On the experimental determination of the migrating defect fraction under cascade damage conditions

ELSEVIER

Journal of Nuclear Materials 210 (1994) 244-253

On the experimental determination of the migrating defect fraction under cascade damage conditions H. Trinkaus a, V. Naundorf

b, B.N. Singh ‘, C.H. Woo d

a Institut fiir Festk~rpe~orschl&n~, Forschun~s~entrl4m JiZich and Association mA-Euratom, Postfach 1913, D-52425 Jiilich, Germany ’ Hahn-Meitner-Institut, Glienicker Strasse 100, D-14109 Berlin, Germany ’ Materials Department, R&i National Laboratory, DK-4000 Roskilde, Denmark d Whiteshell Laboratories, AECL Research Pinawa, Manitoba, Canada ROE IL0

Received 29 November 1993; accepted 7 February 1994

Abstract Information on the fraction of defects surviving intracascade recombination, escaping the cascade volume and migrating until their annihilation (“migrating defect fraction”, MDF) can be obtained from the analysis of radiation enhanced diffusion (RED) or radiation induced segregation (RIS) and maximum swelling rates. RED and RIS yield the ratio of the MDF over the effective sink strength whereas maximum swelling rates give lower bound estimates of the MDF. The basic assumptions made in the previous analysis of RED (RIS) and swelling are critically examined in the light of the present understanding of defect production in displacement cascades. MDF values deduced previously from RED are found to be clearly below the lower bound estimates obtained from maximum swelling rates. The discrepancy becomes even larger if the conventional monodefect dislocation bias is used in the analysis of swelling. Possible reasons for this discrepancy are discussed: (1) differences in the contribution of mobile defect clusters produced in cascades to RED and swelling, and (2) an underestimation of both the sink strength evolving during cascade damage conditions and the driving force for the swelling. We argue that the conventional method to deduce the sink strength from sink densities observed in TEM and the application of the conventional monodefect dislocation bias indeed yield only lower bound estimates for the sink strength and for the swelling rate, respectively. If the MDF were established by some other method RED (or RIS) could be used to measure the sink strength.

1. Introduction Irradiation of solid materials results in changes of their macroscopic properties which are generally detrimental. This macroscopic “radiation damage” depends on a multitude of parameters including those which characterize the initial material state such as chemical composition and structure as well as those which characterize external influences such as temperature, stress and irradiation conditions. Even the irradiation conditions alone are characterized by a number of parameters such as the displacement rate, the displacement dose and the details of the recoil energy spectrum. It is virtually impossible to cover such a large parameter space by quantitative experiments. Thus, designing ma~22-3115/94/$07.~

terials depends strongly on interpolating and extrapolating experimental data. For this, modelling is required in particular for conditions which are not yet fully accessible by experiments such as fusion reactor conditions. For modelling the large scale and long term defect accumulation under irradiation, however, more specific quantities than displacement rate and dose are required to characterize the “primary displacement damage” occurring under cascade damage conditions. A key quantity which has received increasing interest during recent years is the fraction of defects which survive intracascade recombination, escape their native cascade, migrate through the crystal lattice and are thus available for intercascade reactions [1,2].

Q 1994 Elsevier Science B.V. All rights reserved

SSDI 0022-3115(94)00103-U

H. Trinkaus et al. /Journal

of Nuclear Materials 210 (1994) 244-253

There are several experimental methods to study this “migrating defect fraction” (MDF). A well established method to study defect production under cascade damage conditions is to measure the residual electrical resistivity and its recovery after low temperature irradiation. (For a review, see Ref. [2]). Methods to study defect production at elevated temperatures are radiation enhanced self- and impurity-diffusion (RED) [3,4] (for a review, see Ref. [5]) and radiation induced segregation (RIS) [6,7]. Swelling data provide lower bound estimates for the fraction of vacancies surviving re~mbination at high temperatures and can be used to test the data deduced from other methods (for a review of swelling in pure metals, see Refs. [1,8]). In the following, interest will be focused on the comparison of data obtained from RED and swelling data. The crucial assumptions used in the analysis of RIS are essentially the same as those used in the analysis of RED. Recent progress in the understanding of cascade damage has led to an intense discussion among the authors of the present paper concerning the basic assumptions made in the previous analysis of RED, RIS and swelling. The present paper represents the state of the current consensus.

2. Present understanding

of defect production

in dis-

placement cascades

The spatial structure of the irradiation induced primary displacement damage depends crucially upon the recoil energy transmitted from a projectile particle to a primary knockon atom (PKA) in the matrix lattice. If the recoil energy is less than about ten times the displacement threshold, mainly single vacancies and self-interstitial atoms (SIAs) are produced. If, on the other hand, the recoil energy is much larger than ten times the displacement threshold (say > 1 keV) defects are produced in the form of multidisplacement cascades. The latter case is of particular interest in the present context. According to molecular dynamics (MD) studies [911] the defect production process by recoils of high energy may be divided into the “collisional” phase and the “thermal spike” phase. In the collisional phase, the energy of the PKA is distributed within a fraction of 1 ps over secondary, tertiary and higher generation recoils in the lattice. In the spike phase, the energy is randomized and thus converted into heat. From the shock-front around the resulting hot liquidlike region, SIAs are ejected into the surrounding lattice, either as single SIAs or as small SIA loops. Upon cooling down, the hot core of the cascade recrystallizes. The defect structure remaining after a few ps is characterized by a vacancy-rich core surrounded by a shell of SIAs. A

