- Email: [email protected]
Investigation of cascade-induced point-defect fluctuations
244
Journal
INVESTIGATION
OF CASCADE-INDUCED
POINT-DEFECT
of Nuclear Materials 117 (1983) 244-249 North-Holland Publishing Company
FLUCTUATIONS
H. GUROL Science Applications,
Inc., 1200 Prospect Street, P.O. Box 2351, La Jolla, California 92038, USA
and W.G. WOLFER Department
of Nuclear Engineering,
University of Wisconsin, Madison,
Wisconsin 53706, USA
High energy irradiation produces collision cascades, which occur randomly in space and time. Any given point in the material will therefore to subject to intermittent arrival of interstitials and vacancies, even under steady irradiation. The conventional rate theory formulation of the point-defect concentrations, which contains spatial and time averaging and a uniform production rate, may not always adequately describe processes such as void nucleation and irradiation creep. We develop in this paper a theoretical approach to describe the cascade-induced point-defect fluctuations by their moments. The first moment, giving the average point-defect concentration, is used in the traditional rate theory. However, the second moment provides the variance of the point defect concentration, and it becomes important for processes such as void nucleation, dislocation climb, and loop growth. An approximate analytical expression is given for the second moment. The obtained concentration variances compare well with numerical results. We demonstrate with the example of climb-induced dislocation glide how the moments can be used to extend rate theory results. Large enhancement in the creep rate due to point defect fluctuations can be obtained if the barrier size and the average climb rate are small and if the cascade size is large.
1. Introduction High energy irradiation produces collision cascades containing a large number of displaced atoms. The occurrence of the cascades is random in both space and time. The theoretical foundations of void nucleation and growth and other macroscopic processes such as irradiation creep have previously relied on the rate theory for calculating average point-defect concentrations in the material. The rate theory inherently contains therefore spatial and time averages and a uniform production rate of point defects. A fundamental question then is: does the inclusion of collision cascades modify current theories of microstructural aevolution? A number of processes, in particular void nucleation, might be expected to be very sensitive to the number of point-defect pairs per cascade, Y, and the magnitude of the cascade-induced point defect fluctuations. Mansur et al. [l] have developed a procedure for calculating the instantaneous (cascade-induced) pointdefect concentrations as a function of time, neglecting recombination, and under steady irradiation. In a later
0022-3 115/83/0000-0000/$03.00
0 1983 North-Holland
publication [2], irradiation creep induced by cascades was considered. It was found that discrete cascade events can contribute to the release of dislocations. Although the computer simulation results of Mansur et al. [l] of the instantaneous point defect concentrations provide the most detailed description, such complete information is cumbersome to use and often not required for evaluating a given physical process such as dislocation climb, void nucleation, etc. Accordingly, we develop in this paper a theoretical approach to obtain the moments of the point-defect fluctuations directly, and we demonstrate with the example of climb-induced dislocation glide how these moments can be used to extend the rate theory approach to microstructural evolution. The approach can be viewed schematically as shown in fig. 1. Irradiation produces cascades randomly. The detailed physics of the point defect diffusion is given by a response function (or Green’s function) h (c, t), which in turn tions, average
gives and
the instantaneous
their
value
moments. (C);
the
The second
point-defect first
concentra-
moment
moment
gives
is just
the
the vari-
H. Gurol, W. G. Worfer / Cascade
Randomly Produced Cascades
t
'
I
Fig. 1. Schematic
1 Equation
+
illustration
for
Process
used to include
I
cascade
ante (SC’) = (C2) - (C)2. Higher moments can also be obtained. These moments can then be used to obtain the time evolution of a particular process as shown in fig. lb. A number of observations about the nature of the process can be made. First, it can be proven [3] that cascade-induced fluctuations will modify the rate theory average value of a process only if it satisfies a non-linear equation. Second, for many physical applications only the first two moments are required. More precisely, the effect of the higher moments is zero for a process that satisfies a Fokker-Planck equation. Processes in irradiated materials, that we have identified, can be adequately described by a Fokker-Planck equation. In section 2, we calculate the variance of the pointdefect concentrations, by describing the concentrations as “shot noise”. We then compare the theoretical results with numerical calculations. In section 4, we apply this theory to analyze the enhancement of climb-controlled glide creep due to cascade pulsing. Conclusions are presented in section 5.
Fig.
la
Fig.
lb
1 Average Evolution
dX = f(X,t) dt of the approach
245
Concentration Fluctuations &C(t), and Moments
+
.
