Amorphization kinetics under electron irradiation
A model is brought forward of the kinetics of amorphization occurring under electron irradiation. The model is based on consideration of the phase transformations controlled by the structural relaxation of small-scale unstable atomic configurations excited by electrons. Наведено модель кінетики амор...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2005 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2005
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| Цитувати: | Amorphization kinetics under electron irradiation / А.S. Bakai, А.А. Borisenko, K.C. Russell // Вопросы атомной науки и техники. — 2005. — № 4. — С. 108-113. — Бібліогр.: 9 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860238998167879680 |
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| author | Bakai, A.S. Borisenko, A.A. Russell, K.C. |
| author_facet | Bakai, A.S. Borisenko, A.A. Russell, K.C. |
| citation_txt | Amorphization kinetics under electron irradiation / А.S. Bakai, А.А. Borisenko, K.C. Russell // Вопросы атомной науки и техники. — 2005. — № 4. — С. 108-113. — Бібліогр.: 9 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | A model is brought forward of the kinetics of amorphization occurring under electron irradiation. The model is based on consideration of the phase transformations controlled by the structural relaxation of small-scale unstable atomic configurations excited by electrons.
Наведено модель кінетики аморфізації під електронним випромінюванням. Модель грунтується на розгляді фазових перетворень, які контролюються структурной релаксацієй дрібномасштабних нестабільних атомних конфігурацій, які збуджуються електронами.
Приведена модель кинетики аморфизации под электронным облучением. Модель основывается на рассмотрении фазовых превращений, которые контролируются структурной релаксацией мелкомасштабных нестабильных атомных конфигураций, которые возбуждаются электронами.
|
| first_indexed | 2025-12-07T18:27:24Z |
| format | Article |
| fulltext |
PACS: 61.80.Az; 61.82.Bg
AMORPHIZATION KINETICS UNDER ELECTRON IRRADIATION
А.S. Bakai1, А.А. Borisenko1, K.C. Russell2
1National Science Center “Kharkiv Institute of Physics and Technology”
61108 Kharkiv, 1 Akademichna st., Ukraine
e-mail: bakai@kipt.kharkov.ua;
2Department of Materials Science and Engineering and Department of Nuclear Engineering,
Massachusetts Institute of Technology, Cambridge, 77 Mass. Ave., MA 02139, USA
e-mail: kenruss@mit.edu
A model is brought forward of the kinetics of amorphization occurring under electron irradiation. The model is
based on consideration of the phase transformations controlled by the structural relaxation of small-scale unstable
atomic configurations excited by electrons.
INTRODUCTION
Electron irradiation can modify the kinetics of phase
transformations in solids not only due to the diffusion
enhancement as a result of the Frenkel pair production,
but also due to irradiation-induced activation processes
occurring on the crystal-amorphous interface. Research
into those processes is of great importance for interpre
tation of the results of simulation tests, employing elec
tron irradiation, of the structural materials to be used in
molten-salt and high-temperature nuclear reactors. This
work proposes a new model of the electron irradiation-
induced phase transformations. We apply it to the de
scription of amorphization of the intermetallic com
pound Zr3Fe, because that process was studied experi
mentally and such theoretical models were proposed for
its description that were based on considerations not ac
cepted by us.
Motta and Olander [1] computed the contribution of
point defects and compositional disordering to the free
energy of the ordering crystal, and they assumed, as the
necessary condition for amorphization, the following
criterion:
〈Gc〉≥G a . 1
Here, 〈Gc〉 is the averaged free energy of irradiat
ed crystal, G a is the amorphous phase free energy.
They calculated the kinetics of accumulation of point
defects and chemical disordering in the ordering inter
metallic compound Zr(Cr, Fe)2 vs. time and found, em
ploying the criterion (1), that the amorphization had to
take place only after total compositional disordering and
the vacancy concentration reaching about 0.009. This
result was not supported by a number of studies, in
which a partial compositional order was observed in
amorphous alloys of the metal-metal type [2, 3].