245

substantial fraction of these SIAs are found to be clustered, mostly in form of small dislocation loops. The latter findings confirm the conclusion drawn from previous diffuse X-ray scattering measurements on metals irradiated at low temperatures with fast neutrons [12]. During the collisional and the thermal spike phases intense atomic mixing takes place. The phase immediately folIowing the cooling phase is characterized by thermally activated intracascade reactions such as recombination and vacancy clustering. Transmission electron microscopy (TEM) has shown that the vacancy-rich cascade core tends to collapse to form a dislocation loop or a stacking fault tetrahedron [13]. In the temperature range of void swelling (between one third and one half of the melting temperature, T,) such vacancy clusters will, however, dissociate into single vacancies. The defects surviving intracascade recombination are available for reactions with defects originating from other cascades (intercascade reactions) and thus control large scale and long term defect accumulation under continuous irradiation. There is wide agreement now to use the number of “displacements” calculated according to the NorgettRobinson-Torrens (NRT) approximation [14] as a reference value for defect production calculations and measurements. The fraction of defects surviving intracascade recombination during and after the cascade cooling phase relative to the calculated NRT number (“‘defect production efficiency”) and the fraction available for intercascade reactions are of particular interest in the present context. Electrical resistivity measurements after low temperature heavy ion irradiation [2] and MD studies [9--111 indicate that the number of defects surviving the cascade quench phase in Cu and Ni is between 20 and 30% of the NRT value. Damage rate and recovery experiments suggest that at around 0.4T,, about half of these defects survive subsequent intracascade recombination [2]. In contrast, the fraction of “freely migrating defects” (“migrating defect fraction” MDF) deduced from tracer and impurity diffusion during heavy ion irradiation at T> 0.4T, [3-51 (and similarly from RIS [5-71) has been reported to be almost one order of magnitude smaller whereas lower limits of this fraction as estimated from maximum swelling rates yield values which are smaller by only about a factor of 2 [l]. Possible reasons for this (apparent) discrepancy will be discussed below.

3. Defect reaction kinetics Conventional modelling of radiation effects in terms of a chemical rate theory approach [15] is based on the assumption that the large scale/long term defect accumulation is due to diffusional reactions of single vacan-

H. Trinkaus et al. /Journal

246

of Nuclear Materials 210 (1994) 244-253

ties and SIAs produced randomly in space while reactions resulting from cluster motion are negligible. Vacancies and SIAs are considered to be annihilated either by mutual recombination or by absorption at stable sinks. The sink strength reached in steady-state is generally considered to depend on temperature but frequently assumed to be independent of the displacement rate. This picture is obviously too simple to describe defect kinetics under cascade damage conditions where defect clusters of various types and with different properties are continuously generated. Since these clusters act as both sources and sinks for point defects, they must be considered explicitly in any theory of radiation damage involving cascades. Treating the local intracascade reactions separately, the large scale intercascade reactions in the medium may be described by a generalized chemical rate theory approach, taking the production as well as possible sink and source reactions of clusters into account (see Fig. 1). Crucial aspects concerning the role of defect clusters in defect kinetics are their stability and mobility. SIA clusters, for instance, are thermally relatively stable in the temperature range of void swelling while vacancy clusters decay quickly under such conditions. This asymmetry in the final form in which SIAs and vacancies survive intracascade reactions, which has been recently introduced as a “production bias” [16], produces a potential source for void swelling, provided

Displacement

that a significant fraction of the stable SIA clusters produced in cascades are annihilated preferentially at extended sinks such as dislocations and grain boundaries [17,18]. Small clusters consisting of two or three vacancies [19] or SIAs [20] are known to be mobile with mobilities similar or even higher than that of the corresponding monodefects. Large clusters tend to form dislocation loops the mobility of which have been ignored in modelling until recently. Such defect clusters may, however, migrate by the slow climb or the fast glide mechanism. Recent MD studies have shown that for some sizes small loops can assume the highly glissile unfaulted configuration [lo]. Such loops are expected to perform an extremely fast thermally activated onedimensional glide motion. The activation energy for this process is estimated by elasticity theory to be below 0.1 eV in pure metals for loops containing less than 100 SIAs [17]. Recovery experiments suggest that SIA clusters of considerable size can move already in annealing stage II [21]. The glide of small vacancy loops, on the other hand, seems to be impeded by the poorly relaxed vacancy platelet located in the loop plane. Such vacancy loops as well as sessile SIA loops may, however, migrate by climb which is most likely controlled by pipe diffusion along the dislocation core characterized by an activation energy comparable with the single vacancy migration energy (“conservative climb” or “self-climb” [22]).

Cascade

Cascade Quench; Annihilation & Mixing

! Quenched Cascade

lntracascada Recombination

& Clustering

ImmobileClusters

c MobileClusters

Mobile Point Defects

I

(b)

(a)

Fig. 1. Flow diagram

t

Long Living Sinks

Stable ImmobileClusters

illustrating

possible

intracascade

(a) and intercascade

reactions

(b) between

point defects

and defect clusters.

