Bias for prismatic dislocation loops in zirconium. Numerical analysis
Using the analytical expression for the energy of elastic interaction of radiation point defects with a prismatic edge dislocation loop of zirconium (Burgers vector bᴰ=1/3⟨1120⟩, {1120} occurrence plane), the bias of loops of different nature (vacancy and interstitial) was calculated by the finite d...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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| Цитувати: | Bias for prismatic dislocation loops in zirconium. Numerical analysis / O.G. Trotsenko, A.V. Babich, P.M. Ostapchuk // Problems of Atomic Science and Technology. — 2023. — № 2. — С. 3-7. — Бібліогр.: 17 назв. — англ. |
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Trotsenko, O.G. Babich, A.V. Ostapchuk, P.M. 2023-12-10T12:43:58Z 2023-12-10T12:43:58Z 2023 Bias for prismatic dislocation loops in zirconium. Numerical analysis / O.G. Trotsenko, A.V. Babich, P.M. Ostapchuk // Problems of Atomic Science and Technology. — 2023. — № 2. — С. 3-7. — Бібліогр.: 17 назв. — англ. 1562-6016 PACS:62.20Dc;62.20.Fe DOI: https://doi.org/10.46813/2023-144-003 https://nasplib.isofts.kiev.ua/handle/123456789/196089 Using the analytical expression for the energy of elastic interaction of radiation point defects with a prismatic edge dislocation loop of zirconium (Burgers vector bᴰ=1/3⟨1120⟩, {1120} occurrence plane), the bias of loops of different nature (vacancy and interstitial) was calculated by the finite difference method. The toroidal geometry of the reservoir was used. It allowed one to calculate biases for loops of any size without any correction of the elastic field in its area of influence. In the dilatation center approximation the dependences of the loop bias on the loop radius were obtained. The principal possibility of coexistence of loops of different nature in the prismatic plane of zirconium is shown. A qualitative concept of the radiation growth (RG) mechanism was formulated within the framework of the classical elastic ideology. Використовуючи аналітичний вираз для енергії пружної взаємодії радіаційних точкових дефектів із призматичною крайовою дислокаційною петлею цирконію (вектор Бюргерса bᴰ=1/3⟨1120⟩, площина залягання {1120}) методом кінцевих різниць пораховано фактор переваги петель різної природи (вакансіонної та міжвузелової). Використовувалася тороїдальна геометрія резервуара, що дозволяє провести розрахунки для петлі будь-якого розміру без будь-якої корекції пружного поля в її області впливу. У наближенні центру дилатації отримані залежності фактора переваги петель від їхнього радіусу. Показано важливу можливість спільного співіснування в призматичній площині цирконію петель різної природи. Сформульовано якісну концепцію механізму радіаційного зростання (РЗ) у рамках класичної пружної ідеології. The research presented in this article was financially supported by the Ukrainian government budget program «Government support for priority scientific research and scientific & technical (experimental) developments» (budget financial Code 6541230). en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Problems of Atomic Science and Technology Physics of radiation damages and effects in solids Bias for prismatic dislocation loops in zirconium. Numerical analysis Фактор переваги призматичних дислокаційних петель у цирконії. Чисельний аналіз Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Bias for prismatic dislocation loops in zirconium. Numerical analysis |
| spellingShingle |
Bias for prismatic dislocation loops in zirconium. Numerical analysis Trotsenko, O.G. Babich, A.V. Ostapchuk, P.M. Physics of radiation damages and effects in solids |
| title_short |
Bias for prismatic dislocation loops in zirconium. Numerical analysis |
| title_full |
Bias for prismatic dislocation loops in zirconium. Numerical analysis |
| title_fullStr |
Bias for prismatic dislocation loops in zirconium. Numerical analysis |
| title_full_unstemmed |
Bias for prismatic dislocation loops in zirconium. Numerical analysis |
| title_sort |
bias for prismatic dislocation loops in zirconium. numerical analysis |
| author |
Trotsenko, O.G. Babich, A.V. Ostapchuk, P.M. |
| author_facet |
Trotsenko, O.G. Babich, A.V. Ostapchuk, P.M. |
| topic |
Physics of radiation damages and effects in solids |
| topic_facet |
Physics of radiation damages and effects in solids |
| publishDate |
2023 |
| language |
English |
| container_title |
Problems of Atomic Science and Technology |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Фактор переваги призматичних дислокаційних петель у цирконії. Чисельний аналіз |
| description |
Using the analytical expression for the energy of elastic interaction of radiation point defects with a prismatic edge dislocation loop of zirconium (Burgers vector bᴰ=1/3⟨1120⟩, {1120} occurrence plane), the bias of loops of different nature (vacancy and interstitial) was calculated by the finite difference method. The toroidal geometry of the reservoir was used. It allowed one to calculate biases for loops of any size without any correction of the elastic field in its area of influence. In the dilatation center approximation the dependences of the loop bias on the loop radius were obtained. The principal possibility of coexistence of loops of different nature in the prismatic plane of zirconium is shown. A qualitative concept of the radiation growth (RG) mechanism was formulated within the framework of the classical elastic ideology.