This work proposes an alternative model of amor
phization caused by electron irradiation. It is assumed
that the amorphous phase nucleation occurs owing to
the aggregation of radiation damage in such regions that
may include point defect complexes, a sector of disloca
tion core, a boundary, a triple junction or some other
structural defects that lead to a considerable local in
crease of the free energy. The amorphous phase growth
is controlled by both the diffusion and relaxation of the
electron irradiation-excited non-equilibrium states of
structural elements on the crystal-amorphous (C-A)
phase boundary. According to observations of the field
ion microscopy, the C-A boundary is of the order of in
teratomic distance wide [4]. The electrons, while being
scattered on the boundary layer, excite and bring groups
of atoms into a non-equilibrium state. The excited atom
ic configurations then relax into either amorphous or
crystalline state. The Addendum considers the elemen
tary phenomenological model that accounts for the na
ture of probabilities of the relaxation of excited atomic
configurations into one phase or the other. Thus, within
the framework of the given model, the amorphous phase
can grow even in such regions where the criterion (1) is
not fulfilled, if the radiation-stimulated diffusion pro
cess occurs faster than the reverse process of relaxation
does into the crystalline state that has a lower free ener
gy. Within the limits of the developed model, the criti
cal electron flux for amorphization at a fixed tempera
ture is calculated. Also, the required dose to amorphiza
tion is calculated vs. the values of electron flux and tem
perature in accordance with the Kolmogorov-Avrami
theory of phase transformations.
A comparative study is made between theory and
experiment on amorphization of the intermetallic com
pound Zr3Fe, occurring under electron irradiation [5].
A MODEL OF MOTION
OF THE INTERPHASE BOUNDARY
The observations made via field ion microscopy [4]
indicated that the C-A boundary was narrow with the
width of the order of the atomic scale. In our model, an
assumption was made that atoms or groups of atoms
near the interphase boundary could move into the other
phase by a way of collective rearrangements with the
average displacement l≤a (where а is the interatom
ic distance), causing the interphase boundary motion in
this region.
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108
mailto:bakai@kipt.kharkov.ua
Contributions to the rate of growth (dissolution) of
precipitates of the amorphous phase are made both by
irradiation and thermodynamic driving force. We shall
designate those contributions in terms of u irr and
u th , respectively:
u≡ dR
dt
=u irruth , 2
where R is the amorphous phase precipitate radius.
The amorphous phase precipitate growth rate due to
the action of the thermodynamic driving force has the
following form:
u th=r−1 DΣ [exp Δgβ −1 ] ,
Δg≈U c−U aξT
3
where r is the size of the atomic group participating in
the elementary act of interphase boundary rearrange
ment (of the order of several interatomic distances),
U c and U a are the potential energy values of this
group in the potential relief minima, corresponding to
crystalline and amorphous states, respectively, ξ –
configuration (at T 0 ) entropy of the group in
amorphous state, β=1/ k BT , k B is the Boltzmann
constant, DΣ is the diffusion coefficient on the inter
phase boundary.
From (3) one can see that the amorphous phase pre
cipitate growth rate under the action of thermodynamic
driving force is positive when Δg0 . When
Δg0 , the value of u th is negative, but the growth
rate can still be positive due to the contribution from
u irr (see, 4 below). It should be taken into account
that the C-A boundary acts as a sink for point defects.
For this reason, the crystalline phase layers adjacent to
the boundary are free from point defects. One can also
expect that, in an ordering alloy, the chemical order in
these layers is higher than in the crystal bulk, far from
the point defect sinks. That is why, it should be consid
ered that Δg0 and u th0 .
