C A English

M L Jenkins
email: mike.jenkins@materials.ox.ac.uk

Molecular ion irradiations of molybdenum

Keywords: defect analysis, radiation damage user supplied): displacement cascades, cascade collapse radiation damage, displacement cascades, cascade collapse

come

Introduction

The mechanical properties of structural materials in nuclear reactors are degraded by microstructural changes caused by fast-neutron irradiation. A fast fission or fusion neutron can impart an energy of several tens of keV to a primary knock-on atom (PKA). The PKA then generates a branching series of collisions termed a displacement cascade. As a consequence several hundred vacancies and self-interstitial atoms (SIAs) may be created within a small highly disordered volume, typically of dimension ~10 nm. Most of the displacements caused by fast neutrons occur in cascades generated by PKAs with energy > 10 keV. Consequently displacement cascades have received a good deal of theoretical and experimental attention over many years. Despite this, some aspects of cascades are still poorly understood.

The main theoretical tool for investigating the primary damage state at the end of the displacement phase of the cascade (which occurs in a few picoseconds) has been molecular dynamics (MD). Simulations of cascades by the MD method have been aided in recent years by the development of better interatomic potentials and more powerful computers, allowing more realistic simulations of larger cascades (see e.g. [1][2][3]). Such simulations have demonstrated that clustering of both vacancies and SIAs can occur at short timescales within cascades. The longer-term evolution of cascades has been studied by Kinetic Monte-Carlo methods, and further clustering may occur (see e.g. [4,5]). Clustering within cascades may be studied experimentally by carrying out low-dose heavy-ion irradiations of thin-foils and analysing the resulting damage structures by transmission electron microscopy (see e.g. the reviews [6,7]). The heavy-ions mimic the PKAs produced by fast neutrons. In such studies it is usually the case that only the vacancy component of the damage is seen. In many materials resolvable vacancy loops are produced within individual cascades by a process of cascade collapse.

Particular attention in recent years has focussed on displacement cascades in α-Fe, primarily due to the likely use of low-activation ferritic steels in future fission and fusion reactors. Fe and ferritic alloys are unusual in that cascade collapse does not occur at low doses in cascades initiated by self-ions [8,9]. However it does occur in cascades initiated by more massive ions. The fraction of cascades which collapsed to form visible loops increases as the mass of the incident ion increases [8][9][10]. From this it has been inferred that the thermal-spike lifetime and the energy density and vacancy concentration in the cascade centre must play an important role in the formation of the vacancy clusters. Further, in contrast to other bcc metals, in α-Fe a mixed population of a/2<111> and a<100> perfect loops was observed.

The effect of the mass of the PKA on displacement cascade evolution in α-Fe has recently been modelled by Calder et al. using MD [3,11]. These authors simulated cascades of energy 5-30 keV with PKA masses chosen to represent C, Fe and Bi. Bi 2 and Bi 3 molecular ion irradiations were simulated by using two or three adjacent Bi atoms as PKAs, and giving each half or a third of the total energy and the same initial direction of motion. The PKA mass was found to have a large effect on the damage produced within individual cascades. The higher the PKA mass, the higher the peak number of displaced atoms. Clustering of both vacancies and interstitials was observed, with the largest clusters in cascades produced by heavier PKAs. Clustering was particularly pronounced in molecular ion simulations. In 30 keV Bi 3 + molecular ion cascades, nearly every event led to the production of a large vacancy loop and one or more interstitial loops. Most of the interstitial loops had Burgers vectors b = a/2<111>, although some loops produced by molecular-ions had b = a<100>. Vacancy loops generally had b = a<100>. The largest loops contained as many as 200 vacancies. The probability for the production of a<100> loops increased with the total cascade energy and was highest for Bi 3 + simulations, lower for Bi 2 + , and lower again for Bi + .

The results summarized in the paragraph above are in qualitative agreement with the early experimental work of Jenkins et al.. [9] in α-Fe mentioned above as well as more recent in-situ experiments in Fe and FeCr alloys [10]. However there have been no other systematic experimental studies in α-Fe of the effect of PKA mass, most likely because of the experimental difficulties of working in a ferromagnetic material with a strong tendency to oxidise. However the new MD simulations of Calder et al. [3,11] have prompted us to resurrect some previously unpublished data in molybdenum, where we carried out irradiations using molecular ions in an attempt to explore systematically the effect of PKA mass and energy on cascade collapse. Mo, like α-Fe, is bcc but there is no difficulty in preparing high-quality foils. In Mo, the defect yield, defined as the fraction of cascades which collapse to give a visible loop, has been found to depend on the material purity, the irradiation temperature and the mass and energy of the ion [12][13][14]. Typically however the defect yields are considerably lower than in pure fcc metals irradiated under similar conditions, and so in this respect Mo somewhat resembles α-Fe.

Molecular ion irradiations provide a convenient method for exploring the effect of important cascade parameters [15][16][17][18]. Molecular ions will dissociate on impact, the atomic components producing cascades which overlap both spatially and temporally, so increasing both the average energy density and vacancy concentration in the region covered by the cascades compared with a single ion of the same total energy. In addition the thermal spike lifetime is likely to be longer since the cascade is more compact. The total disorder created by molecular implants has been shown to be greater than that created by the equivalent atomic implants in GaAs [15] and in Si and Ge [16][17][18]. Antimony ions were chosen for these experiments because of their suitable atomic mass and the fact that beams could be produced free from charge-exchange products [START_REF] Beanland | English in "Application of Ion Beams to Materials[END_REF].

The specific aims of the present experiments were as follows:

1. To examine the production of vacancy loops in atomic and molecular ion cascades in Mo, by measuring the fraction of ions which produced visible loops, and the dependence of the size distributions on the ion species, and thereby gain information on the role of different cascade parameters on cascade collapse.

2.

To study the distribution of the Burgers vectors between the two possible perfect loop Burgers vector types, a/2<111> and a<100> in the molecular ion implants.

Experimental Procedures

Specimen preparation

The starting material was Climet arc cast molybdenum rod supplied by the Climax Molybdenum Company. High purity single crystals were prepared by electron beam zone refining to produce seeded single crystals with two orientations, with normals close to either the [011] or the [001] pole. The zone refining and subsequent heat treatment procedures to reduce the carbon and gaseous impurities to a few ppm level have been described elsewhere [START_REF] Capp | [END_REF]. The purity of the crystals was monitored by measuring their resistivity ratios. The measured values of R 293K /R 4.2K were 6600 + 200 for the (011) crystal and 8800 + 300 for the (001) crystal.

Finally, the single crystal rods were cut into 3mm discs and then electro-polished to produce perforated specimens suitable for irradiation and electron microscopy.

Irradiation procedure

The thinned discs were irradiated at room temperature in a low-energy ion accelerator with beams of Sb + , Sb 2 + and Sb 3 + ions. Three total energies per ion were employed: 60, l20 and 180 keV. Most (011) foils received a dose of 10 16 ions m -2 , while most (001) foils received a dose of 4 x 10 15 ions m -2 . Some lower dose (10 15 ions m -2 ) molecular ion irradiations were also performed 2 .

Electron Microscopy

After irradiation the foils were examined in a conventional electron microscope operated at 100kV. The foil was located in a double-tilt holder with the damage surface adjacent to the electron exit surface, and suitable areas were imaged under several different diffraction conditions. The conditions employed for each crystal orientation are summarised in table 1. The foil thickness was ~5ξ g , where ξ g is the two-beam extinction distance for a {011}-type reflection, equal to 23.2nm [21].

In the present work all measurements were made from micrographs of magnification 3 x 10 5 obtained under dynamical imaging conditions, i.e. with deviation parameter s g = 0, where small defects located within 1.25 ξ g of the surface exhibit a characteristic black-white contrast [22] (see also [START_REF] Jenkins | Characterisation of Radiation Damage by Transmission Electron Microscopy[END_REF]). The defect densities were measured by comparing the same area imaged in several reflections and noting the total number of defects visible. The criterion for positive identification was that a defect should appear in at least two conditions. The defect sizes were measured using micrographs taken with g = 1 01 and s g = 0. Calculations by Saldin et al.. [START_REF] Saldin | [END_REF] suggest that for these imaging conditions the ratio of image size to true defect size is approximately unity.