H. Trinkauset al. /Journal of Nuclear Materials 210 (1994) 244-253 Conceivable intercascade defect reactions are indicated in Fig. lb. Cluster migration will lead to a number of reactions in addition to the ones considered in conventional defect kinetics. Thus, for instance, the coalescence of clusters of unlike defects will result in a recombination-like defect annihilation, while coalescence of clusters of like defects will result in cluster coarsening. A clear distinction between the two typical annihilation mechanisms, i.e. by mutual recombination and by absorption at fixed sinks (for instance on the basis of differences in their damage rate dependence), is no longer possible. Note that defect annihilation cannot be dominated by recombination in the conventional sense [4] when the production rates of single vacancies and SIAs are substantially different. An important aspect for the data analysis is the (quasi-Isteady-state. It is reached when the production of defects is balanced by their annihilation. Due to the reactions discussed above the defect concentrations and the associated sink strengths that evolve under cascade damage conditions are temperature as well as displacement rate dependent.

4. Radiation-enhanced

diffusion

One of the prominent experimental methods to study the defect production under irradiation is the measurement of radiation enhanced selfttracerjand impurity diffusion (RED). Atomic mixing occurring in the collisional and thermal spike phases of displacement cascades makes an important contribution only at low temperatures (below stage Vl, defects escaping the cascades control atomic rearrangements at medium temperatures, whereas defects in thermal equilibrium (mainly monovacancies) govern diffusion at high temperatures (above OX,). Thus, the diffusion coefficient, D, consists of irradiation-induced and thermal contributions, Dir1 and D,,, respectively. The irradiation-induced contribution, Dir,, may be further divided into contributions from cascade mixing, Dmix, and from long-range defect diffusion, Drad, because of the distinctly different time scales associated with these processes. Hence, one may write [3,5] D = D,,, + D,,, = Dmix+ Drad + Dth.

(1)

The radiation-enhanced diffusion coefficient, Drad, which is of interest in the present context, consists of contributions from all mobile defects and defect clusters produced in cascades. Thus, for instance, the radiation enhanced tracer diffusion coefficient of pure metals may be written in the form

where

n = 1, 2, . .

is the number

of vacancies

or SIAs

247

per contributing defect cluster, fvn,in are correlation factors related to a single defect in a cluster, Dvn,in are the diffusion coefficients of the clusters and AC,, in their atomic concentrations in excess of the corrksponding thermal equilibrium concentrations. Correlation factors have been calculated for monodefects [3,5]. For clusters, they are not known quantitatively but it may be safely assumed that their values decrease with increasing cluster size in particular for one-dimensionally migrating glissile loops. Accordingly, the expression at the right-hand side of Eq. (21 will be dominated by the monodefects and possibly small defect clusters. In (quasi-Isteady-state, the defect concentrations Ac,,,,~, are determined by the balance between the respective production rates and annihilation rates. To keep the discussion transparent we consider the partial of NRT-displacements annihilated as fractions n:+ mobile monodefects or in the form of mobile clusters, vn and in, at less mobile sinks with sizes beyond the size range of primary cascade defects, i.e. the defect fractions remaining after intra- as well as intercascade recombination. Note that qznin includes clusters resulting from reactions of primary clusters but not exceeding the size range of such clusters, and 7: includes vacancies which are temporarily stored in thermally unstable clusters. The total fractions of vacancies and SIAs annihilated at extended sinks are equal and are given by 77* = C?7Zfl = C77$ n n

(3)

n* represents a lower bound for the defect fraction n escaping intracascade recombination (MDF) and approaches the latter quantity in the “sink controlled” regime of high temperatures and low displacement rates. With decreasing temperature and/or increasing displacement rate, n* decreases depending upon the sink strength because of the increasing importance of recombination-like reactions between primary cascade defects. In an extended rate theory approach, the concentration AC,, ,n of mobile defect clusters v,Ji, is determined by the partial defect production rates qc+K (where K is the NRT-displacement rate) their removal rate (assumed here, for simplicity, to be cluster migration controlled [23]) and the corresponding sink strength k:,,i,, i.e. Acv,,i,

=q~~,inK/(nD,,,in’,2,,in).

Using Eqs. (3) and (41, Eq. (2) may be written D,‘,d = n*K( (fv,/kZ,)v:n

+ (ji,/k?n),T”},

(4) as (5)

where ( >II:,,. indicates averaging weighted by the corresponding partial fractions qzn,in. It is useful to define a “single Frenkel pair equivalent”, qc,i and a “single

248 vacancy equivalent”, (5) according to

H. Trinkaus et al. /Journal

qz, to the right-hand

of Nuclear Materials 210 (1994) 244-253

side of Eq.

