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Point defect trapping and void growth
Journal of Nuclear Materials 78 (1978) 354-361 0 North-Holland Publishing Company
POINT DEFECT TRAPPING AND VOID GROWTH A.D. BRAILSFORD Scientific Research Stafi
Ford Motor Company, Dearborn, MI 48121, USA
Received 21 April 1978
Trapping of point defects at solute atoms or on the surfaces of coherent precipitates is known to reduce the void swell-
ing of irradiated materials. It is shown here, however, that the dominant physical processes in these two situations are different. For atomic traps, defect capture is mainly balanced by defect emission in the steady state. For precipitate trapping, on the other hand, the balance is mainly achieved by on-site recombination with the anti-defect. Isomorphisms between interstitial and vacancy trapping are derived in the two cases through the development of a semiquantitative analytical model. Substantial agreement is obtained with the prior numerical analyses of other workers.
1. Introduction
tial trapping, MY find little effect until the binding energy exceeds the difference between the free vacancy and free interstitial migration energies. BH, however, find essentially the same effect with either interstitial or vacancy binding [9] on large precipitate-matrix interfaces. While one possible source of disparity could have been the different choices of input parameters, (void concentrations defect migration energy and so on), it was our opinion that the different qualitative nature of the void swelling suppression might bear closer scrutiny. For this reason we undertook the investigation to be reported here. We have found that, in fact, the influence of small and large traps is of entirely different character even though both, of course, lead to a swelling reduction. This difference can be traced back to the physical processes leading to the steady state occupancy the traps ultimately achieve during irradiation. For it is shown that whereas atomic-sized traps attain steady state through a balance of point-defect capture and thermal release, large trap point-defect capture is mainly opposed by on-site recombination with the anti-defect (trapped vacancy with mobile interstitial, for example). It is thus not surprising that the swelling suppression is different in the two regimes. A semi-quantitative analytic expression for the swelling, valid below the peak swelling temperature,
It has long been recognized that the trapping of intrinsic point defects at solute atoms [ 1] or at coherent precipitate-matrix interfaces [2] is an effective means of reducing void swelling in irradiated materials. In each instance the essential physical process is the same. Trapping impedes the migration of the defect to other sinks (e.g. voids and dislocations) in the system, thereby enhancing the probability of recombination with the anti-defect. The greater the preponderance of such events, the smaller the void swelling. Because of its obvious technological importance, several attempts [l-8] have been made to investigate the trap effect, either quantitatively or qualitatively. Some authors [ 1,3-5,7] have focused upon solute atom traps, others upon precipitates of varying size [2,8]. In both cases the potent effect of trapping in reducing void swelling has been amply demonstrated. The most detailed results available, however, betray subtle differences which require explanation. For example, in their study of solute atom trapping, Mansur and Yoo (hereafter MY) find a significant decrease of swelling with vacancy trapping at all temperatures of interest. Bullough and Hayns (hereafter BH) on the other hand,‘find a significant effect of large (i.e. precipitate) traps at intermediate and high temperatures only. Again, for solute atom-intersti354
355
A.D. Brailsford /Point defect trapping and void growth
is developed for each trap-size domain. This is shown to account for the differences in behavior found by the MY and BH analyses.
for vacancy trapping: Kv=K+Kvv,
(4)
Ki=K, 2. Preliminary
(where K is the basic damage rate in dpa/s), while the condition for no matter accumulation at the traps is
Theory
During irradiation, the steady state interstitial and vacancy concentrations, ct and c, respectively, are determined by the following relations: K, - Dvkfc, - Ocic, = 0 9 Ki - Dikfci - Lycicv= 0 e
(1)
(2)
kT=ki2tkZvi,
we find, for vacancy trapping:
K - D&$z, - (@Cc,+ Dik$i) Ci = 0 9 (WV +
(6)
Dik+i) ci = 0 .
It is evident, therefore,
that if
01cv< Dik5i 3
(7)
intrinsic recombination is unimportant compared to trapping effects. Let us assume this is the case. Then from the pair of eqs. (6) we find:
kz
1
where PD is the dislocation density, Zi and Z, capture efficiencies for interstitials and vacancies, and kz is the lowest order void (cavity) sink strength (kz = 4nr,C,, with C, the volume concentration of voids, re their common radius). The sink strengths * k$, and kti account for vacancy capture at an unoccupied trap and interstitial-vacancy recombination at an occupied trap. They will be further specified in section 3. We shall assume throughout this study that thermal emission of a point defect from sinks other than occupied traps is negligible. This is a reasonable approximation below the peak swelling temperature as a rule. Thus if Kvv is the vacancy thermal release rate,
* In an attempt at simplification we have abbreviated the notation of ref. (61: k& is equivalent to k&., k5i to k2i”T of this work. An analogous change is made later for interstitial trapping.
(8)
c V = Kkz/D&‘k’ 1 i v,
kZi
9
(3) iPD ’ k?
eqs. (1-5)
which are consistent This requires:
where k: and Ef are the sink strengths from other than traps:
p-Z I
Combining
(5)
pi = K/DikF ,
k; = k,2 t kc,,
vPD + k;
D&$&v - Dik$ici - Kvv = 0 a
K - DikcTCi-
Here Di and D, are the diffusivity of the interstitial and vacancy, respectively, Ki and Kv their respective production rates in displacements per atom per second (dpa/s); (Yis the intrinsic recombination coefficient. The quantities k,” and kf are referred to as total sink strengths for the vacancy and interstitial, respectively. With vacancy traps for example [6],
P,Z ”
161:
i
(
1+;+
solutions if eq. (7) is satisfied.
>;,
(9)
i 1
with n = (~QK/D.D$~~~) I i v , or (k&i/k;) > [( 1 + ~)r’~ - I]/2 . Since the time rate of change of the fractional ing is given by S = k%(D$v - Dici) y
(10) swell-
(11)
when vacancy emission from voids is negligible, providing the condition (10) is satisfied, it follows that 3 = kzK(af - E$)/ktit:
.
(12)
This may be simplified further in two cases. Namely when either the trapping effect is small or when the sink strengths from other than traps are dominated by a constant, built-in dislocation density, (as in coldworked materials, for example). In this instance, to a first approximation EF and rz’will be independent of the trapping and we can relate s to the swelling rate,
A. D. Brailsford 1 Point defect trapping and void growth
356
s(O), without traps. Since [ 101 S(0) = [k$(@
- E;)/EFk;4] F(n) ,
cited eq. (17) applies if hv is replaced by hr where (13)
where F(7)) = 2Kl + rl)“2 - 11/n,
(14)
and S = 4nrzCJ3, for the same void concentration we find:
s 7-b S(O)
S
113
I I s(o)
.