The directions of Burgers vector b and loop normal n of the small loops were analysed by comparing the symmetry of the observed dynamical images with the theoretically calculated images from known loops. The details of this approach have been discussed by Häussermann, Rőhle and Wilkens [25], Eyre Maher and Perrin [26,27] and Holmes, Eyre, English and Perrin [28] and are summarized in [START_REF] Jenkins | Characterisation of Radiation Damage by Transmission Electron Microscopy[END_REF]. English, Eyre and Holmes [29] discussed in detail the method of analysing the geometry of a mixed population of edge and non-edge perfect loops in bcc metals, and this forms the basis of the analysis employed here. Essentially, the experimental images obtained under the standard imaging conditions are compared with the images computed for the possible loop geometries at these imaging conditions. By image matching it is possible to determine both the Burgers vector and to estimate the loop normal in a wide variety of cases.

Particularly useful images are those formed when g.b > 1or g.b = 0. Some examples are illustrated in figure 1, which are image simulations made using the program described in ref. [30] for the case of a b = a/2[ 1 ], which corresponds to a likely nucleation plane according to the Eyre-Bullough mechanism [31], and the edge configuration with n = [ 1 11 ]. It can be seen that the g.b = 1 and g.b = 2 images retain the same basic form for all loop normals although the angle of the black-white interface rotates with n, while for g.b = 0 the symmetrical 'butterfly' image obtained for the pure edge loop is progressively distorted as n approaches [ 1 01 ]. The characteristic interface structure of the g.b = 2 images is particularly valuable for defect analysis. A full description of the form of the images from non-edge loops can be found in [27], see also [29] and pp 46-49 in reference [START_REF] Jenkins | Characterisation of Radiation Damage by Transmission Electron Microscopy[END_REF].

Results

Figure 2 illustrates the damage microstructures in (011) foils produced by irradiation with Sb + and Sb 3 + ions at total ion energies of 60 and 180 keV. In all cases small black-white images of dislocation loops are visible. Qualitatively it can be seen that the loop number density increased either as the number of atoms in the ion increased at constant ion energy, or as the total ion energy increased at constant ion mass.

Figure 3 shows similar irradiations of (001) foils. For Sb + implants, no damage was observed at an ion energy of 60 keV, while increasing the ion energy to 180 keV resulted only in a very low loop density. For molecular Sb 3 + ions, loops were observed after all the implantations, with an increase in loop density and sizes when the ion energy was increased from 60 to 180 keV. For each irradiation condition illustrated it is clear that the loop density in the (001) foils is significantly lower than in the (011) foils. This is only partly due to the lower dose (see section 3.1 below). The majority of the loop images in both figures 2 and 3 are consistent with layer 1 vacancy loops. The loop depths and consequently their nature were confirmed by stereo microscopy for selected irradiations. For the 120 and 180 keV Sb + implants of (011) foils, some layer 2 vacancy loops were present. In molecular ion irradiations at low doses (10 15 ions m -2 ) where cascades are well separated, we looked for evidence for multiple loops arising from wide separation of the atoms in the molecular ion. No such evidence was found.

We now present quantitative results, which are most conveniently described in terms of the defect yields, the loop geometries and the image size distributions.

Defect yields

Tables 2a and2b show the measured defect yields for the (011) and (001) foils respectively. Consider first the (011) foils. Each row in table 2a gives the yields at a constant energy per ion. It can be seen that the yields increased on going from Sb + to Sb 2 + to Sb 3 +

; typically the yield tripled on going from Sb + to Sb 3 + . Each column in table 2a shows the effect of increasing the ion energy at constant ion mass. In each case the yield increased on going from 60 to 120 keV by a factor of two or more. A further increase in ion energy to 180 keV resulted in a smaller increase in yield, and indeed the yield appeared to saturate at ~0. ) shown on the diagonal of table 2a, the yields increased in the ratio 1 : 3.5 : 7, whereas the number of atomic 60 keV cascades increased only in the ratio 1:2:3. This result highlights the important influence of overlapping cascades produced by the molecular ions.

In the (001) foils (table 2b), the low density of defects in the Sb + and Sb 2 + irradiations meant that there was considerable error in estimating the defect density. However, the Sb 3 + results show that the yield was approximately proportional to the ion energy.

It has been found in this study (see section 3.2 below) and in other studies of bcc metals [32,10] that suitably orientated perfect loops produced close to the specimen surface may glide out of the foil. This would lead to a decrease in the measured yield, which would be dependent on the local foil orientation and loop morphology. No correction has been made for this effect in tables 2a and 2b, while an estimate of its probable magnitude is made in section 4.3.1.

Loop geometry analysis

(011) foils

For all irradiation conditions the Burgers vectors of a large majority of loops at this foil orientation were distributed between the two a/2<111> variants with b in the plane of the foil, that is, b = a/2 [ 1 11 ] and b = a/2 [ 1 of 65-70% of the visible loops by noting if and how such images were distorted from the symmetric "butterfly" generated by edge loops. In this way non-edge loops could be identified and their habit planes estimated, and a determination made on which {110} plane they had likely nucleated (see figure 1 and references [29] and [START_REF] Jenkins | Characterisation of Radiation Damage by Transmission Electron Microscopy[END_REF]). The results of an analysis of this sort for the case of a foil irradiated with 120 keV Sb 2 + ions is shown in table 3, which gives detailed results of Burgers vector determinations and probable loop nucleation planes for three areas (a-c) of the same foil. The great majority of loops exhibited distorted g.b = 0 images typical of non-edge loops, with fewer than 3% showing the symmetrical 'butterfly' of pure edge loops. There was no significant variation in either Burgers vector or loop normal distributions between the three areas. Loops were evenly distributed between the two variants, and 60-70% had loop normals lying between [ 1 01 ] and the direction of the Burgers vector, implying an edge-on ( 1 01 ) nucleation plane. Note that at this orientation it was not possible to distinguish between loops with b = a/2[ 1 11 ] which had originated on the ( 1 10 ) or the (110) plane, or between loops with b = a/2[ 1 11 ] which had originated on the ( 1 10 ) or the ( 10 1 ) plane.

The general pattern seen in table 3 was found in most other cases at this foil orientation, with the exception of foils irradiated with 120 and 180 keV Sb 3 + ions to a dose of 10 l6 ions m -2 . In these cases the loop densities were high, such that the loop-loop separation was of the order of the loop diameter, and it was observed that there were local inhomogeneities in the distributions of Burgers vectors and loop normals. . The detailed analysis is given in table 4, and confirms that the ratio of edge-on Burgers vectors was indeed different in each area. The distribution of loop normals was approximately the same in areas a and b, but in area c far fewer loops seemed to have nucleated on the edge-on ( 1 01 ) plane and most had nucleated on either ( 0 1 1 ) or (101). The defect yield in area c was also lower than areas a and b. This complex behaviour should be compared with the results at a lower loop density presented in table 3 for 120 keV Sb 2 + ions, where little variation in b or n was found.

In foils implanted with molecular ions some loops were found that could not be analysed as having a/2<111> Burgers vectors. This is illustrated in figure 5, which shows an area of a foil irradiated with 180 keV Sb 3 + ions and imaged in dynamical dark-field conditions in the three diffraction vectors g = 200, 1 21 and 1 1 2 . The arrowed loops show black-white contrast with interface structure typical of |g.b| > 1 in all three images. This pattern can only arise from loops with a[100] Burgers vectors [27],where |g.b| = 2 for all three reflections. Table 5 shows the fraction of a[100] loops for different irradiation conditions. It can be seen that some a[100] loop formation occurred with all the molecular ions. The fraction of a[100] loops increased with ion energy, and at a given energy was higher for irradiations with Sb Detailed inspection of the a[100] loops arrowed in figure 5 and of similar images taken in different diffraction conditions revealed that the black-white interfaces were not all parallel, indicative of different loop normals n. From the degree of rotation of the black-white interface of images from the pure edge configuration it was possible to determine the approximate direction of n [29]. More than 50% of the loops had n within ~15 degrees of the edge configuration. Many of the others had n close to <011>. In all conditions where a[100] loops were present, their images were consistent with nucleation on an {011} plane, followed by rotation on the glide cylinder partially or completely towards the pure edge configuration.