The peculiarities of electron irradiation-induced
damage in the vicinity of the interphase boundary are
somewhat different from those in the bulk of the crys
talline or amorphous phase. In the bulk of a solid, the
electrons produce stable or unstable Frenkel pairs. The
threshold energy of the Frenkel pair production in met
als is usually about 30 eV. In the boundary layer the ir
radiation generates different unstable atomic configura
tions possessing the free energy higher than that in the
amorphous state. The structural relaxation of these con
figurations may bring to either non-crystalline or crys
talline structure. If σ e is the cross-section of formation
of the unstable atomic configuration in the boundary
layer under the impact of electron irradiation, while
J e is the electron flux, then the generation rate of
these configurations would be
dN e
dt
=4πR2 dJ e σ e n , 5
where N e is a number of unstable configurations, n
is the atomic concentration in the boundary layer and
d is the layer width. The unstable atomic configura
tion may overlap both the crystalline and non-crystalline
regions. Its structural relaxation leads to formation of
the crystalline or amorphous phase. Let us denote by
wa and wc the probabilities of the unstable configu
ration relaxation into the amorphous or crystalline state
respectively and, assuming that d≈r , we come to
u irr≈ J e σ e n wa−w c r 4 . 6
As noted above, the amorphous phase precipitate
growth rate u can be positive, even if u th0 , if
u irr0 and
u irr−u th 7
or
J e J
e ¿
=
DΣ [1 −exp Δgβ ]
σ e wa−wc nr5 . 8
The derived expression (7) is the criterion of amor
phization under the impact of electron irradiation.
KINETICS OF AMORPHIZATION UNDER
ELECTRON IRRADIATION
The critical amorphous phase nuclei appear in crys
tal as heterophase fluctuations with concentration
w k ~ exp −ΔG k ⋅β
ΔG k =Ga k −Gc k , 9
where Ga k and G c k are the free energies of a
group of k atoms in amorphous and damaged crystalline
state respectively.
The rates of crystal-amorphous and amorphous-crys
tal transformations of k atoms are proportional to the ap
propriate self-diffusion coefficients:
I ca ~ exp−G c k ⋅β
I ac ~ exp−Ga k ⋅β
10
In a perfect crystal I ca << I ac and the amorphous
nuclei are unstable, but if there are stable structural de
fects in a crystal, then the value ΔG k can be small
or even negative in a defect region that includes, for in
stance, point defect complexes, a dislocation core sec
tor, an intergrain boundary, a triple junction or some
other structural defect. In such regions, equilibrium pre
cipitates of the amorphous phase may be formed. So, if
the condition (1) is met locally, in a region whose size is
greater than the critical size of the nucleus, then a het
erogeneous amorphous phase nucleation may take
place. Quite obviously, this nucleation cannot bring
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ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2005. №.4.
Серия: Физика радиационных повреждений и радиационное материаловедение (87), с. 108-113.
109
about any noticeable amorphization of a sample, if the
volume fraction of the regions where the condition (1) is
met is small and the growth rate is controlled just by
thermodynamic driving force. However, as noted above,
the electron irradiation can have a considerable effect on
the amorphous phase growth rate.
Let us designate as V d /V =nd the volume frac
tion of the regions where the condition (1) is met and
where the amorphous phase nucleation may occur (
V d is the volume occupied by extended defects or
(and) irradiation-produced point defect complexes
where the condition (1) is met, V is the total volume).
If one designates the critical amorphous phase nucleus
volume as V a , then the value
N d=nd /V a 11
is the volume density of the amorphous phase nucle
ation centers. The value N d is time-dependent:
dN d t
dt
=−I ca N d t I ac [N d t −N d 0 ]hd
, 12
where hd is the generation rate of point defect com
plexes under the impact of electron irradiation.
Equation (11) has the following solution:
N d t =N 0e I ac− I ca t
I ac N 0−hd
I ac−I ca
{1 −e
Iac− I ca t}
N 0≡N d 0 .
13
Expressions obtained for the nucleation center con
centration (12) and amorphous phase precipitate growth
rate (2), (3), (5) can be used to describe the irradiation-
affected amorphization kinetics within the framework of
the Kolmogorov-Avrami theory. To simplify the final
expressions, we impose the condition ∣I ca∣>>∣I ac∣
for nucleation centers and consider two cases.