I
AC(t), (6C2,
Impulse Response Function of the Medium h(r,tl _
-inducedpoint - defect fluctuations
pulsing
in the evolution
of a particular
process.
sponse function is a function of the medium only. If h,({, t) is known, then the average concentration (C,) and the variance (SC:) = (CJ) - (CJ2 can be calculated. The variance is a measure of the magnitude of the fluctuations. In the previous rate theory, (SC:) is set equal to zero. The concentration and the variance at a given point due to cascades a distance { away is [4] (C,)I,
= X(S)qaJm
-m
h,(S,
t)dt
(2)
and (SC,2)/,
= X(l)q,2Jrn
--oo
h:(S,
t)dt,
(3)
where A({) is the average rate of cascade production in a differential spherical volume element about {. The total concentration, and variance can be obtained by performing an integration over l. The simplest response function is obtained from the spatially averaged equation for the concentration, neglecting recombination and under steady-irradiation (i.e., (C,) is a constant) [5]:
2. Theory
~+D&z~(t)=F(t). The point-defect concentration at a given point in space due to cascades occurring a distance 3 away can be described by a superposition of the individual response functions, and it is given by
C,(l, 1) = C%JL(L t - t,). this expression a = i for interstitials and a = v for vacancies, h,({, t) is the impulse response function, and q, is the effective magnitude of each cascade. q, depends on the net number of point-defects per cascade v, on the amount of cascade overlap, and on mutual recombination. q, is a quantity to be determined from the value of the average concentration (C,). The reIn
(4)
Here F(t) is the instantaneous rate of cascade production. It can be broken up into the sum of an average rate (P) and a fluctuating rate f( t). (P) is just the rate theory production rate, and by definition (f) = 0. D, is the diffusion coefficient, and S, is the total sink strength. The response function for this equation is given by h,(t)
= e -D-s4q
t),
(9
where U(t) is the unit step function. used to obtain the unknown q, as 4, = DSL(CJ where
TV= X-’
E@. (2) can now be
7F7 is the average
(6) time between
cascades.
H. Gurol, W. G. Wolfer / Cascade-induced point -defectfluctuatmw
246
Using this value of q, and eq. (3) the variance becomes: (7)
(SC?> = (7,/2)Q&CJ2.
This expression for (SC:) can be expected to give a lower bound value because of the fact that the spatial integration in effect dilutes the strength of the cascades. A more complete description of the physics is obtained by using the spatially-dependent response function of the diffusion equation (again neglecting recombination and under steady irradiation) as employed by Mansur et al. [l]: ~(~,f)--D,V*c~+D,S,C,=F,(I,f). The function h,([,
h,(c,
(8)
t) is derived
t) = [4sD,t]-3’2
in ref. 6 and given by:
exp[ -[*/4D,t]
X exp[ -D,S,t]u(t).
(9)
The average and the variance can now be evaluated by performing a spatial integration of eqs. (2) and (3). A simple approximation is to use the spatially averaged value for the cascade production rate x in these equations. As before q, can be determined. The result is q, = 12.4 r,(C,)DJ
exp(@/*).
(‘0)
rc = l/X. where rc is the mean time between cascades. Using this value of q,, the variance can be computed [using eq. (3)] as (SCz)]b
= 0.64 T,(C,)*D,S,
exp[2LSJ/*]
K, [2SSl/2].
(11) where K, is the modified Bessel function of the second kind of order two. Expression (11) is the variance of the point-defect concentration due to cascades at position 5. Mansur et al. [ 1] have shown that the cascade screening distance (beyond which cascades have no effect) is about 7S-I/*. The total variance can then be evaluated u approximately by summing the contributions over distance out to about 7s; ‘/* . If one chooses seven concentric shells, each of thickness S;‘/*, for example, the variance be-comes (6C,z) i= 0.64 T,( CJ2D,S,
i
e2kK2(2k).
(12)
k=l
The sum can be evaluated (SC;)
= 3.4 r,(C,)*D,S,.
sponse function used will only modify the numerical constant, and that (SC,‘) given by eq. (13) describes the correct scaling of the fluctuations with the cascade size. It should be noted that this equation gives the variance in terms of (Ca)*, which can be determined by conventional rate theory. Furthermore, (6Ci) is linearly proportional to the net number of defects per cascade when the damage rate TV-’Y is constrained to be a constant. Otherwise, the variance is quadratic in v. In the next section, the theoretical value of the variance is compared with numerical calculations.
to give (13)
This expression varies from expression (7),which relies on a much simpler description of the physics, by a numerical factor only. In fact, we expect that the re-
3. Comparison with numerical calculations We have performed numerical calculations of the point-defect variances by using the Green’s function solution [eq. (9)] of the spatially-dependent diffusion equation for the point-defect concentrations. The Monte Carlo method was used to select discrete random cascade distances from the reference point. Cascades were placed within a volume of radius 7s: ‘j2. This numerical procedure is the same as that presented by Mansur et al. [I], except that they distribute the cascades within spherical shells instead of selecting an arbitrary random distance. Both approaches were found to agree, however, producing the rate theory average value of the concentration. The time dependence of the vacancy and interstitial concentrations was calculated for a fixed damage rate of 10e6 dpa/s, T= SWC, and for values of v ranging from 50 to 500. For consistency in the counting statistics the total number of cascades sampled by the code was the same for all v values: about 1500 cascades. This corresponds to a frequency of about one cascade per 0.7 s for v = 500 and one per 0.07 s for v = 50. The material parameters used were: 0,” = 0.014 cm*/s, Do = 0.008 cm*/s, ET= 1.4 eV, Eim=0.15 eV, S, = 1.1 x 10” cm-*, S, = 1 x 10” cm*. A fractional survival rate of 20% was assumed. Table 1 shows the average point-defect concentrations and a comparison of the theoretical and numerical values of the variances as a function of v. The average vacancy and interstitial concentrations are within about 2 and 4%, respectively, of the rate theory concentrations. The theoretical values of (SC:) were computed using eq. (13). First, it is seen that the theoretical variances compare very well with the numerical values, for all v; that is, the analytical theory presented here shows the correct scaling of the variances with the net number of displacements per cascade. The scaling of (SC:) is linear with the parameter v, as predicted by eq. (13) for fixed damage rate. Therefore, as an example,
241
H. Gurol, W. G. Worfer / Cascade - induced point -defect fluctuations Table 1 Comparison Y
of the theoretical
variances
using eq. (13), as a function
Cci>
(SC,‘) ( x 1033)
(X
(X lo* cmw3)
Theoretical
Numerical
Theoretical
Numerical
1.75 1.75 1.74 1.74 1.I4 1.75
3.1 2.5 1.9 1.24 0.62 0.31
3.3 2.6 2.0 1.33 0.7 0.32
3.5 2.8 2.1 1.4 0.7 0.35
4.0 3.2 2.4 1.6 0.8 0.4
1Ol6 cm-3)
1.57 1.57 1.57 1.56 1.55
even though the damage rate is lop6 dpa/s for all Y, the variances ((SC:), (X2)) at Y = 500 are five times the value at Y = 100. This is just another way of saying that the vacancy and interstitial concentration peaks above their average values are greater for more damaging irradiation. For a given physical process then, the theory presented can be used to ascertain the regime of physical parameters where random cascade-induced effects are significant.