(21)
The central parameter in this development, therefore, is the relative sink strength of occupied traps (for the respective anti-defect) to the sink strength for this anti-defect as determined by all other homogeneously distributed sinks in the system. This is determined in section 3.
(15)
,
with
3. Trap sink strengths
Xv1 = F(q) [l + (&i/k?)]
s
(161
Hence, for Xv a constant, it follows that s = AV2S(0).
(17)
This relation, obviously approximate, will be further explored in section 4. The domain where it can be rigorously justified is clearly rather limited. Nevertheless it appears to have wider qualitative usefulness, as will be demonstrated later. It should be emphasized, however, that it is only to be used when condition (10) is satisfied. The latter provides a lower bound upon ~~~i/~~~ for which eq. (15) is even approximately valid. It is interesting to observe that, for this lower bound, hv = 1. Smaller values of (k
w-9 and Ki=K+KIi)
(19)
IC,=K,
To continue the development, we require the trap sink strengths under conditions where intrinsic recombination is negligible. These have been derived by Brailsford and Bullough [6] for both interstitial and vacancy trapping. For vacancy traps situated on the surface of a spherical particle of radius rv their results * are k$i = @.&/Vv + s) 1 &,=k%l
- KIi = 0 m
It is thus only necessary to change the subscript I to V and interchange i and v. For the conditions already
(22)
-fv+s), 1 - fv> 3
where fv is the occupation probability of a trap site and c$ the relative rate of thermal release from an occupied site to capture by an unoccupied site. The parameter kc is defined by k$ = 4nrvCv ,
(23)
with Cv the volume concentration of particles, while s = a2/ZwQ2rv .
(24)
Here t a is the lattice constant, 51,the atomic volume, Z the number of matrix sites adjacent to a trap and w the surface density of trap sites. The probability fv is determined [6] by substitution of eq. (22) into eq. (5). This leads to: f+ [K@? - E2) + DvcBE2(E2 + k2 )] -K@=O,
(20)
-fv)/(l
.Kvv = ~vD”k~~~/(
- f:{K(E;
in place of eqs. (2) and (4). Similarly, eq. (5) is replaced by DiktiCi - DJ$~v
Xi’ = F(n})Il + fkf,lK;S)]
- i$-
;K&
t i<,
+ &c,B~f~;]
s} (25)
* It wilI be assumed here that [6] k,,rv << 1. t The quantity (az/Zwn) has been used in place of the Burgers’ vector b of ref. [6] In order to show the explicity dependence on surface trap density in general.
351
A.D. Brailsford /Point defect trapping and void growth
[which is eq. (90) of ref. [6] when pv = 01. fv is the positive root of eq. (25) between zero and unity. The results for interstitial trapping have an analogous form. In place of eq. (22) Brailsford and Bullough [6] find *: k:” = k:fi/(fr
+ s) 3 (26)
~~i=k:(l-fI)/(l-fIts), KIi = f$iktiC8/<
1 - f~> 2
where kf = 4nrrCr and fr is the occupation probability of an interstitial trap site. All symbols have an analogous meaning to the vacancy trapping case, the defect and anti-defect being now the interstitial and vacancy, respectively, instead of vice-versa. Consequently the equation determining fr is determined from eq. (25) by changing the suffix V to I and interchanging the suffices i and v: fi {K(kf - kz) - Dtc~k~(k~
+ kf)}
- fr {K(Ef - k:) + [K(kf t 5:) t DicpEfFt] +K&;s=O.
S} (27)
This is eq. (87) of ref. [6] for their !?t = 0. Particular solutions for fv and f, will now be obtamed for the two limits of small and large traps. 3.1. Atomic traps The spherical trap model, with w-l = 4nr$ and rv roughly one half the nearest-neighbor separation [ 111, leads to s = 1. We follow ref. [6] in taking s precisely rqual to unity as descriptive of discrete solute atom trapping t . In this limit eq. (25) becomes:
(28) where &, = Kk%/D,.cBk2E2 v i v*
(29)
* We assume kp-I << 1. t The precise equality is merely an algebraic aid; it is not CIUCM.
Thus, for k: = k: (i.e. for neglect of the dislocation bias effect) one finds fv = (Ef/2k$)[(4&
t 1)“2 - l] ,
(30)
if k$ >> kf, or kc >> /3&, whichever is the larger. The first of these inequalities is a condition on the solute-atom concentration, namely QCv >> (nk:/ 4nrv), the latter being of the order of 2 X 10e6 for kf = 5 X lOlo /cm 2. Hence the first inequality will be satisfied for all but the most dilute solutions. The second is of importance when /Iv exceeds unity; it implies that even though this be the case, the trap occupation probability is still small nevertheless. Consequently, when eq. (30) is valid, eq. (22) may be approximated by k&E
Gfv 3 k;, L lk2 2 V?
Kvv = -&D&c,B
(31)
.
If one now uses eq. (8) to determine ct and c, and considers the separate contributions to eq. (5) it is found that the dominant physical processes are vacancy capture and thermal release. Thus a balance is achieved by the same mechanisms that determine thermal equilibrium *, although the occupation probability is different under irradiation. The entire development rests upon intrinsic recombination being unimportant. Combining eqs. (30) and (31) and inserting in the inequality (9) we find this obtains if 40~ > Q, or xv > 1, where xv=-
40~ 17
DiG =ac,” .
(32)
Now in the materials of interest, the interstitial is much more mobile than the vacancy; thus [ 121 01= 30 Di/a’, (for the split-interstitial configuration). Also, for vacancy trapping in an fee. lattice [ 121, c,” = (l/ 12) exp(-eT/kZ), where E: is the binding energy of the trap-vacancy complex, k is Boltzmann’s constant and T the absolute temperature. Hence eq. (32) may also be expressed as 96n xv==RCv
5 exp(e?/kq ( a 1
.
(33)
* For thermal equilibrium one cannot neglect thermal emission processes from vacancy sinks other than traps, as we have done here. In fact eq. (90) of ref. [6] yields the carrect result, f:q = cg(ct + ct>, for K = 0, only if their zt (= DV@$
is retained.
358
A.D. Brailsford /Point defect trapping and void growth
The lower bound upon xv, i.e. xv > 1, is thereby equivalent to a lower bound upon e$. Binding energies less than this bound, which depends upon the solute concentration, have essentially no effect upon the swelling according to our model. Larger binding energies reduce the swelling according to eq. (17) with Xv evaluated from eqs. (16), (30) and (3 1): A,’ = +F(n)[(nxv
+ l)l’2 + l] .