Використовуючи аналітичний вираз для енергії пружної взаємодії радіаційних точкових дефектів із призматичною крайовою дислокаційною петлею цирконію (вектор Бюргерса bᴰ=1/3⟨1120⟩, площина залягання {1120}) методом кінцевих різниць пораховано фактор переваги петель різної природи (вакансіонної та міжвузелової). Використовувалася тороїдальна геометрія резервуара, що дозволяє провести розрахунки для петлі будь-якого розміру без будь-якої корекції пружного поля в її області впливу. У наближенні центру дилатації отримані залежності фактора переваги петель від їхнього радіусу. Показано важливу можливість спільного співіснування в призматичній площині цирконію петель різної природи. Сформульовано якісну концепцію механізму радіаційного зростання (РЗ) у рамках класичної пружної ідеології.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/196089 |
| citation_txt |
Bias for prismatic dislocation loops in zirconium. Numerical analysis / O.G. Trotsenko, A.V. Babich, P.M. Ostapchuk // Problems of Atomic Science and Technology. — 2023. — № 2. — С. 3-7. — Бібліогр.: 17 назв. — англ. |
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ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №2(144) 3
SECTION 1
PHYSICS OF RADIATION DAMAGES AND EFFECTS IN SOLIDS
https://doi.org/10.46813/2023-144-003
BIAS FOR PRISMATIC DISLOCATION LOOPS IN ZIRCONIUM.
NUMERICAL ANALYSIS
O.G. Trotsenko, A.V. Babich, P.M. Ostapchuk
Institute of Electrophysics and Radiation Technologies of the NAS of Ukraine,
Kharkiv, Ukraine
E-mail: ostapchuk@kipt.kharkov.ua
Using the analytical expression for the energy of elastic interaction of radiation point defects with a prismatic
edge dislocation loop of zirconium (Burgers vector Db 1/ 3 112 0 , 0211 occurrence plane), the bias of loops
of different nature (vacancy and interstitial) was calculated by the finite difference method. The toroidal geometry of
the reservoir was used. It allowed one to calculate biases for loops of any size without any correction of the elastic
field in its area of influence. In the dilatation center approximation the dependences of the loop bias on the loop
radius were obtained. The principal possibility of coexistence of loops of different nature in the prismatic plane of
zirconium is shown. A qualitative concept of the radiation growth (RG) mechanism was formulated within the
framework of the classical elastic ideology.
PACS: 62.20Dc;62.20.Fe
INTRODUCTION
The development of the domestic nuclear power
industry at present is mainly based on the use of light-
water reactors with the use of zirconium materials in the
core that are low-absorbing in the thermal neutron
spectrum [1, 2]. Ensuring the operational reliability,
specified service life and safety of nuclear installations,
taking into account economic and environmental
factors, is feasible with compliance with very strict
requirements for these materials, taking into account the
degradation of their physical and mechanical properties
under the influence of high temperatures, neutron
irradiation and corrosion in the coolant. An example of
such requirements is the high radiation [3] and
mechanical [4] durability of the structural materials of
fuel elements and fuel assemblies, in particular, their
dimensional stability. This problem has many aspects.