(001) foils

It was only in foils irradiated with Sb 3 + ions that the loop densities were high enough for a detailed analysis of the loop geometries to be made. The analysis is illustrated in figure 6, which shows a specimen irradiated with 120 keV Sb 3 + ions and imaged in diffraction vectors g = 200 (figure 6a) and g = 020 (figure 6b to loops B. This assignment was confirmed by comparison with computed images of loops with these Burgers vectors [29]. This conclusion that most loops in (001) foils irradiated with Sb 3 + ions have a<100> Burgers vectors was also found at other ion energies. In the 120 keV Sb 3 + irradiation illustrated in figure 6 fewer than 2% of the loops had b = a/2<111>. For 180 keV Sb 3 + irradiations the fraction of a/2<111> loops was <5%. In the case of the loops produced by 60 keV Sb 3 + ions, 30% of the loops were too small for analysis, but the remainder were analysed as having a<100> Burgers vectors. Thus the defect yields given in table 2b for (001) foils in fact demonstrate how the production of edge-on a<100> loops is determined by ion energy and mass. Within experimental error the Burgers vectors were equally distributed between a[100] and a [010]. No evidence of a[001] Burgers vectors was found in (001) foils imaged at z = [011], where computations show that loops with this Burgers vector would produce strong images.

Again loop normals were determined by comparing the experimental images with computed images of loops with loop normals between <110> and the Burgers vector [29]. The distribution of loop normals was determined for the loops formed in the two higherenergy irradiations. The loop normals of the great majority of the loops were consistent with nucleation on the edge-on (110) and ( 0 1 1

) planes and then rotation towards the edge configuration. For one foil irradiated with 120 keV Sb 3 + ions, fewer than 8% of the loops had n within l0 o of the edge configuration and more than 60% had n within 20° of the [110] or [ 0 1 1 ] directions. Mean image diameters are given in table 6, while selected image size distributions are presented in figure 7. In the case of (011) foils, an increase in the ion energy at constant ion mass resulted in a shift in the peak in the distribution to larger sizes and an increase in the maximum loop size, causing an increase in the mean image size (see columns in table 6). This trend was most pronounced for irradiations with Sb 3 + ions and is illustrated in figure 7a-c. The same trends were seen on increasing the ion mass at constant ion energy (see the rows in table 6), and are illustrated for the most pronounced case of 180 keV ions in figure 7d-f. In (001) foils, it was only possible to obtain distributions for irradiations with Sb 3 + ions. Here, the mean image diameter (last column of table 6) increased with ion energy. There were no significant differences between the size distributions in (001) foils (figure 7g-i) and (011) foils irradiated with the same ions (figure 7a-c). For 60 keV Sb 3 + ions, the larger mean image diameter in the (011) foils was caused by a greater fraction of loops with diameters > 4.0 nm. Finally, a detailed analysis of the loops formed after the 180 keV Sb 3 + irradiations revealed that there were no significant differences between the size distributions in (011) foils of the a<001> loops and the a/2<111> loops, although no a[001] loops were identified with image diameters less than 2.5 nm. This last observation may simply be due to the low number densities with consequent bad statistics.

Image size distributions

Discussion

Consistency of observations with the Eyre-Bullough mechanism for loop formation in bcc metals

In bcc metals it has long been considered that loops may form by the Eyre-Bullough mechanism [31]. This mechanism envisages that point-defects aggregate initially on {110} planes to form faulted a/2<111> loops. The very high stacking fault energy of molybdenum causes such loops to unfault , most probably when the loop diameter is still < 1nm, and so below the resolution limit(~ 2nm) for TEM. The mechanism is described in a typical case by the shear reaction:

a/2[ 1 01 ] + a/2[100] → a/2[ 1 11 ]
In section 3.2.1 it was demonstrated that a significant number of non-edge a/2<111> loops were observed, and that it was possible to determine on which {110} plane they had likely nucleated, i.e. the data in the present paper are consistent with the mechanism in the case of a/2<111> vacancy loops.

In addition found that that our observations of the loop habit planes of a<100> vacancy loops in Mo are also consistent with this mechanism (see section 4.3). As in the case of a/2<111> loops, the normals found for a<l00> loops suggested nucleation on {110} planes. The shear is described by the reaction of the type: There is also evidence in the literature for the Eyre-Bullough mechanism, at least in the production of vacancy loops. For example, Häussermann [12] observed a/2<110> faulted loops in ion-irradiated tungsten, and many a/2<111> loops had large shear components, consistent with habit planes close to {110}. Several other studies have found a/2<111> loops in ion-irradiated bcc metals with normals close to <110> directions, or with normals lying between <110> and the direction of the Burgers vector, e.g. [12,29]. However, we have also to note that in recent MD simulations of the configurations of self-interstitial clusters in Fe, Tapesa et al [33] found that the habit plane of a/2<111> loops corresponding to the minimum energy configuration at 0K tends towards the {110} plane. Thus, the observation of <110> loop normals can only be said to be consistent with the Eyre-Bullough mechanism, not proof of its occurrence. However, we note that the effect of surface image forces on the resultant loop configurations is also consistent with the Eyre-Bullough mechanism. For example, Jäger and Wilkens [34] have shown in heavy-ion irradiation experiments in tungsten that their results may be explained if the Eyre-Bullough mechanism is operative, and the unfaulting reaction is influenced by the close proximity of the surface, which is within 10 nm of the damage region. Unfaulting reactions which result in loops with Burgers vectors pointing towards the surface tend to be favoured.

a/2[ 011 ] + a/2[ 1 1 0 ] → a[001]
The Eyre-Bullough mechanism has not yet been identified explicitly in MD simulations of vacancy loops e.g. [1][2][3]. Indeed Marian et al. [32] claim that it does not occur for a<100> interstitial loops, and proposed an alternative mechanism whereby such loops could be produced. In contrast, Tapesa et al [33] found in their MD study of α-Fe evidence for transformation of faulted a/2<110> interstitial loops to a<100> loops at 0K. However, dynamical simulations suggested that small a/2<111> interstitial loops are more stable than other loops despite the lower energy of a/2<110> loops at 0K.

There could be several reasons why the mechanism has not been seen for vacancy loops in MD simulations. First, it is intrinsically difficult to interpret MD data in terms of mechanisms. In particular, vacancy loops form in the highly-disordered cascade core as the thermal spike cools, and the morphology of such loops at an early stage in their evolution is difficult to discern in the MD visualizations. Second, the simulations may not have been run for long enough for the mechanism to be recognised. For example, in recent simulations by Calder et al.(see [11] and section 4.4 of this paper) vacancy clusters first appeared quite late in the displacement phase, typically after >10-15 ps, whereas interstitial clusters appeared after 1-2 ps. Recognisable vacancy loops did not form before 20s, while the simulations were terminated at ~30 ps. Third, it might simply be the case that the Eyre-Bullough mechanism does not occur in these simulations.

Formation of a<100> loops in molecular ion irradiations

We have found that in (001) foils, a larger fraction of the loops had Burgers vectors a<l00> in Sb 3 + irradiations compared with Sb 2 + irradiations at the same energy. Further, at constant ion mass the number density of a<100> loops increased with ion energy for both foil orientations. At first sight the observation of a<100> loops in molybdenum is unexpected, because such loops are expected to have much larger energies than a/2<111> loops. Little et al. [35] have calculated for several bcc metals the probability that an a/2<110> interstitial loop unfaults to a loop with Burgers vector a<100> rather than to a/2<111>. They employed-a simple continuum model to represent the propagation of the unfaulting partial dislocation across the faulted loop, and then used the relative magnitude of the energy barriers to give the relative probabilities. For molybdenum at room temperature, the probability of a<100> loop formation is extremely low (1 in 10 19 ). It is likely that the unfaulting probabilities for vacancy loops are similar.

Molecular ion cascades will occur closer to the surface than single ion cascades of the same energy. However, it is not likely that the occurrence of a<100> loops is due to the influence of the surface. Elastic interactions with the surface will reduce the energies of both a/2<111> and a<100> loop types, but by only a small amount in comparison with the large energy difference between the two geometries (see the discussion in Section 4 of Jenkins et a1. [9]).