(34)
This will be discussed further in section 4. The analysis of eq. (27) for interstitial trapping at solute atoms parallels the above completely. For, with PI = Kk:/Dici”ki’k~ , the re-expression
k:
Pl]tf1($1+ 1)
has identical form to eq. (28) when the term in (kf - k,?) is neglected. With the restriction on 0 given after eq. (30) this term is indeed negligible since (Zi - Zv) < 1. Thus the corresponding solution for fI is: fI -(k;/2k$[(vxI
+ 1)“2 - l] ,
1 - kT ln(D$k:/u>~k$)
An example of this isomorphism
.
(41)
is given in section 4.
3.2. Traps on coherent precipitates
Af2-(B-Cs)f-Ds=O, (36)
3
eB=en+em_em ” ” L
When the traps reside on a surface of large radius, s becomes small compared to unity for any non-zero w. Eq. (25) has the generic form:
of eq. (27):
+ 1 _K(kT-q)
_-k2 ki PI = 0
(35)
xv = XI, then the effects on the swelling are identical. We may express this isomorphism in terms of binding energies as follows. For split-interstitial capture [ 141, cp = (l/6) exp(-$/k7’), where e? is the binding energy of the trapped interstitial. Let D, = 0: exp(-er/ko and Di = Dp exp(-$‘/kT) where ~7 and E/” are the motion energies for thevacancy and interstitial. Then the binding energy of interstitial traps which yields the same effect as vacancy traps is:
(37)
where A, B, C and D are independent roots of eq. (42) for small s are: j-(l) = B/A ,
f12’ = -Ds/B .
(42) of s. The two
(43)
All the capital letter parameters are positive, and B/A is less than unity by inspection of eq. (25). Thus fl’) is the desired ro o t [6]: (44)
where (38)
Eq. (27), for interstitial trapping, has the same form as eq. (42), but now D is negative while B/A is greater than unity. Hence f12) is here the physically acceptable root [6] :
(39)
(45)
This limit also leads to: k?v =
k?fI ,
k&‘v Ik* 2 13
KIi ‘I $fIDik:cB ,
in analogy with eq. (3 1). Thus steady state occupation may be shown to result now from a balance between interstitial capture and release. Similarly, the effect on the swelling is here determined by the parameter hr: Xi’= $F(n)[(~xr
t 1)li2 t l] .
(40)
Thus there is a complete isomorphism between the effects of vacancy and interstitial trapping at solute. atoms in the sense that if the respective concentrations, trapping radii and binding energies are such that
We would emphasize two points. First, the isomorphism established for atomic traps is lost at this point. For strong binding the vacancy traps are essentially full, the interstitial traps essentially empty. Mathematically this arises because we have to switch from one branch of the double-valued function f to the other. Physically it arises because dislocations preferentially attract interstitials. Thus the vacancy traps accommodate as much of the excess vacancy flux as they can, while the interstitial trap is continuously emptied by the same excess flow. Second, as the
359
A. D. Brailsford /Point defect trapping and void growth
vacancy binding energy increases, eventually bothfv and fr become independent of temperature (or equivalently, of binding energy). In this limit fv = 1 so that, from eq. (22) k$i-kZ,,
(46)
4)
5-
0 0
while from eq. (26): k& ‘v k;(k;/kf) The condition fore: (k$/kf)
‘v kf . of self-consistency,
> [(l t #I2
X
?-
- 1]/2
(47) eq. (lo), is there-
(48)
(for vacancy trapping). This mainly dictates what the precipitate dispersion must be. Interstitial trapping leads to an essentially similar result with k$ replaced by kf, while the swelling reduction parameters, hv and hr:
indicate the equivalence of either trap type if now k+ = kf. This isomorphism is precisely the result found by Hayns [9]. To reach these results we have assumed fv = 1. Evidently, from eq. (44) this is valid for binding energies, Ef, such that:
(50) assuming cB = exp(-Ef/kT) for surface trapping. For ~~=l,k$~~~,(k~-k~)~3XlO~/cm~anda=3A the right hand side of eq. (50) is 1.2 eV. This crude estimate indicates that the isomorphism between interstitial and vacancy trapping at coherent precipitates, as embodied in the relation kf = kc, can be attained with reasonable values of the vacancy binding energy. 4. Implications for void growth In this section some numerical examples of the preceding discussion are given and the results compared with prior work of other authors. 4. I. Atomic traps In section 3 we established the isomorphism between interstitial and vacancy trapping at substi-
600°
0
l
c X
l-
I
I
,I
.2
I
I
I
I
.3 4 5 .6 A E (eV) Fig. 1. Void growth rates at constant irradiation dose as a function of effective binding energy for vacancy (a) and interstitial (X) trapping. Results from ref. [7]. 0
tutional solute atoms given by eq. (41). Examples of this are provided by the recent calculations of MY. They use a trapping model which, for atomic traps, is identical in all essential respects to that of ref. [6]. (This matter is discussed in detail in the Appendix.) Their fig. 8 gives the void growth rate (dr,/dt, in our notation) as a function of temperature, for various interstitial and vacancy trapping energies. From these data we have obtained the growth rate as a function of an effective binding energy, AL’, where: AE= e,B
(vacancy trapping) ,
AE = $ - E: t E? + kT In(D~/D~) trapping) * .
(51) (interstitial (52)
Results ** for two temperatures below that at which the growth rate is a maximum are given in fig. 1. (We recall that we have ignored thermal vacancy emission from other than vacancy traps.) The isomorphism is clearly evident in that the results for either type of * Eq. (41) reduces to (52) for kf = k$ if the factor of 2 in the argument of the logarithm is discarded. This factor was not incorporated by MY; it has been dropped for the purposes of comparison. ** The point defect parameters are as given by MY.
A. D. Brailsford /Point defect trapping and void growth
360
.- -----
\
0 .
\
0
lnletstlllol
\
x
\
.
600°C 650°C Traps x 600°C 650°C
VOCOflCY Traps
\
\ 0
\ \
.