One of them is related to the radiation growth effect
(RG). This term refers to the shape-change of crystalline
solids under conditions of irradiation by energetic
particles without the application of an external load. The
volume of the material does not change, in contrast to
the effect of radiation swelling. The RG phenomenon is
particularly characteristic of anisotropic materials, in
particular zirconium (hcp). So, the fuel cladding and fuel
assemblies of all conventional nuclear reactors that
generate power from the fission of uranium by thermal
neutrons are made from zirconium alloys because of
their low thermal neutron absorption cross-section. The
stability of their dimensions, as well as the ability to
predict changes in their shape, is very important to the
designers and operators of such reactors, since
deformation affects the operability and service life of
the reactor core. Therefore, RG of zirconium and its
alloys has been the subject of intensive research since
the second half of the last century [5, 6]. It has been
found that zirconium during growth expands in the a -
direction and shrinks along the c -axis [7, 8]. Such its
behavior is associated with the idea of Buckley S.N.
[5, 6] that interstitial loops are formed predominantly on
the prismatic planes of zirconium, {1010}, and vacancy
loops on the basic (0001). The reason is the stresses
caused by the thermal peaks of the collision cascades
and the corresponding thermal expansion along the
different zirconia axes. Although the physics of the RG
mechanism has changed over the years, Buckley's
general concept has remained the same: vacancy loops
nucleating and growing on the base planes “eating” the
crystal along the c -axis, while growing interstitial
loops, forming additional extra planes in the a -
direction, increase its size. Moreover, it turned out that
dislocation loops of different nature (vacancy and
interstitial) can coexist on prismatic planes [7, 8]. And
that does not fit at all into the standard concept of the
dislocation bias (EID – elastic interaction difference)
[9], since it is believed [10] that the dislocation loop
bias does not depend on its nature. Therefore, the joint
coexistence of interstitial and vacancy loops, as well as
the growth of basic vacancy loops within the standard
elastic ideology seems inexplicable.
The most popular version of the cause of radiation
growth of zirconium is anisotropic diffusion of radiation
point defects (PD) between its planes (DAD theory –
diffusional anisotropy difference) [9, 11]. Its main
assumption is as follows / /a c a c
i i v vD D D D . Here a
mD
is the diffusion coefficient of PD of m-type in the basal
plane of zirconium, c
mD is the coefficient of diffusion in
c -direction (subscript v and i refer to vacancies and
self-interstitial atoms (SIA) respectively). However,
there is no experimental confirmation of this inequality
to date. Moreover, numerical calculations [12] have
shown that the inequality in the range of reactor
temperatures (T < 800 K) is just the opposite
mailto:ostapchuk@kipt.kharkov.ua
4 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №2(144)
/ /a c a c
i i v vD D D D . Therefore, a physical cause of
RG associated only with anisotropic diffusion of
radiation PD seems doubtful.
In [13, 14], within the framework of the classical
elastic ideology (EID), the bias for basic zirconium
loops (edge dislocation with the Burgers vector
Db 1/ 2 0001 and mixed with DSb 1/ 6 2023 )
of different nature was numerically calculated
depending on their radius. It was shown that the
existence of vacancy type loops is possible only in the
presence of an uncompensated source of vacancies into
the basal plane. It was suggested that such a source
could be interstitial loops on prismatic planes. In order
to verify it, in this work, a similar problem is solved for
a prismatic edge dislocation loop (the Burgers vector
Db 1/ 3 112 0 , the 0211 } occurrence plane [8]),
using the analytical expression for the energy of its
elastic interaction with the PD.
LOOP BIAS
The sink bias is determined by a relation of the form
1 / v iB Z Z . Here subscripts v and i correspond to
vacancies and SIA respectively. If B>0, one says that
the loop has a preference to SIA. The dimensionless
quantity
,v iZ is called the absorption efficiency of the
PD by the sink. It appears as a result of calculating the
PD diffusion flux to a specific sink. Under the
assumption of diffusion isotropy of the medium
(Dij = Dδij) the PD flux J to the dislocation loop is
found by solving the following diffusion problem in its
region of influence using the quasi-stationary
approximation:
0div j r ; ( ) ( )DC j r r r ;
1/ Bk T ; (1)
( )
ln exp ( )
e
C
E
C
r
r r ;
S
J dn j r .