4. Application to climb-controlled glide creep As a specific application of this theory we consider here the climb-controlled glide of dislocations. Only the outline of the method is discussed here. A detailed discussion of cascade-induced creep will be presented in a forthcoming paper [7]. Consider an arbitrary edge dislocation segment whose glide plane is initially a distance x,, from a parallel reference plane. With time, the climb distance x(r) varies randomly with the arrival of the point-defects. Rapid positive increases in x(t) are caused by the arrival of interstitials from nearby collision cascades. The slower decreases in x( t ) are due to the more steady arrival of vacancies. On the average, the climb distance x(l) exhibits a net increase or decrease, depending on the net bias AZi of the particular edge dislocation segment relative to all other dislocations and sinks. since the number of point-defects reaching the dislocation is random, the climb distance x(t) is a stochastic variable, with an associated probability distribution function p (x, t). The probability that x is in the interval x to x + dx at time 1, given that x(0)=x,,, is then p(x, r)dx. It can be shown that p(x, t) satisfies approximately a Fokker-Planck equation with “drift” and “diffusion” terms, v and a given by V=
lim -(Ax) At-0
of Y
CC”)
1.57
500 400 300 200 100 50
and numerical
At
’
(14)
(sc;)(xIo*S)
% =
lim At-0
((Ad> . At
(15) \
,
I
where vis the net average climb rate. The quantity q is the “diffusion coefficient” associated with the random climb of the dislocation segment. @ now can be related to the variance of the point-defect concentrations, using standard methods [4]. The result is
where 52 is the atomic volume, b is the Burgers vector, and pd is the dislocation density. In this expression we have neglected cross-correlation terms such as (SC; XV), making eq. (16) a lower bound value for a; that is the effect of cascade-induced fluctuations will be greater than predicted by eq. (16). Suppose that the dislocation segment is pinned by an obstacle of “effective” height h at a climb distance x,, from the lower glide plane not intersected by the obstacle. Since not all portions of the length of the dislocation climb at the same instantaneous rate, the effective height h must be defined as if all points along a segment of length - h climb at the same rate. The height h will therefore always be somewhat greater than the physical barrier height. We now ask for the mean escape time 7(x0, h) required for the dislocation (starting at position xc) to climb either over or under the obstacle. This escape time is known in the statistical mechanics literature as the “first passage time” of the stochastic variable x(t) defined over the interval 0 c x(t) Q h. The first passage time satisfies the following differential equation [8]
+
Vd7+1=0 dx,
’
(17)
7(0,h)=7(h,h)=O, which
is associated
with the Fokker-Planck
equation
248
H. Gurol, W. G. Wolfer / Cascade-induced
for the probability distribution function p(x, t). The solution of this equation for 7(x,,, h) is readily obtained [7] and represents the mean escape time for a dislocation held up initially by an obstacle of height h at a distance x,, from the lower end of the obstacle. Since we can safely assume that the dislocations are held up by the obstacles with equal probability at any distance x0 between 0 and h, the mean escape time, averaged over all initial positions, is given by
T(h)
=$(x,,,
The result of the integration
q+&
(,_e-e)-l_$_;
V
(
_-
(18)
h)dx,.
point-defect
Table 2 Calculation of the ratio of cascade-induced creep to the rate theory creep for specific values of h, bias factor AZ, , and 6’. The parameter 0 is an indication of the importance of statistical fluctuations to dislocation climb: as 0 decreases, the effect of fluctuations is greater. Eq. (22) was used. T = 500°C, P = 10mh dpa/s.