\ \
0
\
\ \ \ \
x0
\
where r,(O) is the void radius, without traps, for the same irradiation dose as that at which Y, is measured. The trend indicated by our model (the broken line) is in good agreement with the numerical analysis of MY and deviations are in the sense expected, in that our assumption of no effect of traps below some critical binding energy obviously furnishes an upper bound. It should be emphasized that the rapid decrease of growth rate with binding energy depicted in fig. 2 arises because without traps, intrinsic recombination is dominant. If the sink strengths, kf and k:, are increased to the point where 71 << 1, then the sensitivity to binding energy is found to be greatly reduced. 4.2. Trapping at coherent precipitates
o-
' 01
I 02
I 03
1 I 04 05 06 AE (eV)
1 07
I, 08
Fig. 2. A comparison of the results of ref. [7] with the upper bound furnished by the present analysis (atomic traps).
trapping fall approximately on a continuous curve at each temperature. MY also noted the larger binding energies required for interstitial trapping to produce effects comparable to vacancy trapping and gave a qualitative argument based upon effective diffusivities. Their numerical analysis led them to suggest an approximate equivalence between vacancy and interstitial binding energies embodied in E! = (e? - er + ef). Our analytical model, leading to the re-expression of their results in fig. 1, shows that the equivalence is more accurately defined by eqs. (51) and (52). In fig. 2 we have illustrated the qualitative effect on void growth as obtained from our analysis, for a temperature of 650°C and dose rate of low3 dpa/s, for solute trapping in Ni. These results were obtained with a recombination radius, ro, (where 4rrrfli = aa) taken equal to the trap radius * and an atomic fraction of traps of 1Oe3. The sink strengths used in evaluating 17 were kt u kz = 5 X 10” /cm2, as used by MY. When eq. (17) is valid we can equally well take (53) * Doubling the trap radius decreases AE by approximately 0.05 eV at this temperature.
In section 3.2, it was shown that hv tends to a value independent of the binding energy. We shall explore the swelling reduction in this limit only. For a fixed volume fraction, 6, of second phase material, k$ = 3$/r;. Thus, combining eqs. (I 7) and (49): (54) where kf is to be regarded as a constant. The quantity 3@/kFr$ must also satisfy eq. (48). Accordingly, if intrinsic recombination is small without traps [i.e. 17<< 1, F(Q) = l] the latter condition is impotent and eq. (54) describes a relation between the swelling reduction and trap size. Numerical results for this domain are provided by the work of BH; those appropriate to 45O”C, now in austenitic steel, are illustrated in fig. 3, for @= 0.8%. To check their form against eq. (54) we fitted to their data point at rv = 20 A. The continuous curve in fig. 3 satisfactorily describes the trend they observed. The relation (44) breaks down, of course, when rv is of atomic dimensions. The analysis of section 4.1 then becomes appropriate. It should also be mentioned that for Cv = lOI cmw3 and rv = 100 A, the swelling calculated by BH is relatively insensitive to intrinsic recombination and vacancy binding energy at the same temperature. This is in accord with the basis we have given for the present discussion to be valid. Fig. 3 shows that above a certain size, coherent precipitate trapping has little effect on the swelling.
A. D. Brailsford
/Point
defect
361
trapping and void growth
Appendix 16 .
t
i
For the convenience of the reader, we document the relation between the notation used recently by Mansur and Yoo and that introduced earlier by Brailsford and Bullough. To be specific, consider vacancy trapping at solute atoms. The analogous form to eq. (22) used by MY is, with their CL =.ti&
ktv = 4nr;C,( 1 -_&,) , Kvv = i2 4
Ctfv exp(-EFlk7’)
(A-1) ,
/
oo,’
1
10
20
30
40
50
f0
70
60
90
I00
rV(A)
Fig. 3. A comparison of the results obtained in ref. [ 81, for swelling at fixed dose and fiied volume fraction of precipitates, with the prediction of the present analysis.
This feature also shows up in the case where intrinsic recombination without traps is dominant (n >> 1). This critical radius, TV, may then be inferred from the inequality (48) to be & - (61$/@7~‘*)“*. (For the parameter choices @= 0.8%, kf = 5 X 1O1’ /cm2 and 17= 1O*, TV is of the order of 30 A.) For CI< YV< TV we find S/S(O) - (rv/rv)3 in this limit.
5. Summary The Brailsford and Bullough theory of point defect trapping has been used to study the effect of trap size upon void growth. A clear physical distinction has been drawn between the processes leading to steady state occupancy when the traps are atomically dispersed as opposed to when they are uniformly distributed over coherent precipitate-matrix interfaces. Different isomorphisms between interstitial and vacancy trapping have been established in the two cases. Examples of each type have been identified in the numerical analyses of other workers. Substantial agreement with this prior work has been established.
Acknowledgements The author has benefited greatly from a free exchange of information with Drs. R. Bullough, M.R. Hayns, L.K. Mansur and M.H. Yoo.
where the trapping radii, rv and r;, for vacancy capture and on-site interstitial recombination are allowed to differ. From comparison with thermal equilibrium MY show that 4nr;b* = 12fi. The correspondence is now readily established if fv is (as we have shown) taken to be small, E,” equated to $, C, to Cv, rt to fr; and rt to rv. References [l] S.D. Harkness and Che-Yu Li, Proc. 1971 Int. Conf. on Radiation-Induced Voids in Metals, eds. J.W. Corbett and L.C. Ianniello (USAEC Conf-710601, 1972) p. 798. [ 21 R. Bullough and R.C. Perrin, Voids Formed by Irradiation of Reactor Materials, eds. SF. Pugh, M.H. Loretto and D.I.R. Norris (Brit. Nucl. Energy Sot., 1971) p.79. [ 31W. Schilling and K. Schroeder, Consultant Symposium on the Physics of Irradiation-Produced Voids (ed. R.S. Nelson, AERE, Harwell Rep. R7934) p. 212. [4] J.S. Koehler, J. Appl. Phys. 46 (1975) 2423. [5] P.R. Okamoto, N.Q. Lam and H. Wiedersich, Proc. Workshop on Correlation of Neutron and Charged Particle Damage (ed. J.O. Stiegler, Oak Ridge Report Conf760673,1976) p. 111. [6] A.D. Brailsford and R. BuIlough, J. Nucl. Mat. 69 & 70 (1978) 434. [7] L.K. Mansur and M.H. Yoo, Oak Ridge National Laboratory Report TM-6134 (1976); submitted to J. Nucl. Mat. [B] R. Bullough and M.R. Hayns, to be published. [9] M.R. Hayns, private communication. [lo] A.D. Brailsford and R. Bullough, J. Nucl. Mat. 44 (1972) 121. [ 1 l] A.C. Damask and G.J. Dienes, Point Defects in Metals (Gordon & Breach, New York, 1963) p. 83. [12] Seeref. [ll] p. 124. [13] Seeref. [ll] p. 92. [14] See ref. [ll] p. 135.