The “fraction of freely migrating defects” (MDF) discussed in previous papers [3-51 corresponds to a “single Frenkel pair equivalent” of the contributions of all mobile defects to RED. According to Eqs. (5) and (6), DLd is, for a given K, a measure of the product of the defect fraction annihilated at extended sinks and the reciprocal sink strength weighted by the corresponding correlation factors. In the sink controlled regime, the T- and K-dependence of the sink of D,‘,d reflects the T- and K-dependence strength. At lower T and/or higher K, the T- and K-dependence of Dzd is also affected by n* via the influence of recombination-like reactions. The key quantities in deducing n* from measured DLd data are the sink strengths for the main defects contributing to D&,, i.e., for single vacancies and SIAs (or the number of jumps performed by a defect between its generation and annihilation which is inversely proportional to the sink strength). In previous work [3-51, an attempt was made to determine independently both the “fraction of freely migrating defects” and the sink strength in self-ion irradiated nickel from RED data measured as a function of K and T. The analysis was performed for a temperature range (850-950 K) where Dmix-=z D such that an accurate knowledge of Dmixis not needed to determine Drad. The deduction was based on a number of commonly accepted assumptions the most important of which are the following: (1) The only defects contributing to RED are single vacancies and SIAs produced in equal amounts. (2) These defects annihilate either by mutual recombination or by getting absorbed by “permanent” sinks the strength of which is allowed to be temperature-dependent but assumed to be independent of the displacement rate. The idea of the analysis was the following. Assuming a rate-independent sink strength, the deviation of the rate dependence of Drad from linearity was considered to be due to recombination. The recombination radius is relatively well known. A fitting procedure would then be able to provide both the MDF (from the recombination regime) and the sink strength (from the sink regime). This procedure yielded a strongly temperature dependent sink strength and a rather low, temperature independent MDF of about 1.5% with an uncertainty within the framework of the analysis of +4.5% and - 1.1% for self-ion irradiated Ni. The relatively good agreement between the sink strengths

obtained from this procedure and the sink strengths deduced from TEM observations, was taken as a confirmation of the underlying assumptions. On the basis of the sink strengths deduced, it has been concluded that at high displacement rates defect annihilation would be recombination-controlled at low as well as at high temperatures [4]. Furthermore, it has been suggested [4] that a substantial fraction of the low value of 1.5% for the MDF could be due to isolated Frenkel pairs produced by the low energy tail of the recoil spectrum. Both the existence of a recombination controlled regime and a predominant production of defects in the form of isolated Frenkel pairs would indeed limit a possible difference in the production of vacancies and SIAs (“production bias”) to small values [41. Similarly low values of MDF have been deduced from RIS measurements, for instance in Ni-Si alloys [6,7]. In this case, “relative efficiencies” have been derived by comparing the effects of irradiation by different ions [7] (light and heavy ions producing Frenkel pairs predominantly in isolated form or in displacements cascades, respectively). An interesting result was that, on one hand, the observed temperature and displacement rate dependencies of RIS in virgin samples irradiated with different types of ions were consistent with the commonly accepted assumptions concerning recombination controlled defect annihilation [7], and that, on the other hand, Kr pre-irradiation of Ni-Si alloys reduced the segregation rate during subsequent He irradiation to about half the value observed without Kr pre-irradiation [6]. On the basis of this doping effect, which most likely originates in the formation of additional defect sinks by the heavy ion pre-irradiation, it must be concluded that defect annihilation at sinks is important for RIS and its consideration is crucial in deducing MDF from RIS, contrary to the assumptions made in Ref. [7].

5. Swelling The biased annihilation of SIA type defects at dislocations and other extended sinks such as grain boundaries and surfaces results in the accumulation of vacancies, void formation and swelling. Since the swelling rate corresponds to the difference in the fluxes of SIA type and vacancy type defects to extended sinks different from cavities an analysis of this quantity can only provide lower bound estimates for the fraction of defects surviving intracascade recombination [l]. Under appropriate conditions, swelling may, however, reflect a substantial fraction of the surviving defect fraction. In the present context, interest should indeed be focused on high swelling rates, i.e. swelling per NRT dpa.

H. Trinkaus et al. /Journal

of Nuclear Materials 210 (1994) 244-253

Using the same definitions as introduced in the preceding section we may write the bulk swelling rate as 3 = ~*K{(k,2,,&,)~:.

- (ki2n,,/k?J~:,},

(7)

where k$,C and k&C are the sink strengths of cavities for the annihilation of vacancy and SIA type defects, respectively, and ( >,,c,., indicates again averaging weighted by the respective partial fractions Y&+. The structure of Eq. (7) is similar to that of Eq. (5) but there are two important differences: (1) RED is determined by the sum of vacancy and SIA contributions whereas the swelling rate is determined by the difference in the vacancy and SIA contributions, and (2) the averages in Eq. (5) contain correlation factors that decrease with increasing cluster size whereas in the averages of Eq. (7) these factors are replaced by the sink strengths of cavities for the annihilation of defect clusters that do not significantly depend upon cluster size. According to Eq. (71, swelling may be affected by both an asymmetry in the production (“production or source bias”) as well as in the annihilation (“annihilation or sink bias”) of vacancy and SIA clusters. In fact, Eq. (7) represents a generalization of previously reported formula for the swelling rate, covering the effects of both the conventional single defect dislocation bias and the production and cluster annihilation bias. In the former case, Eq. (7) may be approximated by 4,, = q*&k:k:/k2,

(8) where p is the monodefect dislocation bias parameter, kz and ki are the sink strengths of cavities and dislocations for the annihilation of single vacancies, respectively, and k* = kz + k$ At temperatures where the lifetime of the primary vacancy clusters is controlled by their dissociation (but vacancy emission from voids and dislocations is neglibiblel the contribution of the production and cluster annihilation bias in Eq. (7) may be approximated by [ 161 s,, = $Yk,2/k4,