_Y
500 500 500
h(nm)
A Z, /Zi
8=hV/9
R
10
0.1
I 1
0.I
9.6 0.96 0.096 0.48 0.048
1.26 6.34 60.0 12.4 126.0
1000
is
1
0.01 0.1 0.01
1 1
1000
)
fluctuations
(19)
where 0 = hV/q, and represents a dimensionless parameter which indicates the relative importance of cascade-induced fluctuations in climb. We see immediately that for the case of no fluctuations, that is for q + 0, one obtains the release time ?(h)+h/2V,
(20)
which of course is just the conventional rate theory result. On the other hand, in the fluctuation-dominant regime (v + 0), one finds
(21)
f(h)
From expressions
(19) and (20), one can obtain the ratio rate with fluctuations to the creep rate given by conventional rate theory. The result is
R of the creep
_ (1
-e-@)-l_;_;
(22)
A plot of this ratio as a function of 0 is shown in fig. 2. For values of B > - 20, the effects of fluctuations are negligible. For 0 < - 4, the ratio R rises very rapidly. We have calculated the enhancement R due to cascade-induced fluctuations for typical ranges of barrier sizes and dislocation bias differences A Zi. The parameter 0 was calculated by using eq. (16) for q, which in turn uses the values of (SC;) and (SC:) given by expression (13). The results are shown in table 2. Large enhancements are observed especially for small barriers - 1 nm and small biases A Zi/Zi - 0.01. Physically, these results are reasonable, since for small barriers nearby cascades can rapidly drive the dislocation
= hV/D
Fig. 2. Plot of the ratio of the cascade-induced
creep rate to the rate theory creep rate as a function
of the parameter
8.
H. Gu~ol, W. G. Wo(fet / Casazde - induced point - deject fhctuations
over the barrier. Reducing the bias has the effect of reducing the average climb velocity, and hence increasing the relative effect of the statistical fluctuations.
5. Conclusions We have outlined an approach to incorporate pointdefect fluctuations as caused by the discrete production in cascades into models for the microstructural evolution under irradiation. The approach is based on the use of moments of the point-defect concentrations, and on the recognition that often only the first two moments are required to study the effect of irradiation on microstructural processes. In particular, the effect of fluctuations caused by “cascade pulsing” can be conveniently represented by the variances (SC:) and (SC~}. Using a spatially-dep~dent response function in which recombination is neglected, these variances are well described by a simple analytical expression [eq. (13)]. We note, that the accuracy obtained for the variances is limited only by the accuracy of the response function h(r, r). The variances for negligible recombination depend linearly on the net number of point defects produced per colIision cascade when the damage rate is constrained to be constant. In agreement with previous studies of creep by climb-controlled glide, large enhancements in the creep rate due to cascades are obtained only when the barrier is - 1 nm in size or less, when the net bias for preferential interstitial absorption is 0.01 or less, and when the cascade produces 500 net point-defect pairs or more. The sensitivity of these results to irradiation temperature and to recombination requires, however, further study. We have chosen cIimb-controIled glide as a particular ~crost~ctural process to demonstrate our method and to compare it with previous work [2]. The approach taken in ref. 2 considered the average release of dislocations due only to single cascades. However, this corresponds to the dislocation responding to a transient arrival rate of point-defects, where only dislocations pinned near the obstacle edge can be relased. It does not account for the eventual release of dislocations which are situated away from the obstacle edge, due lo the cumulative effects of cascades under steady-state irradiation. Furthermore, ref. 2 does not treat the simultaneous arrival of interstitials and vacancies. The basis of the method used in ref. 2 for relating the climb height to the cascade position was presented in a previous paper [9], where the dislocation climb distance due
249
to a single cascade was calculated. The particular application considered in this paper shows that there exists a well-defined and rigorous procedure to incorporate point-defect fluctuations into processes related to point defect absorption once the moments for the point defect concentrations are known, It should be noted that this approach includes the simultaneous arrival of vacancies and interstitials, and the overlapping of the vacancy concentration contributions from different cascades.
Note added in proof
During the course of the preparation of this manuscript we became aware of similar work being done by A.D. Marwick on the calculation of the pointdefect variances [lo].
Acknowiedgements
We would like to thank Mr Timothy Naughton of the University of California at Santa Barbara for his efforts in obtaining the numerical results of section 3. This research was supported in part by the National Science Foundation under grant number CPE-8025300 with the University of California, and by the Office of Fusion Energy, Department of Energy, under contract DE-AC02-82ER52082 with the University of Wisconsin.
References [I] L.K. Mansur, W.A. Coghlan and A.D. Bra&ford, [2] [3] [4] [S] [6] [7] [8] [9] [IO]
J. Nucl. Mater. 85/86 (1979) 591. L.K. Mansur, W.A. Coghlan, T.C. Reiley and W.G. Wolfer, J. Nucl. Mater. 103/104 [1981] 1257. H. Gurol, J. Nucl. Mater. 90 (1980) 133; J. Appl. Phys. 50 (1979) 2705. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw Hill, New York, 1965) Ch. 16. H. Gurol, J. Appl. Phys. 51 (1980) 15. P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. 1 (McGraw-Hill, New York, 1953) p. 191. W.G. Wolfer and H. Gurol, to be published. W.A. Darling and A.J.F. Siegert, Ann. Math. Stat. 24 (1953) 624. H. Guroi, N.M. Choniem and W.G. Wolfer, J. Nucl. Mater. 99 (1981) 1. A.D. Max-wick, IBM Research Report, RC9276 (#40768) 1982: also to be submitted to J. Nucl. Mater.