A more likely explanation for the increased probability for the formation of a<100> loops is the increased average energy density and thermal spike lifetimes in the centre of molecular ion cascades. The calculated probability of a<100> loop formation only approaches the observed frequency if a temperature close to the melting point is achieved in the cascade region (when the calculated probability of 10 -4 compares with the observed probability of 10 -3 -10 -4 ). Within cascades with a high density of deposited energy, the effective temperature of the region may be considerably higher than the host lattice temperature, leading to a "thermal spike". Sigmund [36] calculated that the cumulative average energy density (expressed as a temperature) decreased with increasing ion energy, but that the lifetime of the thermal spike increased. A value for the cumulative average energy density (eV/atom) for atomic ion cascades can be arrived at from this work. For molecular ion cascades, a value can be obtained by assuming that the average energy density in a cascade created by an Sb m + ion with total energy mE is m times that of an atomic Sb + ion with energy E. For Sb 3 + ions with energy between 60 and 180 keV, the estimated energy density is always of the order of the sublimation energy (6.8 eV/atom) for molybdenum, supporting the supposition that high temperature regions occur within Sb 3 + cascades.

It is therefore possible to account for a<100> loop formation by postulating that there are high temperature regions within some of the molecular ion cascades, which persist during the initial formation and growth of the faulted loop. Within such regions the probability of unfaulting to a<100> is significant. The increased lifetime of these zones at higher ion energies allows a greater number of faulted loops to nucleate, grow and unfault.

An interesting comparison can be made between the a<l00> loop formation in molybdenum and that previously observed in α-Fe [8,9]. In contrast to the present results in (011) foils, where even in the most favourable case <2% of the loops had Burgers vectors a<l00>, in (011) foils of α-Fe about 50% of the loops were identified as a<l00> after 80 keV W + ion irradiation [9]. It is interesting to note that the calculations presented in ref. [35] show that the probability of a/2<110> loops unfaulting to a<100> loops is considerably higher in α-Fe than in molybdenum, in agreement with the observed trends. a<100> loops appear to be more stable in α-Fe than in other bcc metals. Very recent calculations by Dudarev et al. [37] have explained the greater relative stability of a<100> loops in α-Fe at high temperatures in terms of an elastic instability, driven by spin fluctuations.

Insight into vacancy loop formation in (011) foils

Estimates of true defect yields

We first consider briefly the possible effect on the defect yield of the loss of loops by glide to the surface. In (011) foils, the perfect loops formed after unfaulting from the edge-on ( 101

) plane have b = a/2[ 1 11 ] or a/2[ 1 1 1 ]
). Since these Burgers vectors lie in the foil plane, none of these loops will be lost to the surface. This is not true for other {110} nucleation planes, where a proportion of loops have a component of b towards the surface and will be lost. An estimate of the true defect yield can be obtained by identifying and counting those non-edge loops which had originated on the edge-on ( 1 01 ) plane (i.e. loops with a loop normal n lying between [ 1 01 ] and the edge-on Burgers vectors (a/2[ 1 11 ]or a/2[ 1 11 ]). These figures are shown in tables 3 and 4. Typically about 60-70% of loops were found to have originated on the ( 1 01 ) plane. About 60-70% of the observed yield therefore derives from loops which nucleated on ( 101 ). If we further assume that nucleation on any of the six {110} planes is equally likely, the true yield is six times this fraction of the observed yield, i.e. the true yield is about four times the observed yield. Such a correction would imply that for 120 and 180 keV Sb 3 + molecular ion irradiations the true yield is close to unity.

In table 7 we present the corrected defect yields, cascade efficiencies, and vacancy retention efficiencies. The cascade efficiency is defined as the fraction of vacancies surviving in each cascade to form a loop, whilst the vacancy retention efficiency is defined as the percentage of calculated displacements which are retained in loop form. It differs from the cascade efficiency by including those cascades which do not form visible loops. The number of displacements in the cascade has been calculated using the internationally accepted formalism due to Norgett, Torrens and Robinson [38].

The assumption in the argument above that loop nucleation on any of the six {110} planes may not be correct. At higher doses the inhomogeneous distributions of b and n (table 4) suggest that loop nucleation is probably influenced by the strain fields of existing nearby loops. It is possible therefore that we have underestimated loop losses to the surface in the estimates above. This would not however affect the conclusion that the true yields under high-energy molecular ion irradiation are close to unity.

Vacancy loop formation in atomic ion cascades

The calculated number of displacements in atomic-ion cascades is proportional to the ion energy. However the mean size of loops is only weakly dependent on the ion 6). The number of vacancies in a loop of mean area increases by only 30% on going from 60 keV Sb + ions to 180 keV Sb + ions. This leads to a decrease in the cascade efficiency with increasing ion energy, which drops by a factor of about two on going from 60 keV to 120 keV, with a further small decrease at 180 keV (table 7). However, the increase in the corrected defect yield with ion energy leads to the interesting result that the fraction of calculated displacements which are retained in loop form is approximately independent of ion energy at ~7%-10%. As the ion energy increases, cascades will become larger and the average vacancy concentration within the cascade volume will decrease. In the cascade region large fluctuations in the vacancy concentration will occur, leading to sub-cascade formation. It is within these regions of local high vacancy concentration that vacancy loops are likely to nucleate.

The combination of increasing defect yield and decreasing cascade efficiency with increasing ion energy implies that the probability of vacancy aggregation in a region of the cascade is greater in high energy implants, but that a smaller proportion of the vacancies created are able to participate in loop growth. Thus we conclude that in high energy cascades, only those vacancies in the immediate vicinity of the growing vacancy cluster are able to diffuse to it.

Vacancy loop formation in molecular ion cascades

As a first estimate, we have assumed in calculating cascade efficiencies that the number of displacements produced in a Sb m + (m = 2 or 3) molecular ion cascade of energy mE is the sum of the displacements produced in m single-ion cascades of energy E. The likelihood of thermal spike effects mean that this is likely to be only an approximate estimate .

Our results show that the production of vacancy clusters is strongly enhanced in cascades created by diatomic and triatomic ions. In particular:

• For 120 keV and 180 keV Sb 3 + ions, the corrected defect yields are close to unity. As no evidence was found of multiple loops in low dose molecular ion irradiations, this implies that the formation of a vacancy loop occurs in nearly all cascades. These yields are considerably higher than those obtained with atomic implants. Such high yields are common in fcc metals but have only been found previously in bcc metals irradiated with very heavy ions, e.g. [9].

• The vacancy retention efficiency is strongly enhanced by increasing the ion mass at constant ion energy. The data in table 7 show that a factor of 5-8 more vacancies are retained in loop form for Sb 3 + ions than for Sb + ions of the same energy.

• At constant energy per atom in the molecular ion, the ratio of defect yields for Sb + , Sb 2 + , and Sb 3 + ions increases more sharply (by 1:4:7) than the ratio of total ion energies (1:2:3). These trends in the data demonstrate that vacancy loop formation in cascades produced by molecular ions is strongly favoured. It is tempting to associate this with the increased energy density in molecular ion cascades. This is because molecular ions are expected to dissociate on impact, the different components creating overlapping displacement cascades. If spatial overlap were complete in the collisional phase of the cascade, then single and molecular ions of the same energy per atom might be expected to produce cascades of similar volumes. However, Jenkins and English [39] have studied the disordered zone sizes produced in ordered Cu 3 Au by Sb + atomic and molecular ions. In such experiments the disordered zones give a measure of the size and shape of the region where many replacements have occurred. The data obtained in these experiments suggested that disordered zone sizes are determined more by the total ion energy than by the energy per atom. For example, disordered zones created by 150 keV Sb 3 + ions were very similar in size to those created by 150 keV Sb + ions, but had mean diameters about 30% larger than those produced by 100 keV Sb 2 + ions and -60% larger than those produced by 50 keV Sb + ions. The shapes of disordered zones produced by 150 keV Sb 3 + ions were however more regular than those produced by 150 keV Sb + single ions. These findings may be evidence that cascade overlap is incomplete. It is nevertheless likely that there are regions of overlap within molecular ion cascades where particularly high energy densities do occur, and these may lead to dynamic events which trigger loop formation. Insight into such processes is now available from MD simulations. This is described in the next section.