In the first case, the initial extended structural de
fects play a decisive role in the amorphous phase nucle
ation, the value hd in 14 being negligible. In this case,
the amorphous phase volume fraction X t is a time
variable in accordance with the following expression
[6]:
X t =
1 −exp[−8πu3 N 0
I
ca3
e−I ca t
−1 I ca t−
I ca t 2
2
I ca t 3
6 ].
. 15
Asymptotically, in the limit I ca t >> 1 , the expres
sion 16 transforms into
X t =1 −exp[−4πN0 u3 t3 /3] . 17
From (14), one can see that the characteristic time of
amorphization in this approximation is:
τ ca ~N
01/3 u−1
, 18
while the dose-to-amorphization is
dpaca ~ σ F J eN
01/3 u −1
, 19
σ F being the stable Frenkel pair generation cross-sec
tion.
In the second case considered here, the initial ex
tended defects play no role and one can put N 0=0 in
(12). Then, in the limit I ca t >> 1 , we have:
N d t ≈hd / I ca 20
and (see [6])
X t =1 −exp{−πhd u3 t4
3 }. 21
From (18), one can deduce the characteristic time of
amorphization in this approximation:
τ ca ~hd u3−1/4
, 22
while the dose-to-amorphization is
dpaca ~ σ F J ehd u3−1/4
. 23
In the both cases considered, we have a lag time of
amorphization t lag ~ 1/ I ca . At t >> t lag , the C-A
transformation is described by the asymptotic expres
sions, (14) or (18).
NATURE OF DIFFUSION PROCESSES
ON INTERPHASE BOUNDARY
As regards the conventional thermally activated dif
fusion, the diffusion coefficient expression has the form:
Dth=r2 ν0exp −Eβ , 24
where r is the length of the jumps (in our case, it is the
size of the atomic group transiting from one phase to the
other), ν0 is the frequency of attempts (of the order of
the Debye frequency), Е is the activation energy value.
Yet, the final value of the experimentally measured
critical electron flux needed for amorphization at low
temperatures, as shown in Fig.1, comes out in favor of a
supposition that there should be a certain supplementary
athermal diffusion process leading to interphase bound
ary rearrangements. We assume that this athermal diffu
sion process should be controlled by quantum-mechani
cal tunneling. In this case, the athermal diffusion coeffi
cient expression acquires the form:
Dath=r 2ν0 D , 25
where the coefficient of transmission under potential
barrier is
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ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2005. №.4.
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110
D=exp−2
ℏ
2 mE l ,
m is the tunneling complex mass, Е is the potential
barrier height, l is the distance between equilibrium
positions, ℏ is the Planck constant.
Another possible contribution to the interphase
boundary diffusion can be made from the radiation-
stimulated processes as a result of the so-called “radia
tion shaking” [7]. This term implies the influence of
phonons produced by the annihilation of the irradiation-
generated unstable Frenkel pairs with the mean radius
being of the order of the interatomic distance and life
time 10-12 – 10-11 s. The radiation-stimulated diffusion
coefficient in this case is determined by the expression
D r=
4πJe Nσ F r2 n
3 K Fac
6π Ea2
3
, 26
where N is the number of unstable pairs relative to that
of stable ones, К is the bulk modulus, F is the vol
ume of dilatation center corresponding to the unstable
Frenkel pair, ac is the volume of dilatation center
corresponding to the unstable atomic configuration on
the interphase boundary.
Then, since the thermally activated, athermal and ra
diation-stimulated diffusion processes are independent,
the expression for the total diffusion coefficient on in
terphase boundary would have the form:
DΣ=D
th2D
ath2D
r 2 . 27
DISCUSSION OF RESULTS
A comparison is given below of the predictions of
our theory with experimental data published in [5]. Note
at once that the experimental data agrees better with the
expressions (19), (20) than with (15), (16), for the rea
sonable choice of the fitting parameters.