Journal
INVESTIGATION
OF CASCADE-INDUCED
POINT-DEFECT
of Nuclear Materials 117 (1983) 244-249 North-Holland Publishing Company
FLUCTUATIONS
H. GUROL Science Applications,
Inc., 1200 Prospect Street, P.O. Box 2351, La Jolla, California 92038, USA
and W.G. WOLFER Department
of Nuclear Engineering,
University of Wisconsin, Madison,
Wisconsin 53706, USA
High energy irradiation produces collision cascades, which occur randomly in space and time. Any given point in the material will therefore to subject to intermittent arrival of interstitials and vacancies, even under steady irradiation. The conventional rate theory formulation of the point-defect concentrations, which contains spatial and time averaging and a uniform production rate, may not always adequately describe processes such as void nucleation and irradiation creep. We develop in this paper a theoretical approach to describe the cascade-induced point-defect fluctuations by their moments. The first moment, giving the average point-defect concentration, is used in the traditional rate theory. However, the second moment provides the variance of the point defect concentration, and it becomes important for processes such as void nucleation, dislocation climb, and loop growth. An approximate analytical expression is given for the second moment. The obtained concentration variances compare well with numerical results. We demonstrate with the example of climb-induced dislocation glide how the moments can be used to extend rate theory results. Large enhancement in the creep rate due to point defect fluctuations can be obtained if the barrier size and the average climb rate are small and if the cascade size is large.
1. Introduction High energy irradiation produces collision cascades containing a large number of displaced atoms. The occurrence of the cascades is random in both space and time. The theoretical foundations of void nucleation and growth and other macroscopic processes such as irradiation creep have previously relied on the rate theory for calculating average point-defect concentrations in the material. The rate theory inherently contains therefore spatial and time averages and a uniform production rate of point defects. A fundamental question then is: does the inclusion of collision cascades modify current theories of microstructural aevolution? A number of processes, in particular void nucleation, might be expected to be very sensitive to the number of point-defect pairs per cascade, Y, and the magnitude of the cascade-induced point defect fluctuations. Mansur et al. [l] have developed a procedure for calculating the instantaneous (cascade-induced) pointdefect concentrations as a function of time, neglecting recombination, and under steady irradiation. In a later
0022-3 115/83/0000-0000/$03.00
0 1983 North-Holland
publication [2], irradiation creep induced by cascades was considered. It was found that discrete cascade events can contribute to the release of dislocations. Although the computer simulation results of Mansur et al. [l] of the instantaneous point defect concentrations provide the most detailed description, such complete information is cumbersome to use and often not required for evaluating a given physical process such as dislocation climb, void nucleation, etc. Accordingly, we develop in this paper a theoretical approach to obtain the moments of the point-defect fluctuations directly, and we demonstrate with the example of climb-induced dislocation glide how these moments can be used to extend the rate theory approach to microstructural evolution. The approach can be viewed schematically as shown in fig. 1. Irradiation produces cascades randomly. The detailed physics of the point defect diffusion is given by a response function (or Green’s function) h (c, t), which in turn tions, average
gives and
the instantaneous
their
value
moments. (C);
the
The second
point-defect first
concentra-
moment
moment
gives
is just
the
the vari-
H. Gurol, W. G. Worfer / Cascade
Randomly Produced Cascades
t
'
I
Fig. 1. Schematic
1 Equation
+
illustration
for
Process
used to include
I
cascade
ante (SC’) = (C2) - (C)2. Higher moments can also be obtained. These moments can then be used to obtain the time evolution of a particular process as shown in fig. lb. A number of observations about the nature of the process can be made. First, it can be proven [3] that cascade-induced fluctuations will modify the rate theory average value of a process only if it satisfies a non-linear equation. Second, for many physical applications only the first two moments are required. More precisely, the effect of the higher moments is zero for a process that satisfies a Fokker-Planck equation. Processes in irradiated materials, that we have identified, can be adequately described by a Fokker-Planck equation. In section 2, we calculate the variance of the pointdefect concentrations, by describing the concentrations as “shot noise”. We then compare the theoretical results with numerical calculations. In section 4, we apply this theory to analyze the enhancement of climb-controlled glide creep due to cascade pulsing. Conclusions are presented in section 5.
Fig.
la
Fig.
lb
1 Average Evolution
dX = f(X,t) dt of the approach
245
Concentration Fluctuations &C(t), and Moments
+
.
I
AC(t), (6C2,
Impulse Response Function of the Medium h(r,tl _
-inducedpoint - defect fluctuations
pulsing
in the evolution
of a particular
process.