(9)

where r& is the total fraction of SIAs produced in the form of clusters and annihilated at sinks different from cavities. The theoretical maximum of the total bias corresponds to a case in which all SIA type defects would be annihilated at sinks different from cavities, i.e., k3,, .-x kfn but k:,,C not -=xk:,. As discussed in the preceding section, a “single defect sink bias equivalent” to the total bias described by Eq. (7) may be introduced formally (not written down explicitly here) which can be substantially larger than the real monodefect dislocation bias. We emphasize here that defect fluxes to extended sinks such as grain and subgrain boundaries, must be treated separately, in particular when long

249

ranging SIA transport by loop glide plays a crucial role [17,18]. Swelling rates observed in the pure fee metals Al, Cu, Ni irradiated with fast fission and fusion neutrons are particularly high at low doses (< 0.1 dpa) where values around l%/dpa have been found ranging from 0.5 to 2%/dpa [1,8]. For oxygen-bearing Cu irradiated with fast fission neutrons [24] and for Ni irradiated with fusion neutrons even higher values up to about S%/dpa have been observed [25]. Accordingly, in these cases we would have to consider surviving defect fractions above 2% or even S%/dpa, even though less isolated Frenkel pairs are produced by neutrons than by ions because of the harder neutron recoil spectrum. Maximum swelling rates would yield particularly high values for MDF if the conventional monodefect dislocation bias were used. According to Eq. (81, swelling is maximum when the cavity sink strength is equal to the dislocation sink strength meaning that v* 2 4S,,/Kp. For commonly assumed values of p below 50% such a procedure would result in values of q* of up to 40% and even more (for fusion neutron irradiated Ni, for instance). The enhanced swelling observed adjacent to grain boundaries 126-291 and in subgrains bounded by dislocation walls [27] in fast fission neutron irradiated metals is of special interest in the present context. The width of these regions amounts to several km and is substantially larger than the diffusional range of vacancies or SIAs. The most detailed analysis has been performed for neutron irradiated Al [27,28]. In this case, the maximum swelling in zones adjacent to grain boundaries and in subgrain cells was found to be about 4 and 7%/dpa, respectively. In neutron irradiated Cu, the maximum swelling close to grain boundaries, has been recently measured to be 2.7%/dpa [29]. TEM observations of voids in neutron irradiated Ni [26] which have not been analysed fully quantitatively indicate similar values as for Cu. Recently, the enhanced swelling adjacent to grain boundaries has been analysed in terms of SIA transport via one-dimensional glide of loops to these planar sinks [18]. Strong evidence for this interpretation is provided by the close correlation between the width of the swelling zone and loop glide range deduced from the void microstructure observed in TEM. It has been shown that at the location of maximum swelling about half of the glissile loops are able to escape to the boundary. Thus, a maximum swelling of 4 and 2.7%/dpa would correspond to 8 and 5.4% defects produced in the form of glissile loops for Al and Cu, respectively. The value of 8% for Al is in good agreement with the value of 7% maximum swelling/dpa in subgrain cells of Al where almost all of the glissile loops are able to escape to the surrounding dislocation walls.

250

H. Trinkaus et al. /Journal

of Nuclear Materials 210 (19941244-253

In view of the fact that the accuracy of swelling data is not much better than within a factor of 2, the lower bound estimate for the surviving defect fraction following from the observed maximum swelling data can only be assigned to be somewhere between 5 and 10% for Al and between 3.5 and 7% for Cu (and Ni). It is worth noticing that these lower bound estimates for the MDF represent substantial fractions of the value (- 12%) deduced from damage rate and recovery experiments

La.

6. Discussion According to the review presented above, we are confronted with two different estimates for the fraction of “surviving” or “migrating” defects (MDF) under cascade damage conditions, one deduced from radiation enhanced diffusion at elevated temperatures, n = (1.5 + 4.5 - l.l)% for Ni (- Cu), the other obtained from maximum swelling rates, 7) 2 (3.5 to 7)% (or even much larger if the conventional single defect dislocation bias were used) for Cu (u Ni). A range reduced to 3.5 to 6% would be consistent with both types of estimates. A narrowing of the range of values for 77 by such a comparison is, however, quite unsatisfactory since both types of estimates obviously differ systematically from each other such that the quoted “most likely” values differ by at least a factor of 3 to 4. Principally, two possibilities are to be considered to resolve this discrepancy: (1) differences in the relation between single defect and defect cluster contributions to RED and swelling, and (2) systematic errors in the analysis of RED and swelling. To examine the first possibility we compare four extreme cases: (1) Both vacancies and SIAs are produced and annihilated as (mobile) isolated defects (as assumed in the original analysis of RED [3-5]), (2) all vacancies are produced and annihilated as mobile single defects while all SIAs are produced and annihilated in the form of clusters which do not contribute to diffusion (maximum “production bias” for vacancies), (3) the opposite to (2), vacancies and SIAs exchanged (this case could be important below stage V) and (4) both vacancies and SIAs are produced in the form of mobile loops such that they contribute much less to diffusion than they would do as isolated defects. The previous analysis [3-5] is based on the first case. In the second case, the fraction of “freely migrating” vacancies, nv* is larger than the corresponding “single Frenkel pair equivalent” 77: by a factor of 1.6 and thus would be 2.4% when as(fv1 +fil)/fvl = suming 7% 1-v= 1.5%. This increase is obviously not sufficient to obtain consistency with the observed maximum swelling rates. In the third case, the fraction of “freely migrating” SIAs would be T$ = 4.2% with 7:” = 1.5%.