Comparison with MD simulations

There have been no MD simulations of molecular ion cascades in Mo, but recently such simulations have been made in "α-Fe" by Calder et al. [3,11]. In these studies, the PKA mass was also varied in single-ion cascades. Strongly enhanced vacancy clustering was found in cascades initiated by heavy PKAs, and was particularly pronounced in molecular ion cascades, with the formation of dislocation loops. Calder et al. followed the atomic exchanges leading to the production of Frenkel pairs. The paths from surviving SIAs to their originating vacancies were followed. Such paths were usually very tortuous. When all such paths were displayed simultaneously, tangled, so-called spaghetti-regions were defined, within which most displacements had occurred. Calder et al. did not find a strong variation in the total volume of these spaghetti-regions for different ions of the same energy, but the shapes of these regions were more spherical for cascades produced by molecular ions than those produced by single-ions of the same energy. Such spaghetti regions should correlate with the disordered zones formed within cascades in ordered alloys, and so these finding are consistent with our results on molecular-ion irradiated Cu 3 Au mentioned above [39]. Qualitatively one might expect the thermal spike lifetime to be longer in a spherical cascade because of its smaller boundary area compared with cascades of more complex shape, so that energy is transferred to the surrounding matrix more slowly. In addition the local regions of high deposited energy density and vacancy concentration would likely be more proximate in spherical cascades than in cascades of more irregular shape. These effects may enhance vacancy clustering. The conclusion that conditions of long thermal spike lifetimes, and high local energy densities and vacancy concentrations favour collapse has also been drawn in previous MD simulations [40][41][42] and experiments [6,9].

The actual process of point-defect clustering in the MD simulations seems not to have been completely elucidated. In Calder et al's simulations, interstitial clusters, including loops with b = a/2<100> and a<100>, formed at the peripheries of the cascade regions within 1-2 ps of the initiation of the cascade by some sort of dynamic process, which seemed to involve the production of a shock wave which propagated out from the cascade core immediately after the displacement phase of the cascade. Interstitial loop formation did not seem to involve conventional nucleation and growth processes, but was reminiscent of loop punching. The formation of large interstitial clusters was a good predictor of strong vacancy clustering at the cascade core later in the evolution of the cascade. Indeed vacancy loop production was always found in association with interstitial clustering. In an earlier MD study of cascades in Fe produced by 50 keV self-ions, Soneda et al. [42] found a single example of vacancy loop production (out of 100 cascades simulated), and in this case too large interstitial clusters formed at the periphery of a high energy-density region early in the cascade evolution. Vacancy loops formed in the thermal spike phase of the cascade after about 20-30 ps, by a process of collapse of the vacancy-rich core in a process similar to the classical picture. These loops all had Burgers vectors a<100>. Although a<100> loops are in reality a particular feature of Fe, the potential used in these simulations of "α-Fe" would not have been expected to favour the formation of a<100>loops over a/2<111> loops. Nevertheless the simulations do predict vacancy loop formation and its dependence on ion mass in good accord with our experiments.

The most marked difference between recent MD simulations and experiments is that in experiments interstitial loops are not seen at low doses. In thin-foil experiments such as those described here, this may be due partially to preferential loss of SIAs and small interstitial clusters to the surfaces. Loss of interstitials to the surface and consequent enhanced vacancy clustering due to reduced recombination have been found in MD simulations of cascade damage in Fe by Stoller [43]. The fraction of vacancies contained in clusters increased as the cascade initiation site approached the surface. A similar result was found by Gao and Bacon for near-surface displacement cascades in Ni 3 Al [44]. Experimental evidence that the surface influences cascade collapse has been presented by Mőller et al. [45]. In Ni 3 Al, the defect yield in cascades created by 50 keV Ta + ions at 573K decreased from about 0.7 for cascades centred close to the ion-entry surface to about 0.2 for cascades centred at a depth of more than 10 nm. There was a similar trend but smaller decrease in yield in Cu 3 Au irradiated at room temperature with 50 keV Ni + ions. It does therefore seem to be the case that the strong vacancy clustering seen in our experiments may be partly attributed to thin-foil irradiations.

Loss of SIAs and interstitial loops to the surface does not however give a complete explanation of the difference between MD simulations and experiments. In Calder et al.'s MD simulations [11] many of the interstitial loops were of a size that would be resolvable in the electron microscope. In an (011) foil some interstitial loops would be expected to have their glide cylinders in the plane of the foil and such loops would not be lost to the surface. However none was found. In earlier thin-foil experiments in ion-irradiated Cu, Stathopoulos [46] analysed the nature of ~ 500 loops of sizes 1-5 nm. by careful measurements of their depths, and found no interstitial loops. Moreover, cascade collapse to vacancy loops with relatively high probability but with few if any small interstitial loops has been found in several low-dose neutron experiments in bulk materials. For example, Muncie et al. [47] found in low-dose (3 x 10 21 neutrons m -2 ) irradiations of Cu at 80 o C that loops of sizes< 4nm were predominantly of vacancy nature, and were present at a number density consistent with the expected concentration of energetic cascades and defect yields found in heavy-ion experiments. Some larger interstitial loops were observed but at concentrations much less than vacancy loops. In similar experiments in dilute Cu-Ge alloys, Shepherd et al. [48] found that the number of interstitial loops depended on the alloy composition, suggesting that they had formed by nucleation and growth involving long-range diffusion rather than a cascade process. In low-dose neutron irradiations of Cu 3 Au, loops were always associated with disordered zones, and were found at cascade cores, again suggested their nature was vacancy [49]. It seems therefore that there is little experimental evidence for the formation directly in cascades of interstitial loops large enough to be resolved in the electron microscope.

Summary and Conclusions

1. Cascade collapse in Mo has been studied by employing molecular ion irradiations.

Collapse was strongly enhanced in molecular ion cascades compared with single ion cascades of the same total energy, with increases both in defect yields and in loop sizes. These findings are in qualitative agreement with recent MD simulations of cascades in α-Fe which also found enhanced clustering in molecular ion cascades.

2. In (011) foils most loops had Burgers vectors b = a/2<111> lying in the plane of the foil. However in molecular ion irradiations a small fraction of loops with b = a[100] was also found. This fraction was higher for Sb 3 + than for Sb 2 + ions and also increased with ion energy a<100> loops were also produced in molecular ion irradiations of (001) foils.

3 .

 3 To determine loop habit planes.

1 1 ]

 1 . At z = [011] these Burgers vectors give g.b = 0 images in g =

  Figure 4 shows three neighbouring areas (a, b and c) of the same foil in a specimen which had been irradiated with 10 l6 180 keV Sb 3 + ions m -2 and imaged in g = 1 21 . It can be seen that the number of |g.b| = 2 images increases from fgure 4(a)-(c), indicating a change in the ratio of loops with b = a/2[ 1 11 ] and b = a/2

  ). A large majority of loops fall into one of two categories. The loops labelled A show typical |g.b| > 1 type contrast in g = 200 while in g = 020 they exhibit characteristic |g.b| = 0 images. The loops labelled B show the reverse contrast, with |g.b| = 0 images in g = 200 and |g.b | >1 images in g = 020. This suggests the assignment of b = a[100] to loops A and b = a[010]

Formatted

Formatted

  provide further evidence for Deleted: we have a new result that Deleted: probably produced by

Formatted

  The strong experimental evidence for the Eyre-Bullough mechanism in vacancy loop formation suggests that its occurrence or absence in MD simulations may be a good indicator of how realistic are the simulations. We recommend that modelers should look for this mechanism explicitly.

Formatted

Formatted

  Figure captions

Figure 1 : 1 1

 11 Figure 1: Simulated black-white contrast images of edge and non-edge perfect loops in Mo. The simulations are for a first-layer loop with b = a/2[ 1 1 1 ] loop in a foil with normal z = [011] and imaged in g = 1 21 , 1 01 and 1 1 2 . Five loop normals n are illustrated lying in the (011) plane between n = [ 1 01 ] and n = [ 1 1 1 ]. Foil thickness 5.25, loop depth 5.10, loop radius 0.1, in units of the extinction distance ξ 110 .

Figure 2 :

 2 Figure 2:Damage microstructures in (011) foils produced by irradiation with Sb + and Sb 3 + ions at total ion energies of 60 keV and 180 keV. In each case the diffraction vector g = 1 01 .

Figure 3 :

 3 Figure 3: Damage microstructures in (001) foils produced by irradiation with Sb + and Sb 3 + ions at total ion energies of 60 keV and 180 keV. In each case the diffraction vector g = 200.