sponse function is a function of the medium only. If h,({, t) is known, then the average concentration (C,) and the variance (SC:) = (CJ) - (CJ2 can be calculated. The variance is a measure of the magnitude of the fluctuations. In the previous rate theory, (SC:) is set equal to zero. The concentration and the variance at a given point due to cascades a distance { away is [4] (C,)I,
= X(S)qaJm
-m
h,(S,
t)dt
(2)
and (SC,2)/,
= X(l)q,2Jrn
--oo
h:(S,
t)dt,
(3)
where A({) is the average rate of cascade production in a differential spherical volume element about {. The total concentration, and variance can be obtained by performing an integration over l. The simplest response function is obtained from the spatially averaged equation for the concentration, neglecting recombination and under steady-irradiation (i.e., (C,) is a constant) [5]:
2. Theory
~+D&z~(t)=F(t). The point-defect concentration at a given point in space due to cascades occurring a distance 3 away can be described by a superposition of the individual response functions, and it is given by
C,(l, 1) = C%JL(L t - t,). this expression a = i for interstitials and a = v for vacancies, h,({, t) is the impulse response function, and q, is the effective magnitude of each cascade. q, depends on the net number of point-defects per cascade v, on the amount of cascade overlap, and on mutual recombination. q, is a quantity to be determined from the value of the average concentration (C,). The reIn
(4)
Here F(t) is the instantaneous rate of cascade production. It can be broken up into the sum of an average rate (P) and a fluctuating rate f( t). (P) is just the rate theory production rate, and by definition (f) = 0. D, is the diffusion coefficient, and S, is the total sink strength. The response function for this equation is given by h,(t)
= e -D-s4q
t),
(9
where U(t) is the unit step function. used to obtain the unknown q, as 4, = DSL(CJ where
TV= X-’
E@. (2) can now be
7F7 is the average
(6) time between
cascades.
H. Gurol, W. G. Wolfer / Cascade-induced point -defectfluctuatmw
246
Using this value of q, and eq. (3) the variance becomes: (7)
(SC?> = (7,/2)Q&CJ2.
This expression for (SC:) can be expected to give a lower bound value because of the fact that the spatial integration in effect dilutes the strength of the cascades. A more complete description of the physics is obtained by using the spatially-dependent response function of the diffusion equation (again neglecting recombination and under steady irradiation) as employed by Mansur et al. [l]: ~(~,f)--D,V*c~+D,S,C,=F,(I,f). The function h,([,
h,(c,
(8)
t) is derived
t) = [4sD,t]-3’2
in ref. 6 and given by:
exp[ -[*/4D,t]
X exp[ -D,S,t]u(t).
(9)
The average and the variance can now be evaluated by performing a spatial integration of eqs. (2) and (3). A simple approximation is to use the spatially averaged value for the cascade production rate x in these equations. As before q, can be determined. The result is q, = 12.4 r,(C,)DJ
exp(@/*).
(‘0)
rc = l/X. where rc is the mean time between cascades. Using this value of q,, the variance can be computed [using eq. (3)] as (SCz)]b
= 0.64 T,(C,)*D,S,
exp[2LSJ/*]
K, [2SSl/2].
(11) where K, is the modified Bessel function of the second kind of order two. Expression (11) is the variance of the point-defect concentration due to cascades at position 5. Mansur et al. [ 1] have shown that the cascade screening distance (beyond which cascades have no effect) is about 7S-I/*. The total variance can then be evaluated u approximately by summing the contributions over distance out to about 7s; ‘/* . If one chooses seven concentric shells, each of thickness S;‘/*, for example, the variance be-comes (6C,z) i= 0.64 T,( CJ2D,S,
i
e2kK2(2k).
(12)
k=l
The sum can be evaluated (SC;)
= 3.4 r,(C,)*D,S,.
sponse function used will only modify the numerical constant, and that (SC,‘) given by eq. (13) describes the correct scaling of the fluctuations with the cascade size. It should be noted that this equation gives the variance in terms of (Ca)*, which can be determined by conventional rate theory. Furthermore, (6Ci) is linearly proportional to the net number of defects per cascade when the damage rate TV-’Y is constrained to be a constant. Otherwise, the variance is quadratic in v. In the next section, the theoretical value of the variance is compared with numerical calculations.
to give (13)
This expression varies from expression (7),which relies on a much simpler description of the physics, by a numerical factor only. In fact, we expect that the re-
3. Comparison with numerical calculations We have performed numerical calculations of the point-defect variances by using the Green’s function solution [eq. (9)] of the spatially-dependent diffusion equation for the point-defect concentrations. The Monte Carlo method was used to select discrete random cascade distances from the reference point. Cascades were placed within a volume of radius 7s: ‘j2. This numerical procedure is the same as that presented by Mansur et al. [I], except that they distribute the cascades within spherical shells instead of selecting an arbitrary random distance. Both approaches were found to agree, however, producing the rate theory average value of the concentration. The time dependence of the vacancy and interstitial concentrations was calculated for a fixed damage rate of 10e6 dpa/s, T= SWC, and for values of v ranging from 50 to 500. For consistency in the counting statistics the total number of cascades sampled by the code was the same for all v values: about 1500 cascades. This corresponds to a frequency of about one cascade per 0.7 s for v = 500 and one per 0.07 s for v = 50. The material parameters used were: 0,” = 0.014 cm*/s, Do = 0.008 cm*/s, ET= 1.4 eV, Eim=0.15 eV, S, = 1.1 x 10” cm-*, S, = 1 x 10” cm*. A fractional survival rate of 20% was assumed. Table 1 shows the average point-defect concentrations and a comparison of the theoretical and numerical values of the variances as a function of v. The average vacancy and interstitial concentrations are within about 2 and 4%, respectively, of the rate theory concentrations. The theoretical values of (SC:) were computed using eq. (13). First, it is seen that the theoretical variances compare very well with the numerical values, for all v; that is, the analytical theory presented here shows the correct scaling of the variances with the net number of displacements per cascade. The scaling of (SC:) is linear with the parameter v, as predicted by eq. (13) for fixed damage rate. Therefore, as an example,
241
H. Gurol, W. G. Worfer / Cascade - induced point -defect fluctuations Table 1 Comparison Y
of the theoretical
variances
using eq. (13), as a function
Cci>
(SC,‘) ( x 1033)
(X
(X lo* cmw3)
Theoretical
Numerical
Theoretical
Numerical
1.75 1.75 1.74 1.74 1.I4 1.75
3.1 2.5 1.9 1.24 0.62 0.31
3.3 2.6 2.0 1.33 0.7 0.32
3.5 2.8 2.1 1.4 0.7 0.35
4.0 3.2 2.4 1.6 0.8 0.4
1Ol6 cm-3)
1.57 1.57 1.57 1.56 1.55
even though the damage rate is lop6 dpa/s for all Y, the variances ((SC:), (X2)) at Y = 500 are five times the value at Y = 100. This is just another way of saying that the vacancy and interstitial concentration peaks above their average values are greater for more damaging irradiation. For a given physical process then, the theory presented can be used to ascertain the regime of physical parameters where random cascade-induced effects are significant.