POINT DEFECT TRAPPING AND VOID GROWTH A.D. BRAILSFORD Scientific Research Stafi
Ford Motor Company, Dearborn, MI 48121, USA
Received 21 April 1978
Trapping of point defects at solute atoms or on the surfaces of coherent precipitates is known to reduce the void swell-
ing of irradiated materials. It is shown here, however, that the dominant physical processes in these two situations are different. For atomic traps, defect capture is mainly balanced by defect emission in the steady state. For precipitate trapping, on the other hand, the balance is mainly achieved by on-site recombination with the anti-defect. Isomorphisms between interstitial and vacancy trapping are derived in the two cases through the development of a semiquantitative analytical model. Substantial agreement is obtained with the prior numerical analyses of other workers.
1. Introduction
tial trapping, MY find little effect until the binding energy exceeds the difference between the free vacancy and free interstitial migration energies. BH, however, find essentially the same effect with either interstitial or vacancy binding [9] on large precipitate-matrix interfaces. While one possible source of disparity could have been the different choices of input parameters, (void concentrations defect migration energy and so on), it was our opinion that the different qualitative nature of the void swelling suppression might bear closer scrutiny. For this reason we undertook the investigation to be reported here. We have found that, in fact, the influence of small and large traps is of entirely different character even though both, of course, lead to a swelling reduction. This difference can be traced back to the physical processes leading to the steady state occupancy the traps ultimately achieve during irradiation. For it is shown that whereas atomic-sized traps attain steady state through a balance of point-defect capture and thermal release, large trap point-defect capture is mainly opposed by on-site recombination with the anti-defect (trapped vacancy with mobile interstitial, for example). It is thus not surprising that the swelling suppression is different in the two regimes. A semi-quantitative analytic expression for the swelling, valid below the peak swelling temperature,
It has long been recognized that the trapping of intrinsic point defects at solute atoms [ 1] or at coherent precipitate-matrix interfaces [2] is an effective means of reducing void swelling in irradiated materials. In each instance the essential physical process is the same. Trapping impedes the migration of the defect to other sinks (e.g. voids and dislocations) in the system, thereby enhancing the probability of recombination with the anti-defect. The greater the preponderance of such events, the smaller the void swelling. Because of its obvious technological importance, several attempts [l-8] have been made to investigate the trap effect, either quantitatively or qualitatively. Some authors [ 1,3-5,7] have focused upon solute atom traps, others upon precipitates of varying size [2,8]. In both cases the potent effect of trapping in reducing void swelling has been amply demonstrated. The most detailed results available, however, betray subtle differences which require explanation. For example, in their study of solute atom trapping, Mansur and Yoo (hereafter MY) find a significant decrease of swelling with vacancy trapping at all temperatures of interest. Bullough and Hayns (hereafter BH) on the other hand,‘find a significant effect of large (i.e. precipitate) traps at intermediate and high temperatures only. Again, for solute atom-intersti354
355
A.D. Brailsford /Point defect trapping and void growth
is developed for each trap-size domain. This is shown to account for the differences in behavior found by the MY and BH analyses.
for vacancy trapping: Kv=K+Kvv,
(4)
Ki=K, 2. Preliminary
(where K is the basic damage rate in dpa/s), while the condition for no matter accumulation at the traps is
Theory
During irradiation, the steady state interstitial and vacancy concentrations, ct and c, respectively, are determined by the following relations: K, - Dvkfc, - Ocic, = 0 9 Ki - Dikfci - Lycicv= 0 e
(1)
(2)
kT=ki2tkZvi,
we find, for vacancy trapping:
K - D&$z, - (@Cc,+ Dik$i) Ci = 0 9 (WV +
(6)
Dik+i) ci = 0 .
It is evident, therefore,
that if
01cv< Dik5i 3
(7)
intrinsic recombination is unimportant compared to trapping effects. Let us assume this is the case. Then from the pair of eqs. (6) we find:
kz
1
where PD is the dislocation density, Zi and Z, capture efficiencies for interstitials and vacancies, and kz is the lowest order void (cavity) sink strength (kz = 4nr,C,, with C, the volume concentration of voids, re their common radius). The sink strengths * k$, and kti account for vacancy capture at an unoccupied trap and interstitial-vacancy recombination at an occupied trap. They will be further specified in section 3. We shall assume throughout this study that thermal emission of a point defect from sinks other than occupied traps is negligible. This is a reasonable approximation below the peak swelling temperature as a rule. Thus if Kvv is the vacancy thermal release rate,
* In an attempt at simplification we have abbreviated the notation of ref. (61: k& is equivalent to k&., k5i to k2i”T of this work. An analogous change is made later for interstitial trapping.
(8)
c V = Kkz/D&‘k’ 1 i v,
kZi
9
(3) iPD ’ k?
eqs. (1-5)
which are consistent This requires:
where k: and Ef are the sink strengths from other than traps:
p-Z I
Combining
(5)
pi = K/DikF ,
k; = k,2 t kc,,
vPD + k;
D&$&v - Dik$ici - Kvv = 0 a
K - DikcTCi-
Here Di and D, are the diffusivity of the interstitial and vacancy, respectively, Ki and Kv their respective production rates in displacements per atom per second (dpa/s); (Yis the intrinsic recombination coefficient. The quantities k,” and kf are referred to as total sink strengths for the vacancy and interstitial, respectively. With vacancy traps for example [6],
P,Z ”
161:
i
(
1+;+
solutions if eq. (7) is satisfied.
>;,
(9)
i 1
with n = (~QK/D.D$~~~) I i v , or (k&i/k;) > [( 1 + ~)r’~ - I]/2 . Since the time rate of change of the fractional ing is given by S = k%(D$v - Dici) y
(10) swell-
(11)
when vacancy emission from voids is negligible, providing the condition (10) is satisfied, it follows that 3 = kzK(af - E$)/ktit:
.
(12)
This may be simplified further in two cases. Namely when either the trapping effect is small or when the sink strengths from other than traps are dominated by a constant, built-in dislocation density, (as in coldworked materials, for example). In this instance, to a first approximation EF and rz’will be independent of the trapping and we can relate s to the swelling rate,
A. D. Brailsford 1 Point defect trapping and void growth
356
s(O), without traps. Since [ 101 S(0) = [k$(@
- E;)/EFk;4] F(n) ,
cited eq. (17) applies if hv is replaced by hr where (13)
where F(7)) = 2Kl + rl)“2 - 11/n,
(14)
and S = 4nrzCJ3, for the same void concentration we find:
s 7-b S(O)
S
113
I I s(o)
.
