінювання. Вивчаються статистичні та геометричні властивості каскадів за різних температур, енергій первинно вибитих атомів і напрямків їх направленого руху. Показано можливість каналювання при еволюції каскадів, що приводить до утворення краудіонів. Досліджено статистичні особливості проходження каскадів і характерний час відпалу. Проведено аналізу процесів формування структур точкових дефектів. В данной работе исследуются процессы формирования каскадов, их прохождение и отжиг в чистых кристаллах циркония при различных условиях облучения. Изучаются статистические и геометрические свойства каскадов при различных температурах, энергиях первично выбитых атомов и направлениях их движения. Показана возможность каналирования при прохождении каскадов, что приводит к образованию краудионов. Исследованы статистические особенности прохождения каскадов и характерное время отжига. Проведён анализ процессов формирования структур точечных дефектов. This work was financially supported by the Sichuan Province International Science and Technology Cooperation and Exchange Research Program (2016HH0014) and the China Postdoctoral Science Foundation (2015M582575). en Інститут металофізики ім. Г.В. Курдюмова НАН України Металлофизика и новейшие технологии Взаимодействия излучения и частиц с конденсированным веществом A Study of Atomic Displacements Produced in Cascades in Irradiated α-Zr by Using Molecular Dynamics Simulations Дослідження атомових зміщень, створюваних у каскадах у опромінюваному α-Zr, з використанням методів молекулярно-динамічного моделювання Исследование атомных смещений, производимых в каскадах в облучаемом α-Zr, с применением методов молекулярно-динамического моделирования Article published earlier |
| spellingShingle | A Study of Atomic Displacements Produced in Cascades in Irradiated α-Zr by Using Molecular Dynamics Simulations Ovcharenko, Yu.M. Kokhan, S.V. Kharchenko, D.O. Wu, X. Wen, B. Wu, L. Zhang, W. Взаимодействия излучения и частиц с конденсированным веществом |
| title | A Study of Atomic Displacements Produced in Cascades in Irradiated α-Zr by Using Molecular Dynamics Simulations |
| title_alt | Дослідження атомових зміщень, створюваних у каскадах у опромінюваному α-Zr, з використанням методів молекулярно-динамічного моделювання Исследование атомных смещений, производимых в каскадах в облучаемом α-Zr, с применением методов молекулярно-динамического моделирования |
| title_full | A Study of Atomic Displacements Produced in Cascades in Irradiated α-Zr by Using Molecular Dynamics Simulations |
| title_fullStr | A Study of Atomic Displacements Produced in Cascades in Irradiated α-Zr by Using Molecular Dynamics Simulations |
| title_full_unstemmed | A Study of Atomic Displacements Produced in Cascades in Irradiated α-Zr by Using Molecular Dynamics Simulations |
| title_short | A Study of Atomic Displacements Produced in Cascades in Irradiated α-Zr by Using Molecular Dynamics Simulations |
| title_sort | study of atomic displacements produced in cascades in irradiated α-zr by using molecular dynamics simulations |
| topic | Взаимодействия излучения и частиц с конденсированным веществом |
| topic_facet | Взаимодействия излучения и частиц с конденсированным веществом |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/112628 |
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