Figure 4 :

 4 Figure 4: Three areas (a, b and c) of the same foil in a specimen which had been irradiated with 10 l6 180 keV Sb 3 + ions m -2 and imaged in g = 1 21 . The Burgers vectors and loop normal distributions are not uniform, see the text for details.

Figure 5 :

 5 Figure 5: An (011) foil irradiated with 180 keV Sb 3 + ions and imaged in dynamical dark-field conditions in the three diffraction vectors g = 200, 1 21 and 1 1 2 . The arrowed loops have Burgers vector a[100].

Figure 6 :

 6 Figure 6: An (001) foil irradiated with 120 keV Sb 3 + ions and imaged in diffraction vectors (a) g = 200) and (b) g = 020. Loops have either a[100] or a[010] Burgers vectors, see text for details.

Figure 7 :

 7 Figure 7: Image size distributions: Top row: (011) foils irradiated with Sb 3 + ions of different energies; Middle row: (011) foils irradiated with single and molecular180 keV ions; Bottom row: (001) foils irradiated with Sb 3 + ions of different energy.

Figure 2 Page

 2 Figure 2

Figure 6 Page

 6 Figure 6

Table Deleted :Table Deleted :

 DeletedDeleted Table

Deleted: , Deleted: Table

Deleted:

Table Deleted :

 Deleted Consistent with earlier work, for

	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y
	Deleted:
	8

Table Deleted :Table Deleted :

 DeletedDeleted 3 + ions than for Sb 2 + ions. No evidence was found for loops with inclined a[010] or a[001] Burgers vectors. /m2

		Deleted: Figure
	F o r P e e r	Formatted: Not Superscript/ Subscript Formatted: Superscript Deleted: Deleted: where Deleted: qualitatively Deleted: Figure
	R e v i e w	Deleted: Table Deleted: T Deleted: Figure
	O n
	l y
	9

Table Deleted :Table Deleted :

 DeletedDeleted In addition, Jäger and Wilkens[36] have shown in heavy-ion irradiation experiments in tungsten that the unfaulting reaction may be influenced by the close proximity of the surface which is within 10nm of the damage region. Unfaulting reactions which result in loops with Burgers vectors pointing towards the surface tend to be favoured.

http://mc.

manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

	energy (table
	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y
	14

Table Deleted :

 Deleted Table

	F o	Deleted: Deleted:
	r
	P
	e e r
		Deleted: The
	R e v i e w	Deleted: likelihood of thermal spike effects means that this is likely to be an underestimate Deleted: . Deleted: . Deleted: .
	O n	Deleted: Table
	l y
	15

Table 4

 4 Loop normal distributions and defect yields in an (011) foil irradiated with 10 16 120 keV Sb 3 + ions m -2 , for the three areas shown in figure4.

	3. Vacancy loops of both a/2<111> and a<100> types may have formed by the Eyre-
	Bullough mechanism, which involves initial nucleation of a faulted loop on a
	{110} plane.
	4. The experiments described here suggest that the thermal spike lifetime, cascade
	energy density and vacancy concentration are important factors in cascade
	collapse.

http://mc.manuscriptcentral.com/pm-pml

Table 5

 5 Fraction of loops with b = a[100] in (011) foils.

		Ion	Sb +		Sb 2	+		Sb 3	+
	keV
	60		0		<0.1%	0.3%
	120		0		<0.1%	0.7%
	180		0		0.2 -0.4%	1.7%
	Table 6 Mean Image Sizes (nm)
	(Error in mean is + 0.1 nm)
	Foil Normal		[011]				[001]
	Ion	Sb +	Sb 2	+	Sb 3	+	Sb 3	+
	keV
	60	3.2	3.6	4.0	3.6
	120	3.3	4.7	5.0	5.0
	180	3.6	4.8	5.7	5.8

Table 7

 7 Corrected defect yields, cascade efficiencies and vacancy retention efficiencies for (011) foils In calculating the cascade efficiency it has been assumed that molecular ions create the same number of displacements as the appropriate number of atomic ions.

		Corrected Defect		Cascade Efficiency #	Vacancy retention
			Yield								efficiency
		Ion Sb +	Sb 2	+	Sb 3	+	Sb + Sb 2	+	Sb 3	+	Sb +	Sb 2	+	Sb 3	+
	keV
	60	0.15 0.29 0.35 0.43 0.53 0.83 0.06	0.15	0.29
	120	0.30 0.55 1.05 0.24 0.49 0.55 0.07	0.27	0.58
	180	0.49 0.82 1.09 0.21 0.36 0.50 0.10	0.30	0.55

#

At these doses little cascade overlap occurs, so that we are looking essentially at individual cascade events. The doses correspond approximately to 0.001-0.01 dpa.

http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

Acknowledgements

We acknowledge with thanks discussions with Dr A F Calder, including discussions of unpublished MD simulations. We thank Dr Zhongfu Zhou for assistance in the preparation of figure 1.


Alexander V Barashev
email: a.barashev@liv.ac.uk

Stanislav I Golubov

Yuri N Osetsky

Roger E Stoller

Reaction kinetics of non-localised particle-trap complexes

Keywords: diffusion, kinetics of reactions user supplied): transition state theory, dissociation reactions, edge dislocations transition state theory, dissociation reactions, complexes, edge dislocations, metals

Reaction kinetics of non-localised particle-trap complexes

Introduction

Neutron irradiation of metallic materials initiates cascades of atomic displacements, which produce the primary damage in the form of point defects, vacancies and self-interstitial atoms (SIAs), and their clusters [1][2][3][4][5]. The SIA clusters are highly mobile small dislocation loops, which migrate one-dimensionally along the close-packed directions of their Burgers vectors [2,[6][7][8][9]. They are observed to decorate edge dislocations in metals such as Ni and Mo [10,11]. As a result, the dislocations become pinned and the materials brittle. The decoration is thus one of the key processes in the microstructure evolution in metallic materials under cascade irradiation, but is still understood poorly. One of the obstacles is the lack of understanding of the kinetics of the interaction between the interstitial clusters and dislocations. A complexity originates from the long interaction range: e.g. at 600K, the distance from the core of an edge dislocation, at which the interaction energy is of the order of the thermal energy, is ~50 nm (for a cluster of 10 SIAs), that is ~200 Burgers vector lengths. This results in a significantly non-exponential distribution of dissociation times, which may destroy applicability of the reaction rate and transition state theories, which generally have broad applications in chemistry and physics [START_REF]Kinetics of nonhomogeneous processes[END_REF].

In the mean-field approach of the chemical-reaction theory [START_REF] Eyring | [END_REF][14][15], the evolution of the concentration of bi-molecular A-B complexes, N(t) , is described by an equation which includes reaction rates for complex formation, F(t) , and dissociation, D(t) :

& N(t) = F(t) -D(t) . (1)
It is usually coupled with equations for the concentrations of unpaired particles, C A (t) and C B (t) , which define the formation rate: F(t) = αC A (t)C B (t) . The rate coefficient α is found using available methods [14]; e.g. for diffusion-controlled coagulation α ∝ 4πrD AB , where r and D AB are the total radius and diffusion coefficient of A and B species, which is the classical result of Smoluchowski [16]. Determining the dissociation rate requires utilising ideas of the seminal transition-state theory [START_REF] Eyring | [END_REF]15] and the principle of microscopic reversibility (detailed balance) for chemical equilibrium [START_REF] Tolman | The Principles of Statistical Mechanics[END_REF]. The result depends on the probability function of dissociation times, P(t) , i.e. the probability that an event occurs before time t . For a random process without memory, P(t) = 1 -exp(-t / τ ) with the mean

D(t) = N(t) / τ . (2)
Equation ( 2) is valid only for the exponential distribution, hence for localized complexes with single binding energy. For non-exponential decay laws, the conventional rate equations cannot be applied [15].

We show in this paper that for complexes with long-range interaction, such as of an edge dislocation and an SIA cluster, the time scales separate and one can apply the equations for first-order reactions with a rescaled mean dissociation time and the cross-section of complex formation. The rescaling coefficient is the Eyring transmission coefficient. Here we derive a general expression for this coefficient through the first two moments of the distribution function of dissociation times and show that it is equal to unity if the dividing surface between 'bound' and 'free' states is defined by equality of the interaction and the thermal energies.