(21)
The central parameter in this development, therefore, is the relative sink strength of occupied traps (for the respective anti-defect) to the sink strength for this anti-defect as determined by all other homogeneously distributed sinks in the system. This is determined in section 3.
(15)
,
with
3. Trap sink strengths
Xv1 = F(q) [l + (&i/k?)]
s
(161
Hence, for Xv a constant, it follows that s = AV2S(0).
(17)
This relation, obviously approximate, will be further explored in section 4. The domain where it can be rigorously justified is clearly rather limited. Nevertheless it appears to have wider qualitative usefulness, as will be demonstrated later. It should be emphasized, however, that it is only to be used when condition (10) is satisfied. The latter provides a lower bound upon ~~~i/~~~ for which eq. (15) is even approximately valid. It is interesting to observe that, for this lower bound, hv = 1. Smaller values of (k
w-9 and Ki=K+KIi)
(19)
IC,=K,
To continue the development, we require the trap sink strengths under conditions where intrinsic recombination is negligible. These have been derived by Brailsford and Bullough [6] for both interstitial and vacancy trapping. For vacancy traps situated on the surface of a spherical particle of radius rv their results * are k$i = @.&/Vv + s) 1 &,=k%l
- KIi = 0 m
It is thus only necessary to change the subscript I to V and interchange i and v. For the conditions already
(22)
-fv+s), 1 - fv> 3
where fv is the occupation probability of a trap site and c$ the relative rate of thermal release from an occupied site to capture by an unoccupied site. The parameter kc is defined by k$ = 4nrvCv ,
(23)
with Cv the volume concentration of particles, while s = a2/ZwQ2rv .
(24)
Here t a is the lattice constant, 51,the atomic volume, Z the number of matrix sites adjacent to a trap and w the surface density of trap sites. The probability fv is determined [6] by substitution of eq. (22) into eq. (5). This leads to: f+ [K@? - E2) + DvcBE2(E2 + k2 )] -K@=O,
(20)
-fv)/(l
.Kvv = ~vD”k~~~/(
- f:{K(E;
in place of eqs. (2) and (4). Similarly, eq. (5) is replaced by DiktiCi - DJ$~v
Xi’ = F(n})Il + fkf,lK;S)]
- i$-
;K&
t i<,
+ &c,B~f~;]
s} (25)
* It wilI be assumed here that [6] k,,rv << 1. t The quantity (az/Zwn) has been used in place of the Burgers’ vector b of ref. [6] In order to show the explicity dependence on surface trap density in general.
351
A.D. Brailsford /Point defect trapping and void growth
[which is eq. (90) of ref. [6] when pv = 01. fv is the positive root of eq. (25) between zero and unity. The results for interstitial trapping have an analogous form. In place of eq. (22) Brailsford and Bullough [6] find *: k:” = k:fi/(fr
+ s) 3 (26)
~~i=k:(l-fI)/(l-fIts), KIi = f$iktiC8/<
1 - f~> 2
where kf = 4nrrCr and fr is the occupation probability of an interstitial trap site. All symbols have an analogous meaning to the vacancy trapping case, the defect and anti-defect being now the interstitial and vacancy, respectively, instead of vice-versa. Consequently the equation determining fr is determined from eq. (25) by changing the suffix V to I and interchanging the suffices i and v: fi {K(kf - kz) - Dtc~k~(k~
+ kf)}
- fr {K(Ef - k:) + [K(kf t 5:) t DicpEfFt] +K&;s=O.
S} (27)
This is eq. (87) of ref. [6] for their !?t = 0. Particular solutions for fv and f, will now be obtamed for the two limits of small and large traps. 3.1. Atomic traps The spherical trap model, with w-l = 4nr$ and rv roughly one half the nearest-neighbor separation [ 111, leads to s = 1. We follow ref. [6] in taking s precisely rqual to unity as descriptive of discrete solute atom trapping t . In this limit eq. (25) becomes:
(28) where &, = Kk%/D,.cBk2E2 v i v*
(29)
* We assume kp-I << 1. t The precise equality is merely an algebraic aid; it is not CIUCM.
Thus, for k: = k: (i.e. for neglect of the dislocation bias effect) one finds fv = (Ef/2k$)[(4&
t 1)“2 - l] ,
(30)
if k$ >> kf, or kc >> /3&, whichever is the larger. The first of these inequalities is a condition on the solute-atom concentration, namely QCv >> (nk:/ 4nrv), the latter being of the order of 2 X 10e6 for kf = 5 X lOlo /cm 2. Hence the first inequality will be satisfied for all but the most dilute solutions. The second is of importance when /Iv exceeds unity; it implies that even though this be the case, the trap occupation probability is still small nevertheless. Consequently, when eq. (30) is valid, eq. (22) may be approximated by k&E
Gfv 3 k;, L lk2 2 V?
Kvv = -&D&c,B
(31)
.
If one now uses eq. (8) to determine ct and c, and considers the separate contributions to eq. (5) it is found that the dominant physical processes are vacancy capture and thermal release. Thus a balance is achieved by the same mechanisms that determine thermal equilibrium *, although the occupation probability is different under irradiation. The entire development rests upon intrinsic recombination being unimportant. Combining eqs. (30) and (31) and inserting in the inequality (9) we find this obtains if 40~ > Q, or xv > 1, where xv=-
40~ 17
DiG =ac,” .
(32)
Now in the materials of interest, the interstitial is much more mobile than the vacancy; thus [ 121 01= 30 Di/a’, (for the split-interstitial configuration). Also, for vacancy trapping in an fee. lattice [ 121, c,” = (l/ 12) exp(-eT/kZ), where E: is the binding energy of the trap-vacancy complex, k is Boltzmann’s constant and T the absolute temperature. Hence eq. (32) may also be expressed as 96n xv==RCv
5 exp(e?/kq ( a 1
.
(33)
* For thermal equilibrium one cannot neglect thermal emission processes from vacancy sinks other than traps, as we have done here. In fact eq. (90) of ref. [6] yields the carrect result, f:q = cg(ct + ct>, for K = 0, only if their zt (= DV@$
is retained.
358
A.D. Brailsford /Point defect trapping and void growth
The lower bound upon xv, i.e. xv > 1, is thereby equivalent to a lower bound upon e$. Binding energies less than this bound, which depends upon the solute concentration, have essentially no effect upon the swelling according to our model. Larger binding energies reduce the swelling according to eq. (17) with Xv evaluated from eqs. (16), (30) and (3 1): A,’ = +F(n)[(nxv
+ l)l’2 + l] .