In the fourth case, RED would have to be attributed completely to cluster motion. The assumption of the third and fourth case that vacancies are not only produced but also annihilated in the form of clusters even in the temperature range of swelling appears unrealistic at first but cannot be ruled out presently. This assumption implies that vacancy loops migrate, probably by conservative climb [22], and annihilate at sinks before they have dissolved thermally. The lifetime of a vacancy loop is expected to be controlled by the former mechanism at low temperatures and by the latter mechanism at high temperatures. The transition temperature depends sensitively upon the relation between the energies for loop migration and dissolution and may be below, within or even above the swelling regime. Transport of vacancies by loop climb would contribute only little to diffusion but would allow high swelling rates provided that a cluster dislocation bias analogous to the single defect dislocation bias exists. At the transition temperature, however, RED would increase substantially with increasing temperature. Since such an increase has not been observed even at the highest temperature (0.55T,) where the visible microstructure is observed to decrease considerably, vacancy transport by loop climb appears to be unlikely in the investigated temperature range. Thus, we must consider the possibility of systematic errors in the analysis of RED and swelling. We begin with discussing the assumption made in the analysis of RED that the nonlinear dependence of the RED coefficient upon the displacement rate is due to the recombination of single vacancies and SIAs. In fact, the data for Ni [3] may be as well described by the power law approximation Drad =A( T)Kh,

(10)

with A = 5.17 x lo-” m* s-’ (dpa/$‘, b = 0.795 for 850 K and A = 1.64 x lo-l7 m* s-’ (dpa/s)-‘, b = 0.704 for 950 K. The question is now whether the deviation of b from 1 is due to mutual recombination or could also be due to the rate dependence of the sink strength. The sink density visible in TEM will at least provide a lower bound estimate for the total sink strength and its rate dependence may be expected to give an indication of the rate dependence of the latter quantity. The rate dependence of the visible sink density evolving in Ni under cascade damage conditions has not yet been studied systematically but the observed trends [30] indicate that above 800 K this could account for the deviation of the rate-dependence of the RED coefficient from linearity. More detailed information about sink densities visible in TEM has been compiled for stainless steel [31,32] which in many respects behaves similar to Ni. In this case, the observed rate and tempera-

H. Trinkaus et al. /Journal

251

of Nuclear Materials 210 (1994) 244-253

ture dependent sink density, which is generally dominated by dislocations, may be described by a power law and reciprocal Arrhenius type expression of the form p = CXKO exp( E,/kT) with (Y= 10’1mP2(dpa/s)-P, p = 0.25, E, = 0.8 eV,

(11)

which covers the rate range lo-’ dpa/s [31]
= 4rr/lln

rr e3/*r2pl.

(12)

In Fig. 2, the conversion factor according to Eq. (12), is plotted versus dislocation density p, assuming different values for r. For dislocation densities around 1015 m-* evolving under self-ion irradiation the conversion factor is above 2 for reasonable trapping radii about 0.5 nm. Using Eq. (10) for Ni, Eq. (11) for stainless steel and Fig. 2 with r = 0.5 nm, we find for nzi values between 4 and 5% and for 77: values between 6 and 8%, which are close to the lower limit values deduced from maximum swelling rates. This k*/p,

Dislocation Density

(6’ )

Fig. 2. Factor for converting the dislocation density p to a sink strength, k2/p versus p calculated by employing a cylindrical cell approximation, for various values of the effective trapping radius of the dislocations.

agreement must, however, be considered with caution since the evolution of sinks in stainless steel is not necessarily representative for Ni. There are, however, additional reasons for an underestimation of the sink strength deduced from TEM. First, part of the dislocation structure may slip out the surface during thinning the sample for TEM observations. Secondly, a substructure invisible in TEM may significantly contribute to the total sink strength [3.5]. If, for instance, the lifetime of primary SIA clusters was controlled by the absorption of excess vacancies, their quasi-steady-state sink strength could be easily a multiple of the visible dislocation density as has been shown recently [18]. A similar conclusion is obtained with somewhat different assumptions [36]. This means that the correlation between Drad and the visible sink strength could be achieved via the invisible substructure. A cascade hitting an existing defect sink is expected to produce fewer defects than a cascade occurring in an undistorted part of the lattice [35]. Consequently, the defect production efficiency must be considered to decrease when the dislocation distance becomes smaller than the effective interaction range of a cascade. Cascade ranges of _ 10 nm commonly found in MD studies correspond to an effective dislocation density around lOI6 rne2. Such high values of the sink density would have to be considered only at the lowest temperature and highest displacement rate included in the previous analysis of RED [3-51. Larger cascade ranges corresponding to lower effective dislocation densities have, however, been suggested recently [37]. In contrast to the microstructure evolving during irradiation, the surface of the sample represent a really