Analysis

Consider a system under what may be non-equilibrium conditions. There are two kinds of such conditions: accumulation of complexes with a nonzero formation rate and annealing from some initial state with nonzero concentration of complexes. The former process is more general, since it involves an extra process of complex formation and is considered below. It will be shown, though, that to a first approximation the final expression for the dissociation rate can be written in general, independent of the process, form, which is applicable to the annealing problem as well. Then, N(t) is given by an integral accounting for all complexes formed and survived by a time t , and D(t) by a similar integral for complexes dissociating at this instant of time:

N(t) = ds F(t -s) 1 -P(s) [ ] 0 ∞ ∫ , (3)
D(t) = ds F(t -s) ′ P (s) 0 ∞ ∫ . ( 4
)
For an exponential distribution, ′

P (t) = τ -1 1 -P(t)
[ ] and equation (2) follows from equations

(3) and (4) immediately. In general, one can obtain an equation for D(t) as a series of progressively higher order n time derivatives of N(t) by representing F(t) as a Taylor series around time t :

F(t -s) = (-1) n s n F (n ) (t) n! n= 0 ∞ ∑ . ( 5
)
Then, from equations ( 3) and ( 4), one obtains that

N(t) = (-1) n t n +1 F (n ) (t) (n + 1)! n= 0 ∞ ∑ , (6)
D(t) = (-1) n t n F (n ) (t) n! n = 0 ∞ ∑ . ( 7
)
By differentiation of equation ( 6) n times one can obtain the following expressions for the derivatives of F(t) :

τ F (n) (t) = N (n) (t) -(-1) k t k +1 F (n + k ) (t) (k + 1)! k =1 ∞ ∑ . ( 8
)
By substituting equation (8) with increasing n into equation (7) and by treating the deviation from an exponential distribution as a perturbation, hence neglecting high order time derivatives of F(t) , one obtains the dissociation rate as a series of progressively higher order, n , time derivatives of N(t) . For n =0, one reproduces equation ( 2) and, for n =1, the first order time derivative enters the equation: where γ = σ t 2 -τ 2 ( ) / 2τ 2 . Note that the formation rate does not enter explicitly the final equation ( 9), which seems thus applicable to the annealing problem as well. The correction term disappears at steady state, when & N(t) = 0 , and/or for an exponential distribution of dissociation times, when γ = 0 . Otherwise, the dissociation rate depends on & N(t) . This is a memory effect and can be understood as discussed below.

D(t) = N(t) / τ + γ & N (t) , (9)
Consider traps with a wider distribution of dissociation times than the exponential (i.e.

γ > 0 ), such as spatially-distributed potential wells for particles diffusing on a lattice (some examples are considered below). In this case, the distribution consists of a rapidly decreasing part at short times and an exponential tail. The tail is defined by the lowest (negative) interaction energy (i.e. the deepest site of the well), while the short-lived part is due to existence of higher energy sites. So, 'young' complexes have shorter lifetime than the 'older' ones and the physical significance of equation ( 9) can be described in terms of the complex 'age'. If & N(t) > 0 , complexes are on average 'younger' than under stationary conditions.

Hence, the mean dissociation time is shorter and the dissociation rate higher; the opposite

condition exists if & N(t) < 0 . When & N(t) = 0 , the 'age' distribution of complexes coincides
with 1 -P(t) . As a result, the mean dissociation time is equal to that of the distribution, i.e.

τ .

By substituting equation ( 9) into equation ( 1), one obtains the rate equation for the concentration of complexes in its conventional form as for the exponential distribution but with a rescaled mean dissociation time, % τ , and a cross section for complex formation, % α

( F ∝ α , hence % F = F % α / α ): & N(t) = % F(t) -N(t) / % τ , (10)
% τ = τκ -1 , % α = ακ , (11)
κ = (1 + γ ) -1 = 2τ 2 / t 2 . ( 12
)
Hence, the evolution of complex concentration with a non-exponential distribution of dissociation times is characterised by the same equilibrium state as for an exponential distribution with the same mean value but with a different characteristic time for approaching this state. The applicability limits of equations ( 9) and ( 10) can be estimated as those when the absolute value of the term for n =2, which is

τ 1 + γ ( ) 2 -δ     && N (t)
, where δ = t 3 / 6τ 3 , is much smaller than that of the term with the first order time derivative, γ & N (t) :

δκ 2 -1 << κ -1 Θ(t) , (13)
where

Θ(t) = & N / % τ &&
N , which is equal to unity for the exponential decay.

Equation (10) describes the exponential decay law. Hence, the rescaling can be viewed as a redefinition of the complex concentration that eliminates particles that do not follow the exponential distribution. Generally, the lowest energy must define the rescaled dissociation time, since the slowest process must govern the transient processes. Therefore, the new definition should limit complex formation events to those which occur when a migrating particle visits a site of the lowest interaction (highest binding) energy. Such a scenario is exactly that of the transition state theory. Hence, the factor κ in equation ( 12) has the meaning of the Eyring transmission coefficient [START_REF] Eyring | [END_REF]15] for complex formation rate and must be equal to the fraction of particles that reach the deep regions of a well and correct for reactive trajectories that re-cross the transition state and return without 'decomposition' (of a 'free' state) and formation of a 'bound' state. (Evidently, for the diffusion-controlled reactions, the transmission factor for the dissociation rate is equal to ατ j , where τ j is the mean time between jumps in perfect crystal. Hence κα is a common factor for both the formation and dissociation rates.)

To verify the validity of this analysis, we performed Monte Carlo (MC) simulations of particle diffusion in a system containing a 1-D triangular potential well of depth E , defined on an interval x ∈[-x max , x max ] :

U (x) = -E 1 -x / x max ( ) . (14)
In these calculations, the particle diffused in the system x ∈[0, x max ] with 'mirror' boundary conditions. The uphill and downhill jump frequencies, ν + and ν -, were calculated as is the mean waiting time before a jump. The calculations were performed for wells of different width (50, 100 and 200 a j ) and depth ( βE =3.87 and 7.74, which correspond to E =0.1 and 0.2 eV, respectively, at T =300K) and for statistics of ~500,000 dissociation events in each case.

ν ± = exp mβEa j / 2x max ( ) / τ j , where β -1 = k B T , k B being
Naturally, an event of complex formation is defined by the jump of a particle into the well and a dissociation event by a reverse jump. In this case, the calculations of the mean dissociation time for wide wells ( x max >> a j β E ) reproduce well the following equation derived by considering the partition function of the canonical ensemble [START_REF] Barashev | [END_REF]:

τ 1-D tri ≈ 2τ j x max / a j β E ( ) exp β E ( ) -1     . ( 15
)
The calculation results for 1 -P(t) , the so-called survival function, are shown in Figure 1 for wells of different width and depth. The range of well width was chosen to represent 1-D migration of an interstitial loop in the stress-field of an edge dislocation [11]. The error bars are small and not visible on the graph. The function 1 -P(t) is shown to decrease steeply at short times, while an exponential dependence evolves at long times. The deviations from the exponential distribution are quite large. For example, consider the curve calculated for β E =3.87 and x max =100 a j . At t = τ , the number of complexes is reduced by ~25 times, almost an order of magnitude higher than e for the exponential distribution. The variance of this distribution, σ t , is about seven times larger than for the corresponding exponential one.

One calculation for a square well (labelled 'Square') is also shown in Figure 1 and it demonstrates a dependence that is close to exponential decay. This is because any square well is characterised by single energy, in contrast to triangular well, in which depth of the well is not constant. In other words the square wells do not contain high-energy states (shallow regions) inside the well, from where a particle can dissociate without visiting deep regions. (Square wells may also demonstrate deviations from the exponential decay for sufficiently wide wells, such that the diffusion time of a particle over distances of the order of well width are comparable with the mean dissociation time from the well.)

[Insert Figure 1 about here] The curve for β E =3.87 and x max =50 a j , which is labelled ' x =0', is the distribution of dissociation times from the bottom of the well, that is, when all trajectories of diffusing particles started from x = 0 , the deepest position in the well. In this calculation, the mean dissociation time was ~15000τ j , more than an order of magnitude longer than 1200τ j , which is obtained from equation ( 15) and the corresponding MC calculation. This compares well with an equation for wide wells derived by the flux-over-population method pioneered by Farkas [START_REF] Farkas | [END_REF] (see Appendix A):

% τ 1-D tri ≈ 2τ j x max / a j β E ( ) 2 exp β E ( ) -1-β E     . ( 16
)
In addition, this calculation demonstrates an exponential decay, which indicates that the deviation from the exponential function is due to trajectories which do not reach the bottom of the well. Thus, by redefining the complex formation event as that occurring when a particle passes through the deepest point of the well, one restores the exponential distribution. This is because the work required to remove a particle from this point is unique.