(34)
This will be discussed further in section 4. The analysis of eq. (27) for interstitial trapping at solute atoms parallels the above completely. For, with PI = Kk:/Dici”ki’k~ , the re-expression
k:
Pl]tf1($1+ 1)
has identical form to eq. (28) when the term in (kf - k,?) is neglected. With the restriction on 0 given after eq. (30) this term is indeed negligible since (Zi - Zv) < 1. Thus the corresponding solution for fI is: fI -(k;/2k$[(vxI
+ 1)“2 - l] ,
1 - kT ln(D$k:/u>~k$)
An example of this isomorphism
.
(41)
is given in section 4.
3.2. Traps on coherent precipitates
Af2-(B-Cs)f-Ds=O, (36)
3
eB=en+em_em ” ” L
When the traps reside on a surface of large radius, s becomes small compared to unity for any non-zero w. Eq. (25) has the generic form:
of eq. (27):
+ 1 _K(kT-q)
_-k2 ki PI = 0
(35)
xv = XI, then the effects on the swelling are identical. We may express this isomorphism in terms of binding energies as follows. For split-interstitial capture [ 141, cp = (l/6) exp(-$/k7’), where e? is the binding energy of the trapped interstitial. Let D, = 0: exp(-er/ko and Di = Dp exp(-$‘/kT) where ~7 and E/” are the motion energies for thevacancy and interstitial. Then the binding energy of interstitial traps which yields the same effect as vacancy traps is:
(37)
where A, B, C and D are independent roots of eq. (42) for small s are: j-(l) = B/A ,
f12’ = -Ds/B .
(42) of s. The two
(43)
All the capital letter parameters are positive, and B/A is less than unity by inspection of eq. (25). Thus fl’) is the desired ro o t [6]: (44)
where (38)
Eq. (27), for interstitial trapping, has the same form as eq. (42), but now D is negative while B/A is greater than unity. Hence f12) is here the physically acceptable root [6] :
(39)
(45)
This limit also leads to: k?v =
k?fI ,
k&‘v Ik* 2 13
KIi ‘I $fIDik:cB ,
in analogy with eq. (3 1). Thus steady state occupation may be shown to result now from a balance between interstitial capture and release. Similarly, the effect on the swelling is here determined by the parameter hr: Xi’= $F(n)[(~xr
t 1)li2 t l] .
(40)
Thus there is a complete isomorphism between the effects of vacancy and interstitial trapping at solute. atoms in the sense that if the respective concentrations, trapping radii and binding energies are such that
We would emphasize two points. First, the isomorphism established for atomic traps is lost at this point. For strong binding the vacancy traps are essentially full, the interstitial traps essentially empty. Mathematically this arises because we have to switch from one branch of the double-valued function f to the other. Physically it arises because dislocations preferentially attract interstitials. Thus the vacancy traps accommodate as much of the excess vacancy flux as they can, while the interstitial trap is continuously emptied by the same excess flow. Second, as the
359
A. D. Brailsford /Point defect trapping and void growth
vacancy binding energy increases, eventually bothfv and fr become independent of temperature (or equivalently, of binding energy). In this limit fv = 1 so that, from eq. (22) k$i-kZ,,
(46)
4)
5-
0 0
while from eq. (26): k& ‘v k;(k;/kf) The condition fore: (k$/kf)
‘v kf . of self-consistency,
> [(l t #I2
X
?-
- 1]/2
(47) eq. (lo), is there-
(48)
(for vacancy trapping). This mainly dictates what the precipitate dispersion must be. Interstitial trapping leads to an essentially similar result with k$ replaced by kf, while the swelling reduction parameters, hv and hr:
indicate the equivalence of either trap type if now k+ = kf. This isomorphism is precisely the result found by Hayns [9]. To reach these results we have assumed fv = 1. Evidently, from eq. (44) this is valid for binding energies, Ef, such that:
(50) assuming cB = exp(-Ef/kT) for surface trapping. For ~~=l,k$~~~,(k~-k~)~3XlO~/cm~anda=3A the right hand side of eq. (50) is 1.2 eV. This crude estimate indicates that the isomorphism between interstitial and vacancy trapping at coherent precipitates, as embodied in the relation kf = kc, can be attained with reasonable values of the vacancy binding energy. 4. Implications for void growth In this section some numerical examples of the preceding discussion are given and the results compared with prior work of other authors. 4. I. Atomic traps In section 3 we established the isomorphism between interstitial and vacancy trapping at substi-
600°
0
l
c X
l-
I
I
,I
.2
I
I
I
I
.3 4 5 .6 A E (eV) Fig. 1. Void growth rates at constant irradiation dose as a function of effective binding energy for vacancy (a) and interstitial (X) trapping. Results from ref. [7]. 0
tutional solute atoms given by eq. (41). Examples of this are provided by the recent calculations of MY. They use a trapping model which, for atomic traps, is identical in all essential respects to that of ref. [6]. (This matter is discussed in detail in the Appendix.) Their fig. 8 gives the void growth rate (dr,/dt, in our notation) as a function of temperature, for various interstitial and vacancy trapping energies. From these data we have obtained the growth rate as a function of an effective binding energy, AL’, where: AE= e,B
(vacancy trapping) ,
AE = $ - E: t E? + kT In(D~/D~) trapping) * .
(51) (interstitial (52)
Results ** for two temperatures below that at which the growth rate is a maximum are given in fig. 1. (We recall that we have ignored thermal vacancy emission from other than vacancy traps.) The isomorphism is clearly evident in that the results for either type of * Eq. (41) reduces to (52) for kf = k$ if the factor of 2 in the argument of the logarithm is discarded. This factor was not incorporated by MY; it has been dropped for the purposes of comparison. ** The point defect parameters are as given by MY.
A. D. Brailsford /Point defect trapping and void growth
360
.- -----
\
0 .
\
0
lnletstlllol
\
x
\
.
600°C 650°C Traps x 600°C 650°C
VOCOflCY Traps
\
\ 0
\ \
.