4. Application to climb-controlled glide creep As a specific application of this theory we consider here the climb-controlled glide of dislocations. Only the outline of the method is discussed here. A detailed discussion of cascade-induced creep will be presented in a forthcoming paper [7]. Consider an arbitrary edge dislocation segment whose glide plane is initially a distance x,, from a parallel reference plane. With time, the climb distance x(r) varies randomly with the arrival of the point-defects. Rapid positive increases in x(t) are caused by the arrival of interstitials from nearby collision cascades. The slower decreases in x( t ) are due to the more steady arrival of vacancies. On the average, the climb distance x(l) exhibits a net increase or decrease, depending on the net bias AZi of the particular edge dislocation segment relative to all other dislocations and sinks. since the number of point-defects reaching the dislocation is random, the climb distance x(t) is a stochastic variable, with an associated probability distribution function p (x, t). The probability that x is in the interval x to x + dx at time 1, given that x(0)=x,,, is then p(x, r)dx. It can be shown that p(x, t) satisfies approximately a Fokker-Planck equation with “drift” and “diffusion” terms, v and a given by V=
lim -(Ax) At-0
of Y
CC”)
1.57
500 400 300 200 100 50
and numerical
At
’
(14)
(sc;)(xIo*S)
% =
lim At-0
((Ad> . At
(15) \
,
I
where vis the net average climb rate. The quantity q is the “diffusion coefficient” associated with the random climb of the dislocation segment. @ now can be related to the variance of the point-defect concentrations, using standard methods [4]. The result is
where 52 is the atomic volume, b is the Burgers vector, and pd is the dislocation density. In this expression we have neglected cross-correlation terms such as (SC; XV), making eq. (16) a lower bound value for a; that is the effect of cascade-induced fluctuations will be greater than predicted by eq. (16). Suppose that the dislocation segment is pinned by an obstacle of “effective” height h at a climb distance x,, from the lower glide plane not intersected by the obstacle. Since not all portions of the length of the dislocation climb at the same instantaneous rate, the effective height h must be defined as if all points along a segment of length - h climb at the same rate. The height h will therefore always be somewhat greater than the physical barrier height. We now ask for the mean escape time 7(x0, h) required for the dislocation (starting at position xc) to climb either over or under the obstacle. This escape time is known in the statistical mechanics literature as the “first passage time” of the stochastic variable x(t) defined over the interval 0 c x(t) Q h. The first passage time satisfies the following differential equation [8]
+
Vd7+1=0 dx,
’
(17)
7(0,h)=7(h,h)=O, which
is associated
with the Fokker-Planck
equation
248
H. Gurol, W. G. Wolfer / Cascade-induced
for the probability distribution function p(x, t). The solution of this equation for 7(x,,, h) is readily obtained [7] and represents the mean escape time for a dislocation held up initially by an obstacle of height h at a distance x,, from the lower end of the obstacle. Since we can safely assume that the dislocations are held up by the obstacles with equal probability at any distance x0 between 0 and h, the mean escape time, averaged over all initial positions, is given by
T(h)
=$(x,,,
The result of the integration
q+&
(,_e-e)-l_$_;
V
(
_-
(18)
h)dx,.
point-defect
Table 2 Calculation of the ratio of cascade-induced creep to the rate theory creep for specific values of h, bias factor AZ, , and 6’. The parameter 0 is an indication of the importance of statistical fluctuations to dislocation climb: as 0 decreases, the effect of fluctuations is greater. Eq. (22) was used. T = 500°C, P = 10mh dpa/s.