According to our interpretation of rescaling, the factor κ -1 should be equal to the ratio of mean dissociation times given by the two definitions of complex creation events: (1) as a jump through x =0, the deepest point of the well, and (2) as a jump over the edge of the well,

x = x max . A comparison of κ -1 with this ratio calculated by MC is given in Figure 2. As seen, κ -1 is high for wide wells, but the redefinition of complex formation makes it close to unity. Before the redefinition, it compares well with the ratio of times, which confirms the interpretation. Note that the calculations suggest the following equation, which satisfies the limiting case of a nil effect for a single-site trap (at x max = a j ) and is equal to the ratio of equations ( 15) and ( 16) for wide wells:

κ 1-D tri -1 ≈ 1+ x max / a j -1 ( ) β E ( ) -1 1-β E exp -β E ( )     , ( 17
)
where the factor in square brackets is close to unity. As seen from Figure 2, equation ( 17) is in agreement with the MC calculations over the entire range of the potential well width used.

It also shows that κ -1 is higher than unity even for narrow wells, which may be important. We verified how the rescaling operates for the accumulation of complexes of diffusing particles with a 1-D triangular well at a constant formation rate, F . For this, we calculated P(t) by MC and performed the integration indicated in equation ( 3). The results are presented in Figure 3 (on the left) for x max / a j = 20 and β E =3.87, where the number of complexes is normalised by the steady-state value N max = τ F . The data on the right are for the annealing of the same complexes. As seen in the figure, the characteristic time for approaching the steady-state concentration is higher by a factor of κ -1 ≈ 6 as compared with the exponential dependence for a single-site trap with the same mean dissociation time,

p(t,τ ) = 1 -exp(-t / τ ) .
The solution of equation ( 10) p(t,κτ ) coincides with the MC calculations everywhere but at short times. The deviation is due to particles that reach only shallow regions of the well, of the order of thermal energy:

U(x max -% x) ≈ -k B T , % x =
x max / βE , before dissociation with the mean time τ th ≈ τ j %

x / a j . Indeed, as seen from Figure 3, the function p(t,κτ ) + p(t,τ th )τ th / τ , which is a superposition of the two processes, is close to the MC calculations.

[Insert Figure 3 about here]

The same conclusion can be reached by considering diffusion in a potential field with a sink at x =0. It can be shown that the same rate of particle absorption is achieved in the absence of the interaction field but for a sink at x = % x , where the interaction energy is -k B T (see Appendix B). Hence, particles passing beyond this energy level have a high probability of reaching the bottom of the well. Since we know from our calculations that dissociation of particles from the bottom of the well obeys exponential distribution, the condition x = % x should define a point of no return, hence such new well boundary, for which the distribution of dissociation times is exponential. The result is almost trivial. Indeed, the condition for the interaction energy between a migrating point defect (vacancy or interstitial atom) and an edge dislocation to be less than k B T is commonly used as absorbing boundary conditions (see, e.g. [20]). And the assumption of point of no return is a prerequisite for the validity of the classical transition-state theory (see, e.g. [15], p.261). To understand why the rescaling works, consider a simplified representation of the distributions in Figure 1 as a superposition of two random processes with the mean times

τ 2 << τ 1 : ′ P (t) = exp(-t / τ 1 ) / τ 1 + A exp(-t / τ 2 ) / τ 2 [ ] / (1 + A)
, where A is a constant. Then, equation ( 13) reads Aτ 2 << τ 1 , i.e. it requires the number of complexes to be defined by particles passing through the deepest site of the well. In other words, the time-scales for escaping from deep and shallow regions must be well separated. The calculations in Figures 2 and3 imply that this is valid despite the fact that such particles constitute only a small fraction of all particles visiting the trap region, i.e. A >> 1 . This is why N(t) is defined by τ 1 , which is higher than the mean time by a factor of 1 + A : τ 1 ≈ t (1 + A) . This factor can be obtained through the parameters of the distribution as κ -1 ≈ 1 + A , which explains the rescaling.

Conclusions

1. The dissociation of non-localised complexes can be described by the first-order kinetics as for random processes without memory. (The memory is transferred to the complex definition. Now the mere existence of a particle within the trap region is not enough to define a complex; for this, one should remember in addition whether it has visited a site of the maximum binding energy.) An amazing fact is that this remains valid even for very low values of κ , i.e. for large ratios of the rescaled and initial dissociation times, hence for such complexes as of an edge dislocation with an SIA cluster. This is due to the Arrhenius dependence of thermally-activated jumps, which makes the time-scales of escaping events from deep and shallow regions well separated.

2. A dividing surface between 'bound' and 'unbound' states defined as that where the interaction energy is equal to the thermal energy corresponds to the transmission factor equal to unity. Hence, particles that pass beyond this energy level with high probability reach the bottom of the well. This fact may be used in simulations to separate the timescales, e.g. of the fast diffusion of an SIA cluster in the matrix and slow dissociation from the region of strong interaction with edge dislocations. We note that, although this result has been obtained for triangular wells and thus needs

Thus, the same rate of particle absorption is achieved in the absence of the interaction field but with a sink at x = % x , where the interaction energy is -k B T . We note that, although this result has been obtained for triangular wells, it is independent of the parameters defining well shape and, hence, should be general.

3 time

 3 〈t 〉 = τ and the variance σ t 2 = 〈t 2 〉 -〈t〉 2 = τ 2 and D(t) assumes the familiar form for a first order reaction as a reciprocal of the mean time

9 [

 9 Insert Figure2about here]

Figure 1 .

 1 Figure captions

Figure 2 .

 2 Figure2. The dependence of κ -1 and the ratio of mean dissociation times for two complex definitions on the interaction range calculated by MC results for 1-D triangular wells.

Figure 3 . 1 Figure 1 . 2 Figure 2 . 3 Figure 3 .

 3112233 Figure 3. Time dependence of the number of complexes of migrating particles with 1-D triangular wells during steady complex formation and annealing.

  the Boltzmann constant and T the

	temperature, and a j is the jump distance. The value < t >= ν + + ν + (	) -1
	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y
		7

http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

Acknowledgments A.V.B. acknowledges a research grant from the UK Engineering and Physical Sciences Research Council. Research at ORNL was sponsored by the Division on Materials Sciences and Engineering (R.E.S. and Y.N.O.) and the Office of Fusion Energy Sciences (S.I.G.), U. S. Department of Energy, under contract no. DE-AC05-00OR22725 with UT-Battelle, LLC.

Consider a stationary 1-D diffusion problem for particles migrating in potential U(x) : ′ j (x) = 0 , (A1)

C(x max ) = 0 , (A3)

where j(x) is the flux, C(x) is the concentration of particles and D = a j 2 / 2τ j is the diffusion coefficient. Equation (A3) defines an absorption boundary, thus providing only first passage particles through it. Equation (A4) defines a constant source of particles at x =0 with % τ being the mean time between particle impingement. According to the flux-over-population method [START_REF] Farkas | [END_REF], % τ is equal to the mean first passage time of particles from the bottom of the well through the well boundary, if on average only one particle is in the well region:

For a triangular potential well U (x) = -E 1 -x / x max ( ) defined on an interval x ∈[0, x max ] , the solution of equations (A1)-(A4) is given by

where A = 2τ j x max / % τa j βE and B = βE / x max . By substituting equation (A6) into equation (A5) one obtains the mean dissociation time %

Consider the same diffusion problem as in Appendix A, equations (A1) and (A2), but with different boundary conditions, defining complete absorption of migrating particles at the bottom of the well at x = 0 :

For the triangular potential well, equation ( 14), the solution is given by

The flux at x = 0 is

In the absence of interaction field and with a sink at distance x max -% x (in other words, with complete absorption at this distance), the flux is

The two fluxes, equations (B4) and (B5), are equal to each other if

for which )