252

H. Trinkaus et al. /Journal

of Nuclear Materials 210 (1994) 244-253

permanent, invariable and well defined sink. A study of RED under conditions where defect annihilation occurs predominantly at the surface is therefore highly recommendable. In fact, one measurement in the study of RED in Ni [3], the one performed at the highest temperature (950 K) and lowest displacement rate (1.2 X 10e4 dpa/s) represents a candidate for such a case. Using the RED coefficient measured under these conditions and assuming that the surface is the dominant sink, we estimate 77:” to be above 3 to 4% and q*” to be above 5 to 6.5%. Unfortunately, under such high temperature conditions, thermal diffusion becomes comparable with or even larger than RED. There might be some additional contribution to defect annihilation from sessile SIA loops the lifetime of which is controlled by the absorption of thermal vacancies under these conditions. When thermal diffusion is just comparable with RED, the sink strength of such clusters can be shown to be comparable with that of the surface. But even if the sink strength of such clusters is neglected the estimate for the MDF obtained from RED at the highest temperature is significantly above the value of 1.5% deduced from the whole set of data. Next we consider possible errors in the analysis of maximum swelling data. Concerning the accuracy of the quantitative TEM analysis, we note that the voids analysed in the bulk [2,8,24,25] and in regions adjacent to grain boundaries [26-291 are mostly in a size range (above a few nm) suited for quantitative microscopy. Even though this TEM analysis is not very accurate (probably not better than within a factor of 2) no reason for a systematic deviation to lower or higher values can be seen. Concerning the deduction of the MDF from maximum swelling data, on the other hand, it appears clear now that an application of the conventional monodefect dislocation bias results in a substantial overestimation of that quantity. In this context it is worth noticing that an inclusion of the sink strength of invisible cascade induced dislocation loops would in many cases be associated with a dominance of dislocation type defects and thus result in even higher estimates for the MDF. However, even if we do not refer to the conventional approach maximum swelling rates observed in the bulk and close to grain boundaries force us to consider MDF values above 5%. Summarizing this discussion section we may say that the analysis of RED/RIS on the one hand and of maximum swelling data on the other hand on the basis of the conventional single defect reaction kinetics result in substantially different MDF values. This diswhich becomes particularly large if the crepancy, swelling data are analysed in terms of the conventional monodefect dislocation bias, may be removed by properly considering the reaction kinetics of clusters of both types of defects produced in cascades.

7. Conclusions In the present paper, possibilities to estimate the fraction of migrating defects produced under cascade damage conditions from radiation enhanced diffusion (RED) and maximum swelling rates have been examined. The main conclusions may be summarized as follows: (1) The continuous production of clusters of both vacancies and SIAs in displacement cascades and their reactions would affect the value of MDF contributing to RED, RIS and swelling. The interpretation of these phenomena must therefore include explicitly intracascade as well as intercascade clustering, cluster dissociation and cluster annihilation, in addition to the production and annihilation of monodefects. (2) For a given nominal (NRT) displacement rate, the RED (RIS) coefficient provides a measure of the ratio of the fraction of defects annihilated in the form of single defects or small clusters at slowly varying sinks to an appropriately weighted sink strength. A knowledge of the latter quantity is crucial in estimating the MDF. The presently available RED/RIS data do not allow an unambiguous determination of both the sink strength and the MDF from RED (RIS) data alone. (3) Maximum swelling rates yield lower bound estimates of the MDF. (4) Values of MDF deduced from RED (RIS) by using the conventional conversion of sink densities observed in TEM to sink strengths are substantially lower than the lower bound estimates obtained from maximum swelling rates, in particular if the swelling data are analysed in terms of the conventional monodefect dislocation bias. (5) Most probably this apparent discrepancy arises from an underestimation of the sink strength evolving under cascade damage conditions and an underestimation of the driving force for swelling in the conventional approach. To obtain agreement between the MDF values estimated from RED and maximum swelling rates both the sink strength and the driving force for swelling must be considered to be substantially higher than commonly assumed. (6) The most important and probable reason for the previous underestimation of the sink strength and the driving force for swelling is the omission of SIA clustering in cascades. (7) A substantial increase in the sink strength above the values assumed in the previous analysis due to cluster production in cascades would shift the recombination controlled regime (if it existed) to higher displacement rates (and lower temperatures) such that at elevated temperatures defect annihilation would be sink controlled even at the highest displacement rates

H. Trinkaus et al. /Journal

of Nuclear Materials 210 (1994) 244-253

considered. (This would be true inspite of the increased point defect production efficiency since the parameter controlling the transition between recombination and sink dominance goes as displacement rate over the square of the sink strength.) (8) A value of the MDF increased to or above 5% would require a substantial contribution from the clusters produced in cascades in addition to the contribution from Frenkel pairs produced in isolated form. With increased values for both the sink strength and the MDF, previously suggested constraints on the magnitude of the difference in the production of vacancies and SIAs (“production bias”) would be removed. (9) A study of RED under conditions where the surface is the dominant sink (RED close to the surface at high temperatures and low displacements rates) appears to be promising for the study of the MDF. (10) If MDF could be measured by some other method, RED (RIS) could be used to estimate the sink strength.

Acknowledgements Stimulating discussions with Profs. I& Averback and H. Wollenberger are gratefully acknowledged.