We choose the tunneling coefficient D to correspond
to the transition of a pair Fe+Zr of atoms between two
noncoincident lattice sites on the interphase boundary,
which are separated by a distance l≈3 . 6 ⋅10−11 m ,
under the potential barrier of 1675 K. We assume that as
soon as such a transition takes place, the whole region
with radius r≈2a becomes unstable and tends to re
lax into the adjacent phase by the means of “radiation
shaking” process. So, the tunneling of a pair of atoms
between two noncoincident sites acts like a trigger for
such region.
The values used for the fitting parameters are given
in Table. They were chosen on the basis of the criterion
of the best agreement with experiment, as given in Fig
ures 1-4.
Values of fitting parameters
Parameter N of formula Value (SI)
U c−U a
28 −2200⋅k B J
ξ 29 6 ⋅k B J⋅K−1
r 30 5 ⋅10−10 m
σ e
31 2 . 5 ⋅10−25 m2
wa−wc
32 0.2
ν0
33 3 ⋅1013 s−1
D 34 e−33 .65
E 35 6700⋅k B J
hd
36 0 .77⋅J e m−3s−1
N 37 100
σ F
38 5 ⋅10−27 m2
n 39 6 . 4 ⋅1028 m−3
K 40 1011 Pa
F
41 1 . 6 ⋅10−29 m3
ac
42 0 .8 ⋅10−29 m3
a 43 2 . 5 ⋅10−10 m
From Fig. 1 one can gather that theory agrees with
experiment satisfactorily in the low temperature region,
where, as we believe, tunneling diffusion plays a major
role in the kinetics of the C-A boundary. In the region
T≥175 K , where the contribution from thermally
activated processes is substantial, a flatter progression
of the experimental curve than calculated calls for addi
tional analysis.
0 50 100 150 200 250
1.5
2.0
2.5
3.0
3.5
4.0
C
rit
ic
al
D
os
e
R
at
e
fo
r A
m
or
ph
iz
at
io
n
(1
023
e
m
-2
s
-1
)
T (K)
Experiment
Theory
Fig. 1. The critical value of electron flux for amorphiza
tion vs. temperature (7)
One can see from Fig. 2 that our model produces the
right results at T=28 K in the region of relatively
large fluxes J e >> J
e ¿
T =28 K .
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ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2005. №.4.
Серия: Физика радиационных повреждений и радиационное материаловедение (87), с. 108-113.
111
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
C
rit
ic
al
D
os
e
fo
r A
m
or
ph
iz
at
io
n
(1
027
e
m
-2
)
Electron flux (1023 e m-2 s-1)
Irradiation temperature
28 K - experiment
200 K - experiment
28 K - theory
200 K - theory
Fig. 2. Dependence of dose-to-amorphization (20) on
electron flux density
The flatter progression of the experimental curve
than predicted theoretically at T≈200 K , as we sup
pose, must have the same explanation that the discrep
ancy of our theory with experiment in this region in
Fig. 1. The dose needed for amorphization, while ap
proaching from the upper level downward to the critical
flux value at the given temperature (see (7) and Fig. 1),
must increase to infinity. We point out the discordance
between the experimental data in Fig. 1 (Fig. 4 in [5]),
where the critical flux values for amorphization at
T=28 K and T=200 K are respectively about
1 . 8 ⋅1023e⋅m−2⋅s−1 and
3 . 2 ⋅1023 e⋅m−2⋅s−1 , and Fig. 2 (Fig. 6 in [5])
showing the finite values of the doses-to-amorphization
at flux values that are smaller than the above-men
tioned.
In the presented theory (Fig. 3), in the high tempera
ture region, the value dpaca⋅ J e σ F does not depend
on dose absorption rate, similarly to the theory [5].