\ \
0
\
\ \ \ \
x0
\
where r,(O) is the void radius, without traps, for the same irradiation dose as that at which Y, is measured. The trend indicated by our model (the broken line) is in good agreement with the numerical analysis of MY and deviations are in the sense expected, in that our assumption of no effect of traps below some critical binding energy obviously furnishes an upper bound. It should be emphasized that the rapid decrease of growth rate with binding energy depicted in fig. 2 arises because without traps, intrinsic recombination is dominant. If the sink strengths, kf and k:, are increased to the point where 71 << 1, then the sensitivity to binding energy is found to be greatly reduced. 4.2. Trapping at coherent precipitates
o-
' 01
I 02
I 03
1 I 04 05 06 AE (eV)
1 07
I, 08
Fig. 2. A comparison of the results of ref. [7] with the upper bound furnished by the present analysis (atomic traps).
trapping fall approximately on a continuous curve at each temperature. MY also noted the larger binding energies required for interstitial trapping to produce effects comparable to vacancy trapping and gave a qualitative argument based upon effective diffusivities. Their numerical analysis led them to suggest an approximate equivalence between vacancy and interstitial binding energies embodied in E! = (e? - er + ef). Our analytical model, leading to the re-expression of their results in fig. 1, shows that the equivalence is more accurately defined by eqs. (51) and (52). In fig. 2 we have illustrated the qualitative effect on void growth as obtained from our analysis, for a temperature of 650°C and dose rate of low3 dpa/s, for solute trapping in Ni. These results were obtained with a recombination radius, ro, (where 4rrrfli = aa) taken equal to the trap radius * and an atomic fraction of traps of 1Oe3. The sink strengths used in evaluating 17 were kt u kz = 5 X 10” /cm2, as used by MY. When eq. (17) is valid we can equally well take (53) * Doubling the trap radius decreases AE by approximately 0.05 eV at this temperature.
In section 3.2, it was shown that hv tends to a value independent of the binding energy. We shall explore the swelling reduction in this limit only. For a fixed volume fraction, 6, of second phase material, k$ = 3$/r;. Thus, combining eqs. (I 7) and (49): (54) where kf is to be regarded as a constant. The quantity 3@/kFr$ must also satisfy eq. (48). Accordingly, if intrinsic recombination is small without traps [i.e. 17<< 1, F(Q) = l] the latter condition is impotent and eq. (54) describes a relation between the swelling reduction and trap size. Numerical results for this domain are provided by the work of BH; those appropriate to 45O”C, now in austenitic steel, are illustrated in fig. 3, for @= 0.8%. To check their form against eq. (54) we fitted to their data point at rv = 20 A. The continuous curve in fig. 3 satisfactorily describes the trend they observed. The relation (44) breaks down, of course, when rv is of atomic dimensions. The analysis of section 4.1 then becomes appropriate. It should also be mentioned that for Cv = lOI cmw3 and rv = 100 A, the swelling calculated by BH is relatively insensitive to intrinsic recombination and vacancy binding energy at the same temperature. This is in accord with the basis we have given for the present discussion to be valid. Fig. 3 shows that above a certain size, coherent precipitate trapping has little effect on the swelling.
A. D. Brailsford
/Point
defect
361
trapping and void growth
Appendix 16 .
t
i
For the convenience of the reader, we document the relation between the notation used recently by Mansur and Yoo and that introduced earlier by Brailsford and Bullough. To be specific, consider vacancy trapping at solute atoms. The analogous form to eq. (22) used by MY is, with their CL =.ti&
ktv = 4nr;C,( 1 -_&,) , Kvv = i2 4
Ctfv exp(-EFlk7’)
(A-1) ,
/
oo,’
1
10
20
30
40
50
f0
70
60
90
I00
rV(A)
Fig. 3. A comparison of the results obtained in ref. [ 81, for swelling at fixed dose and fiied volume fraction of precipitates, with the prediction of the present analysis.
This feature also shows up in the case where intrinsic recombination without traps is dominant (n >> 1). This critical radius, TV, may then be inferred from the inequality (48) to be & - (61$/@7~‘*)“*. (For the parameter choices @= 0.8%, kf = 5 X 1O1’ /cm2 and 17= 1O*, TV is of the order of 30 A.) For CI< YV< TV we find S/S(O) - (rv/rv)3 in this limit.
5. Summary The Brailsford and Bullough theory of point defect trapping has been used to study the effect of trap size upon void growth. A clear physical distinction has been drawn between the processes leading to steady state occupancy when the traps are atomically dispersed as opposed to when they are uniformly distributed over coherent precipitate-matrix interfaces. Different isomorphisms between interstitial and vacancy trapping have been established in the two cases. Examples of each type have been identified in the numerical analyses of other workers. Substantial agreement with this prior work has been established.
Acknowledgements The author has benefited greatly from a free exchange of information with Drs. R. Bullough, M.R. Hayns, L.K. Mansur and M.H. Yoo.
where the trapping radii, rv and r;, for vacancy capture and on-site interstitial recombination are allowed to differ. From comparison with thermal equilibrium MY show that 4nr;b* = 12fi. The correspondence is now readily established if fv is (as we have shown) taken to be small, E,” equated to $, C, to Cv, rt to fr; and rt to rv. References [l] S.D. Harkness and Che-Yu Li, Proc. 1971 Int. Conf. on Radiation-Induced Voids in Metals, eds. J.W. Corbett and L.C. Ianniello (USAEC Conf-710601, 1972) p. 798. [ 21 R. Bullough and R.C. Perrin, Voids Formed by Irradiation of Reactor Materials, eds. SF. Pugh, M.H. Loretto and D.I.R. Norris (Brit. Nucl. Energy Sot., 1971) p.79. [ 31W. Schilling and K. Schroeder, Consultant Symposium on the Physics of Irradiation-Produced Voids (ed. R.S. Nelson, AERE, Harwell Rep. R7934) p. 212. [4] J.S. Koehler, J. Appl. Phys. 46 (1975) 2423. [5] P.R. Okamoto, N.Q. Lam and H. Wiedersich, Proc. Workshop on Correlation of Neutron and Charged Particle Damage (ed. J.O. Stiegler, Oak Ridge Report Conf760673,1976) p. 111. [6] A.D. Brailsford and R. BuIlough, J. Nucl. Mat. 69 & 70 (1978) 434. [7] L.K. Mansur and M.H. Yoo, Oak Ridge National Laboratory Report TM-6134 (1976); submitted to J. Nucl. Mat. [B] R. Bullough and M.R. Hayns, to be published. [9] M.R. Hayns, private communication. [lo] A.D. Brailsford and R. Bullough, J. Nucl. Mat. 44 (1972) 121. [ 1 l] A.C. Damask and G.J. Dienes, Point Defects in Metals (Gordon & Breach, New York, 1963) p. 83. [12] Seeref. [ll] p. 124. [13] Seeref. [ll] p. 92. [14] See ref. [ll] p. 135.