References [II S.J. Zinkle and B.N. Singh, J. Nucl. Mater. 199 (1993) 173. 121W. Schilling and H. Ullmaier, Mater. Sci. Tech., eds. R.W. Cahn and P. Haasen, vol. 10 (VCH Verlag, Weinheim/New York, 1993). 131 A. Miiller, V. Naundorf and M.-P. Mach& J. Appl. Phys. 64 (1988) 3445. 141 V. Naundorf, J. Nucl. Mater. 182 (1991) 254. V. Naundorf, M.-P. Macht and H. Wollenberger, J. Nucl. Mater. 186 (1992) 227. [51 V. Naundorf, Int. J. Mod. Phys. B 6 (1992) 2925. [61 R.S. Averback, L.E. Rehn, W. Wagner and P. Ehrhart, J. Nucl. Mater. 118 (1983) 83. 171 L.E. Rehn, P.R. Okamoto and R.S. Averback, Phys. Rev. B 30 (1987) 985. [81 B.N. Singh and S.J. Zinkle, J. Nucl. Mater. 206 (1993) 212. [91 T. Diaz de la Rubia and M.W. Guinan, J. Nucl. Mater. 174 (1990) 151; Phys. Rev. Lett. 66 (1991) 2766. [lOI A.J.E. Foreman, CA. English and W.J. Phythian, Philos. Mag. A66 (1992) 655. 1111 A.J.E. Foreman, W.J. Phythian and C.A. English, Philos. Mag. A66 (1992) 671. 1121 B. von Guerard, D. Grasse and J. Peisl, Phys. Rev. Lett. 44 (1980) 262; R. Rauch, J. Peisl, A. Schmalzbauer and G. Wallner, J. Nucl. Mater. 168 (1989) 101.

253

[13] C.A. English and M.J. Jenkins, Mater. Sci. Forum 15-18 (1987) 1003. M. Kiritani, Mater. Sci. Forum. 15-18 (1987) 1023. [14] M.J. Norgett, M.T. Robinson and I.M. Torrens, Nucl. Eng. Des. 33 (1976) 50; ASTM Standards E 521-83 (1983). [15] A.D. Brailsford and R. Bullough, J. Nucl. Mater. 44 (1972) 121; A.D. Brailsford, J. Nucl. Mater. 60 (1976) 257. 1161 C.H. Woo and B.N. Singh, Phys. Status Solidi B159 (1990) 609; Philos. Mag. A 65 (1992) 889. [17] H. Trinkaus, B.N. Singh and A.J.E. Foreman, J. Nucl. Mater. 199 (1992) 1. [18] H. Trinkaus, B.N. Singh and A.J.E. Foreman, J. Nucl. Mater. 206 (1993) 200. [19] R.W. Balluffi, J. Nucl. Mater. 69&70 (1978) 240. [20] H.R. Schober and R. Zeller, J. Nucl. Mater. 69&70 (1978) 341. [21] P. Ehrhart and R.S. Averback, Philos. Mag. A60 (1989) 283; R. Rauch, J. Peisl, A. Schmalzbauer and G. Wallner, J. Phys. Condens. Matter 2 (1990) 9009. 1221 B.L. Eyre and D.M. Maher, Philos. Mag. 24 (1971) 767. [23] H. Trinkaus, B.N. Singh and C.H. Woo, Proc. ICFRM-6, J. Nucl. Mater., to be published. [24] K. Yamakawa, I. Mukouda and Y. Shimomura, J. Nucl. Mater. 191-194 (1992) 396. [25] M. Kiritani, T. Yoshiie, S. Kojima, Y. Satoh and K. Hamada, J. Nucl. Mater. 174 (1990) 327. [26] J.O. Stiegler and E.E. Bloom, Radiat. Eff. 8 (1971) 33. [27] A. Horsewell and B.N. Singh, ASTM-STP 955 (1988) 220. [28] A.J.E. Foreman, B.N. Singh and A. Howewell, Mater. Sci. Forum 15-18 (1987) 895. [29] B.N. Singh and A. Horsewell, Proc. ICFRM-6, J. Nucl. Mater., to be published. [30] J.E. Westmoreland, J.A. Sprague, F.A. Smidt and P.R. Malmberg, Radiat. Eff. 26 (1975) 1. J.A. Sprague, J.E. Westmoreland, F.A. Smidt, Jr. and P.R. Malmberg, in Properties of Reactor Structural Alloys After Neutron or Particle Irradiation, ASTM-STP 570 (ASTM, Philadelphia, PA, 1975) p. 505. [31] R.E. Staller and G.R. Odette, ASTM STP 782 (1983) 275; R. Bullough and SM. Murphy, J. Nucl. Mater. 133&134 (1985) 92. 1321 J. Tenbrink, Thesis, Technische Universitit Berlin, D33 (1987); J. Tenbrink, R.P. Wahi and H. Wollenberger, ASTM-STP 1046 (1989) 543. 1331 A.D. Brailsford and R. Bullough, Philos. Trans. R. Sot. London 302 (1981) 87. [34] B.C. Skinner and C.H. Woo, Phys. Rev. B30 (1984) 3084. 1351 H. Wiedersich, Nucl. Instr. and Meth. B59/60 (1991) 51; J. Nucl. Mater. 205 (1983) 40. 1361 C.W. Woo and A.A. Semenov, Philos. Mag. A67 (1993) 1247. C.W. Woo, A.A. Semenov and B.N. Singh, J. Nucl. Mater. 206 (1993) 170. [37] V. Naundorf and H. Wollenberger, Proc. ICFRM-6, J. Nucl. Mater., to be published.