The experimental observation of the dependence of
this value on the dose absorption rate calls for additional
analysis for the right explanation to be made.
0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045
0.01
0.1
dp
a ca
*(
J eσ F)0.
5
1/T (1/K)
Dose rate
(1023 e m-2s-1)
3.4 - experiment
6.0 - experiment
3.4 - theory
6.0 - theory
Fig. 3. Dependence of product “dose-to-amorphization
(dpa) * square root of dose absorption rate (dpa/s)” vs.
reverse temperature
0 50 100 150 200
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
Dose rate
(10-3 dpa/s)
1.04 - experiment
1.68 - experiment
1.04 - theory
1.68 - theory
Irr
ad
ia
tio
n
do
se
re
qu
ire
d
fo
r a
m
or
ph
iz
at
io
n
(d
pa
ca
)
T (K)
Fig. 4. Dependence of dose-to-amorphization (dpa)
on temperature (20)
In the low temperature region, our theory agrees
well enough with the experimental data on the dose-to-
amorphization (Fig. 4). The flatter progression of the
experimental curve than theoretically predicted, espe
cially at low fluxes, at high temperatures, must be ac
counted for in the same way that the discrepancy of our
model with the experimental data in the high tempera
ture region in Fig. 1.
ADDENDUM. DYNAMICS OF OSCILLATOR
WITH FRICTION IN TWO-WELL POTEN
TIAL UNDER ACTION OF RANDOM COER
CIVE FORCE
Let us formulate the simple model in order to ac
count for the relaxation dynamics of the atom on the C-
A boundary. Let us consider the 1D particle motion in
the two-well potential, as depicted in Fig. A1а. We shall
also take into account the effects of thermal noises, dis
sipation and rare strong perturbations that imitate the ra
diation ambience. The phase portrait of this system in
the absence of the dissipation and perturbations is given
in Fig. A1b.
Here, L1 and L2 are the widths of the left- and right-
hand separatrices, respectively; Es and ΔE are the
depth of the right-hand well and difference in the depths
of the right- and left-hand wells, respectively; S1 and S2
are the phase space areas limited by the appropriate sep
aratrices.
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ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2005. №.4.
Серия: Физика радиационных повреждений и радиационное материаловедение (87), с. 108-113.
112
S1 S2
x
v
L1 L2
∆E
U(x)
x
ES
E
2
vU(x)
2
=+
a)
b)
Fig. A1. Potential energy (а) and phase portrait (b)
of 1D two-well potential
For the particle with unitary mass the equation of
motion will be written as follows:
d 2 x
dt2 2γ dx
dt
dU x
dx
=ϕ t ∑
i
gδ t−t i
. 44
Here, γ is the friction coefficient; ϕ t is the
random coercive force; g is the intensity of rare (on
the scale of the period of oscillations) strong pulses im
parting to the particle the energy E >> E S . The equa
tion (25) differs from the Langevin one by the presence
of the last term in the right-hand part that simulates rare
strong pulses from electrons.
The value ϕ t describes the “thermal noise” and
has the following statistical properties:
〈ϕ t 〉=0 ; 〈ϕ t ϕ t ' 〉=κδ t−t '
κ ~ T .
45.
In the absence of the right-hand part in the equation
(25), the phase trajectory of the system will, depending
on the initial conditions, tend to one of the focuses on
the phase portrait.
At the condition g=0 (over large times the sys
tem is in equilibrium with the thermostat) and
T << E S (the system spends most of its time in the
regions limited by the separatrices), the probabilities for
the system to be found in the regions S1 and S2 (in the
left- or right-hand well, respectively) obey the following
well-known relationship:
w1
e /w2
e=exp −ΔE /k BT , 46
the time spent by the system in one or the other well
(lifetime) being
τ 1,2 ~ exp const⋅γE s/ω1,2 κ . 47
Here, ω1,2 is the frequency of oscillations in the
left- or right-hand well, respectively.