Here ( )C r is the concentration of migrating PD; j r ,
( ) r – their flux density and chemical potential,
respectively; ( )E r – their interaction energy with the
loop; eC – equilibrium thermal concentration of PD in
the crystal in the absence of a stress field E . The
integral is taken over an arbitrary surface containing the
loop with the outer normal n. Equation (1) should be
supplemented with boundary conditions. The inner
surface
CS is usually chosen in the form of a torus
containing a dislocation line. The torus minor radius rc
corresponds to the dislocation core radius. The
boundary condition on it has the form:
( ) ( |exp 0)
CSC E r r . (2)
The condition has the traditional form and corresponds
to the value of the PD chemical potential at the
dislocation core | 0
CS , when we neglect the linear
tension of the loop and the effect of coalescence of
loops of the same nature during annealing. Outer
surface
extS following the authors [10] we choose in the
form torus coaxial with
CS with generating circle radius
extR which corresponds to the radius of the loop
influence region. By analogy with (2) we formulate it
for the chemical potential in the form:
| ln( / )e
Sext
C C . Here C is the average PD
concentration in an effective medium that simulates the
influence of all sinks. This is the standard form of the
PD chemical potential in an effective medium, where
the influence of a particular sink is neutralized by the
others. Then:
( ) ( )ex |p Sext
C E Cr r . (3)
The central element of system (1)-(3) is the interaction
energy of PD with the loop. According to [13], in
zirconium in the case of a dilatation center, it has the
form:
( ) ( )ijV PE Spur = r ,
2
13 11 12 33
13 33 11 12
2 ( )
4 2 ( )
C C C C
P
C C C C
. (4)
Here Cij is the crystal elastic moduli, V - the change in
the volume of the finite crystal, associated with PD;
( )iju r – deformation field caused by the loop in the
point where the PD is situated. In contrast to the basic
loop (axial symmetry); there is only one option for
calculating the deformation field ( )iju r – through the
tensor Green's function ijG (TGF) equations of
equilibrium of a given elastic medium, which allows
calculating the displacements created by a loop with any
Burgers vector
D
i jD D
i jklm m l
kS
G
u C b n dS
x
r r
r . (5)
Here Cjklm is the tensor of elastic moduli of the medium,
modeling crystal; bm
D
is the m – component of the
dislocation Burgers vector; nl
D
is l – component of the
normal vector to an arbitrary surface SD, based on the
dislocation line; r is the observation point coordinate;
r’ is the surface point coordinate SD. TGF is calculated
by the method Lifshitz-Rosenzweig [15]. For a -loop,
the direction of the Burgers vector 1/ 3 1120=D
b and
the normal n to the plane of occurrence 1120 of the
loop coincide, so it is natural to choose the axis “x” of
the Cartesian coordinate system in the same direction.
As a result, for the vacancy a -loop we have [12]:
2 2
2 2 2 2
3 11 12 1 3 33 3 2
3
( ) ( ) ( ) 3 ( ) 2
4
D D
D
S S
b d r d r dY
E Q C C Y
d
V P
r r r r
r = ; (6)
2 2 2 2 2 2 2 2 2
3 3 12 3 13 3 3 3 12 3 13 3 11 12 32
3
( ) (1 3 ) ( ) ( ) 2 (1 ) ( ) ( ) ( ) ( )
d
Q C Y C C Y C C C Y
d
;
ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №2(144) 5
2 2 2
3 3 3( ) ( ) ( )V W ; 2 2 2
3 3 3( ) ( ) ( )Y V ;
1 /x r r ;
3 ( ) /z z r r .
The functions 2
3( ) , 2
3( )W , 2
3( )V are quite
complicated. Their explicit expressions are given in
[16]. It is important that they all depend on only one
variable 2
3 .