_Y
500 500 500
h(nm)
A Z, /Zi
8=hV/9
R
10
0.1
I 1
0.I
9.6 0.96 0.096 0.48 0.048
1.26 6.34 60.0 12.4 126.0
1000
is
1
0.01 0.1 0.01
1 1
1000
)
fluctuations
(19)
where 0 = hV/q, and represents a dimensionless parameter which indicates the relative importance of cascade-induced fluctuations in climb. We see immediately that for the case of no fluctuations, that is for q + 0, one obtains the release time ?(h)+h/2V,
(20)
which of course is just the conventional rate theory result. On the other hand, in the fluctuation-dominant regime (v + 0), one finds
(21)
f(h)
From expressions
(19) and (20), one can obtain the ratio rate with fluctuations to the creep rate given by conventional rate theory. The result is
R of the creep
_ (1
-e-@)-l_;_;
(22)
A plot of this ratio as a function of 0 is shown in fig. 2. For values of B > - 20, the effects of fluctuations are negligible. For 0 < - 4, the ratio R rises very rapidly. We have calculated the enhancement R due to cascade-induced fluctuations for typical ranges of barrier sizes and dislocation bias differences A Zi. The parameter 0 was calculated by using eq. (16) for q, which in turn uses the values of (SC;) and (SC:) given by expression (13). The results are shown in table 2. Large enhancements are observed especially for small barriers - 1 nm and small biases A Zi/Zi - 0.01. Physically, these results are reasonable, since for small barriers nearby cascades can rapidly drive the dislocation
= hV/D
Fig. 2. Plot of the ratio of the cascade-induced
creep rate to the rate theory creep rate as a function
of the parameter
8.
H. Gu~ol, W. G. Wo(fet / Casazde - induced point - deject fhctuations
over the barrier. Reducing the bias has the effect of reducing the average climb velocity, and hence increasing the relative effect of the statistical fluctuations.
5. Conclusions We have outlined an approach to incorporate pointdefect fluctuations as caused by the discrete production in cascades into models for the microstructural evolution under irradiation. The approach is based on the use of moments of the point-defect concentrations, and on the recognition that often only the first two moments are required to study the effect of irradiation on microstructural processes. In particular, the effect of fluctuations caused by “cascade pulsing” can be conveniently represented by the variances (SC:) and (SC~}. Using a spatially-dep~dent response function in which recombination is neglected, these variances are well described by a simple analytical expression [eq. (13)]. We note, that the accuracy obtained for the variances is limited only by the accuracy of the response function h(r, r). The variances for negligible recombination depend linearly on the net number of point defects produced per colIision cascade when the damage rate is constrained to be constant. In agreement with previous studies of creep by climb-controlled glide, large enhancements in the creep rate due to cascades are obtained only when the barrier is - 1 nm in size or less, when the net bias for preferential interstitial absorption is 0.01 or less, and when the cascade produces 500 net point-defect pairs or more. The sensitivity of these results to irradiation temperature and to recombination requires, however, further study. We have chosen cIimb-controIled glide as a particular ~crost~ctural process to demonstrate our method and to compare it with previous work [2]. The approach taken in ref. 2 considered the average release of dislocations due only to single cascades. However, this corresponds to the dislocation responding to a transient arrival rate of point-defects, where only dislocations pinned near the obstacle edge can be relased. It does not account for the eventual release of dislocations which are situated away from the obstacle edge, due lo the cumulative effects of cascades under steady-state irradiation. Furthermore, ref. 2 does not treat the simultaneous arrival of interstitials and vacancies. The basis of the method used in ref. 2 for relating the climb height to the cascade position was presented in a previous paper [9], where the dislocation climb distance due
249
to a single cascade was calculated. The particular application considered in this paper shows that there exists a well-defined and rigorous procedure to incorporate point-defect fluctuations into processes related to point defect absorption once the moments for the point defect concentrations are known, It should be noted that this approach includes the simultaneous arrival of vacancies and interstitials, and the overlapping of the vacancy concentration contributions from different cascades.
Note added in proof
During the course of the preparation of this manuscript we became aware of similar work being done by A.D. Marwick on the calculation of the pointdefect variances [lo].
Acknowiedgements
We would like to thank Mr Timothy Naughton of the University of California at Santa Barbara for his efforts in obtaining the numerical results of section 3. This research was supported in part by the National Science Foundation under grant number CPE-8025300 with the University of California, and by the Office of Fusion Energy, Department of Energy, under contract DE-AC02-82ER52082 with the University of Wisconsin.
References [I] L.K. Mansur, W.A. Coghlan and A.D. Bra&ford, [2] [3] [4] [S] [6] [7] [8] [9] [IO]
J. Nucl. Mater. 85/86 (1979) 591. L.K. Mansur, W.A. Coghlan, T.C. Reiley and W.G. Wolfer, J. Nucl. Mater. 103/104 [1981] 1257. H. Gurol, J. Nucl. Mater. 90 (1980) 133; J. Appl. Phys. 50 (1979) 2705. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw Hill, New York, 1965) Ch. 16. H. Gurol, J. Appl. Phys. 51 (1980) 15. P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. 1 (McGraw-Hill, New York, 1953) p. 191. W.G. Wolfer and H. Gurol, to be published. W.A. Darling and A.J.F. Siegert, Ann. Math. Stat. 24 (1953) 624. H. Guroi, N.M. Choniem and W.G. Wolfer, J. Nucl. Mater. 99 (1981) 1. A.D. Max-wick, IBM Research Report, RC9276 (#40768) 1982: also to be submitted to J. Nucl. Mater.