To derive the expressions describing the probabili
ties to find the system in the region S1 or S2 at the condi
tion g 2>> E S , we consider two limiting cases:
1) Strong friction case (non-periodic regime)
γω1,2 , 〈∣t i−t i1∣〉 << τ1,2 . 48
In this case, the probabilities of staying in the left- or
right-hand well are proportional to the widths (over the
coordinate х) of the appropriate separatrices:
w1,2≈
L1,2
L1L2
. 49
The obvious generalization of this result to the 3D
case under consideration leads to the formula:
wa−wc≈
ρc
ρa
−1 , 50
where ρc and ρa are the densities of the substance
in crystalline or amorphous phase, respectively.
2) Weak Friction Case (Oscillatory Regime)
γ << ω1,2 , 〈∣t i−t i1∣〉 << τ1,2 . 51
In this case, the probabilities of staying in the left- or
right-hand well are proportional to the phase space areas
limited by the appropriate separatrices [8, 9]:
w1,2=
S 1,2
S1S 2
. 52
In practice, instead of the phase volume, inside of
which the system is confined, the entropy of the system,
which is the value equal to the logarithm of the latter, is
oftener used. In terms of the entropy the expression (33)
acquires the form:
w1,2=
exp s1,2
exp s1exp s2
, 53
where s1 and s2 are the entropies of the system in
the regions 1 and 2. In this case, in order to compute the
probabilities of the system staying in one or the other re
gion, one needs to know the difference of its entropies
in these regions.
In the specific case under consideration, the en
tropies of the amorphous and crystalline phases that are
under the same external conditions differ by the value
ς ~ 1 , the co-called configuration entropy of the amor
phous state, that has to do with the absence of the long-
range order in the amorphous state. Then
wa−wc=tanh ς
2 . 54
From Table 1 one can see that the value of the fitting
parameter wa−wc=0 . 2 conforms better to the
weak friction case, because the opposite case of the
strong friction yields wa−wc << 1 , since the densi
ties of the amorphous and crystalline phases differ not
more than by several percent.
CONCLUSIONS
_________________________________________________________________________________
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2005. №.4.
Серия: Физика радиационных повреждений и радиационное материаловедение (87), с. 108-113.
113
This paper proposes a model of a polymorphous
transformation under electron irradiation, taking into ac
count both the processes of inhomogeneous nucleation
of a new (in our case, amorphous) phase and the contri
bution from the relaxation of atomic configurations, ex
cited on the interphase boundary by electron irradiation,
into the process of the growth of precipitate of the new
phase. The example of the simple dynamic system is
used for clarification on the nature of probabilities of
the structural relaxation. The quantitative comparison of
the predictions of the proposed theory with experimen
tal data on amorphization of the intermetallic Zr3Fe un
der electron irradiation results in a reasonable agree
ment of theory with experiment. However, certain quan
titative discrepancies with experimental data call for ad
ditional analysis.
ACKNOWLEDGEMENT
This research was partially supported by Science &
Technology Center in Ukraine (STCU) within the
framework of Project # 294.
REFERENCES
1. A.T. Motta, D.R. Olander. Electron-irradiation-in
duced amorphization //Acta Met. (38). 1990,
p. 2176–2185.
2. S. Steeb and P. Lamparter. Structure of binary
metallic glasses //Journal of Non-crystalline Solids
(156-158). 1993, p. 24–33.
3. Ma. Nuding, P. Lamparter, S. Steeb and R. Bellis
sent. Atomic structure of melt-spun amorphous
Ni56Dy44 studied by neutron diffraction //Journal of
Non-crystalline Solids (156-158). 1993, p. 169–172.
4. A.S. Bakai, I.M. Mikhailovsky et al. Atomic struc
ture of interfaces between amorphous and crystalline
phases in tungsten //Fizika Nizkih Temperatur (25).
1999, #3, p. 282–289.