CALCULATION PROCEDURE
Consider a circular vacancy loop of radius R , lying
in the zirconium prismatic plane ( 0x ) of a
cylindrical system coordinates (r, φ, x). All calculations
are performed using dimensionless coordinates
Dbrr / ; 2 2 2r y z ; cosry ; sinrz ;
Dbxx / ;
2 2 2 22 cos( )x r rr r r r ;
( ) sin sinz z r r . The integration in (6) is
carried out over the area of the loop. Figs. 1 and 2 for
loop with radius 100R in plane x = 40 illustrate
dependency of the dimensionless energy E V P/ (6)
on the azimuth angle for two regions: inner 50r
(Fig. 1) and external 150r (Fig. 2). Wherein
experimental values of the elastic moduli of zirconium
according to [17] (Mbar) look like: C11 = 1.554;
C12 = 0.672; C13 = 0.646; C33 = 1.725;
C55 = C44 = 0.363.
Fig. 1. Dependence of the interaction energy
( )E V P / of the vacancy loop and SIA
on the azimuth angle for R = 100; x = 40
and the inner region of the loop r = 50
Fig. 2. Dependence of the interaction energy
( )E V P / of the vacancy loop and SIA on the
azimuth angle for R = 100; x = 40
and outer region of the loop r = 50
Note that in (4) for zirconium, the value 0P .
Therefore, for SIA ( 0V ), the inner region of the
vacancy loop must be the attraction region
( ( ) 0V PE r / ), and the outer region, the repulsion
region ( ( ) 0V PE r / ). It is this behavior of energy
that takes place in Figs. 1, 2. For a vacancy ( 0V ),
the illustration will be reversed. We note a very weak
dependence on the azimuth angle , which does not
change the nature of the interaction (sign) in each
region. The interaction changes sign only when passing
from the inner region of the loop ( 50r ) to the outer
one ( 150r ). However, this dependence greatly
complicates the calculations for numerical calculations.
Therefore, as in [14], we will eliminate it by averaging
the right side of (6) over the azimuthal angle , making
the problem isotropic in the “ yz ” plane. And one more
note. In the expression for energy (6), the dependence
on the variable “ x “ is quadratic, i.e. replacement
xx doesn't change anything. Therefore, as in the
case of the basic loop, Db 1/ 2 0001 numerical
calculations can be carried out only in one part of the
half-space 0x .
In terms of a variable
( , ) ( , )exp ( , ) /r x C r x E r x C the diffusion problem
(1)–(3) in dimensionless cylindrical coordinates taking
into account isotropy after averaging has the form:
2 2
2 2
1
0
E E
r x r r r x x
(7)
with boundary conditions ( , ) 0r x on the inner
toroidal surface
c cR r r R r , (8)
( , ) 1r x on the outer toroidal surface
ext extR R r R R for extR R ; 0 extr R R
for extR R .
Then for the flux and absorption efficiency
,v iZ we
have:
2 , , c ext
D
J R Z r R R
C
,
1
, , ( , ) ( ,ex )p
2
c ext
S
Z r R R E r x r x d
R
n . (9)
The diffusion problem (7), (8) was solved
numerically by the finite difference method. Fig. 3
shows a cross section of a toroidal reservoir containing
a loop [14], taking into account the reflection symmetry
in the plane 0x and symmetry (после усреднения
по ) about rotation around the x -axis. The specified
symmetry imposes additional boundary conditions:
/ 0x on DA, BC, OA, corresponding to zero flux
through the plane 0x , and / 0r on DO (axis
of symmetry). An arbitrary inner surface S in (9) is
chosen for the convenience of calculations in the form
of a rectangle of rotation. In Fig. 3, this is the contour L.
6 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №2(144)
a b
Fig. 3. Coordinate system for a toroidal reservoir: a – R>Rext, b – R<Rext
RESULTS
Fig. 4 shows the dependences of the bias of the
prismatic loops of zirconium B on their radius (in Db )
for three values of the outer size of the toroidal
reservoir: D
extR 200 b , which corresponds to the
dislocation density 9 28 10 cm (see Fig. 4,а),
D
extR 120 b ( 10 22 10 cm , see Fig. 4,b) and
D
extR 60 b ( 10 28.4 10 cm , see Fig. 4,c). Curves
with a maximums refer to interstitial ( ) loops, with a
minimum to vacancy ( ).
Fig. 4. Dependences of the bias of the prismatic loops of
zirconium, B , on their radius for three values of the
outer size of the toroidal reservoir:
a – D
extR = 200 b ; b – D
extR = 120 b ; c – D
extR = 60 b .