5. A.T. Motta, L.M. Howe, P.R. Okamoto. Amorphiza
tion of Zr3Fe under electron irradiation //Journal of
Nuclear Materials (270). 1999, p. 174–186.
6. J.W. Christian. The theory of transformations in
metals and alloys. Pergamon Press, Oxford, New
York, 1965.
7. V.L. Indenbom. New hypothesis on mechanism of
radiation-stimulated processes //Pis’ma v ZhTF (5).
1979, #8, p. 489–492. (In Russian).
8. V.I. Arnold //Uspekhi Matematicheskih nauk (18).
1963, #6, p. 91. (In Russian).
9. A.S. Bakai, resonant phenomena in nonlinear sys
tems //Diff. Eq. (2). 1966, #4, p. 479–491. (In Rus
sian)/
КИНЕТИКА АМОРФИЗАЦИИ ПОД ЭЛЕКТРОННЫМ ОБЛУЧЕНИЕМ
А.С. Бакай, А.А. Борисенко, К. Расселл
Приведена модель кинетики аморфизации под электронным облучением. Модель основывается на рассмотрении фазовых превра
щений, которые контролируются структурной релаксацией мелкомасштабных нестабильных атомных конфигураций, которые возбу
ждаются электронами.
КІНЕТИКА АМОРФІЗАЦІЇ ПІД ЕЛЕКТРОННИМ ВИПРОМІНЮВАННЯМ
О.С. Бакай, О.О. Борисенко, К. Расселл
Наведено модель кінетики аморфізації під електронним випромінюванням. Модель грунтується на розгляді фазових перетворень,
які контролюються структурной релаксацієй дрібномасштабних нестабільних атомних конфігурацій, які збуджуються електронами.
_________________________________________________________________________________
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2005. №.4.
Серия: Физика радиационных повреждений и радиационное материаловедение (87), с. 108-113.
114
|
| id | nasplib_isofts_kiev_ua-123456789-80565 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:27:24Z |
| publishDate | 2005 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Bakai, A.S. Borisenko, A.A. Russell, K.C. 2015-04-19T13:50:25Z 2015-04-19T13:50:25Z 2005 Amorphization kinetics under electron irradiation / А.S. Bakai, А.А. Borisenko, K.C. Russell // Вопросы атомной науки и техники. — 2005. — № 4. — С. 108-113. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 61.80.Az; 61.82.Bg https://nasplib.isofts.kiev.ua/handle/123456789/80565 A model is brought forward of the kinetics of amorphization occurring under electron irradiation. The model is based on consideration of the phase transformations controlled by the structural relaxation of small-scale unstable atomic configurations excited by electrons. Наведено модель кінетики аморфізації під електронним випромінюванням. Модель грунтується на розгляді фазових перетворень, які контролюються структурной релаксацієй дрібномасштабних нестабільних атомних конфігурацій, які збуджуються електронами. Приведена модель кинетики аморфизации под электронным облучением. Модель основывается на рассмотрении фазовых превращений, которые контролируются структурной релаксацией мелкомасштабных нестабильных атомных конфигураций, которые возбуждаются электронами. This research was partially supported by Science & Technology Center in Ukraine (STCU) within the framework of Project # 294. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Amorphization kinetics under electron irradiation Кінетика аморфізації під електронним випромінюванням Кинетика аморфизации под электронным облучением Article published earlier |
| spellingShingle | Amorphization kinetics under electron irradiation Bakai, A.S. Borisenko, A.A. Russell, K.C. |
| title | Amorphization kinetics under electron irradiation |
| title_alt | Кінетика аморфізації під електронним випромінюванням Кинетика аморфизации под электронным облучением |
| title_full | Amorphization kinetics under electron irradiation |
| title_fullStr | Amorphization kinetics under electron irradiation |
| title_full_unstemmed | Amorphization kinetics under electron irradiation |
| title_short | Amorphization kinetics under electron irradiation |
| title_sort | amorphization kinetics under electron irradiation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80565 |
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