“ ” – refer to interstitial loop; “ ” – to vacancy
First, it should be noted that prismatic dislocation
loops in zirconia are biased sinks, which absorb SIAs
more efficiently than vacancies, since B>0 for both
types of loops (see Fig. 4). The bias depends on the loop
radius and the sink density. And this is consistent with
the general conclusions [10]. However, in [10] there is
an important conclusion that the bias does not depend
on the nature of the loop. In our case, this is not the
case.
Secondly, and this is the most important, there is
always a region in space of the size, in which the bias of
interstitial loop
intB ( ) is higher, than bias vacansion
one
vacB ( ). Those, in this region, SIAs are mainly
absorbed by interstitial loops. The remaining in excess
vacancies can be absorbed by the existing vacancy loops
or migrate to the basal plane and contribute to the
nucleation and further growth of vacancy basis loops. In
our opinion, this makes it possible in principle to
explain two experimentally observed points: the joint
coexistence of loops of both types in the prismatic plane
of zirconium, as well as the nucleation and subsequent
growth of vacancy loops in the basal plane. Note that
the lower boundary of this region
int vacB (R ) B (R )
moves towards smaller sizes with an increase in the
total dislocation density (or a decrease in
extR ). So,
at 10 22 10 cm (see Fig. 4,b) DR 60b or
20 nm. And these are quite real loops visible in a
microscope. In the region R R the existence of loops
within the framework of EID is difficult to explain,
since here
int vacB B . Those SIAs should be
predominantly absorbed by vacancy loops and dissolve
them. Excess vacancies should, accordingly, dissolve
the emerging interstitial loops. Therefore, this area is the
subject of research in the theory of nucliation. Thus, the
following picture emerges. At a low dislocation density
(the initial stage of irradiation), the loops in the
prismatic plane cannot grow due to their large value R
(see Fig. 4,а). As the dislocation density increases, it
moves towards lower values. There is a real possibility
of diffusion growth of emerging interstitial loops. And
they, in turn, stimulate the nucleation and growth of
vacancy loops, first on the prismatic and then on the
basal planes of zirconium. However, here we must also
include the bias of the basis loops and only then
formulate a qualitative possible RR mechanism based
on the classical elastic ideology. In conclusion, we note
that understanding the physics of the RR mechanism
may be useful for predicting the behavior of fuel
cladding during long-term storage of spent fuel in dry
reservoir.
ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №2(144) 7
SUMMARY
1. Prismatic loops in zirconia are biased sinks, which
absorb SIAs more efficiently than vacancies, since for
both types of loops (see Fig. 4) 0B .
2. The principal possibility of coexistence of loops
of different nature in the prismatic plane of zirconium is
shown.
3. A qualitative concept of the RG mechanism was
formulated within the framework of the classical elastic
ideology.
ACKNOWLEDGEMENTS
The research presented in this article was financially
supported by the Ukrainian government budget program
«Government support for priority scientific research and
scientific & technical (experimental) developments»
(budget financial Code 6541230).
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Article received 07.03.2023
ФАКТОР ПЕРЕВАГИ ПРИЗМАТИЧНИХ ДИСЛОКАЦІЙНИХ ПЕТЕЛЬ У ЦИРКОНІЇ.
ЧИСЕЛЬНИЙ АНАЛІЗ
О.Г. Троценко, А.В. Бабіч, П.М. Остапчук
Використовуючи аналітичний вираз для енергії пружної взаємодії радіаційних точкових дефектів із
призматичною крайовою дислокаційною петлею цирконію (вектор Бюргерса Db 1/ 3 112 0 , площина
залягання 0211 ) методом кінцевих різниць пораховано фактор переваги петель різної природи
(вакансіонної та міжвузлової). Використовувалася тороїдальна геометрія резервуара, що дозволяє провести
розрахунки для петлі будь-якого розміру без будь-якої корекції пружного поля в її області впливу. У
наближенні центру дилатації отримані залежності фактора переваги петель від їхнього радіусу. Показано
важливу можливість спільного співіснування в призматичній площині цирконію петель різної природи.
Сформульовано якісну концепцію механізму радіаційного зростання (РЗ) у рамках класичної пружної
ідеології.
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