Now, we consider the formation of an interstitial dislocation loop along 3 in spinel. In spinel, O anion layers are fully dense triangular atom nets stacked in a ccp, ABCABC… geometry (A, B, and C are all distinct layer registries). Between adjacent O layers, 3/4 dense Al and MgAlMg layers are inserted, with regis­tries labeled a, b, and c in Table 1 (a cations have the same registry as A anions; likewise, b same as B, c same as C). For stacking fault layer stacking assessments in spinel, it is conventional to simplify the layer notation for the cations (see, e. g., Clinard et al6). The successive kagome Al layers are labeled a, p, g, while the MgAlMg mixed atom slabs are each pro­jected onto one layer and labeled a0, p0, g0. With these definitions, the registry of cation/anion stacking in spinel follows the sequence: a C p0 A g B a0 C p A g0 B.

As with alumina, when extra pairs of cation and anion layers are inserted into the spinel 3 stacking sequence, a C p0 A g B a0 C p A g0 B, a fault in the stacking sequence is introduced. One can demon­strate how this works by inserting a 1/6 (111) p A block into the stacking sequence described above (this is equivalent to the upper Burgers vector for spinel shown in Table 1, which uses a kagome Al cation layer). We obtain:

a C p0 A g B a0 C p A g0 B a C p0 A g B a0 C p A g0 B (before) a C p0 A g B a0 C p A g0 B p A a C p0 A g B a0 C p A g0 B (after) a C p0 A g B a0 C p A g0 B p A a C p0 A g B a0 C p A g0 B (after, showing stacking fault positions) [3]

Notice in eqn [3] that after block insertion, both the anion sublattice (CABCAB… stacking is not preserved) and the cation sublattice are faulted. Also, notice that the cation and anion stacking sequences are faulted on both sides of the inserted p A block (the layer sequences are broken approach­ing the block from both the left and the right). Thus, the p A block actually contains two stacking faults, on either side of the block. The positions of these stack­ing faults are denoted by vertical red lines in eqn [3]. The dislocation loop formed by 1/6 (111) block insertion in spinel is an extrinsic, cation+anion faulted, sessile interstitial Frank loop.

We can also consider inserting a 1/6 (111) p’ A block into the spinel stacking sequence (i. e., the lower spinel Burgers vector shown in Table 1, which uses a mixed MgAlMg cation slab). We obtain:

a C p’ A g B a’ C p A g’ B a C p’ A g B a’ C p A g’ B (before)

a C p’ A g B a’ C p A g’ B p’ A a C p’ A g B a’ C p A g’ B (after) a C p’ A g B a’ C p A g’ B | p’ A | a C p’ A g B a’ C p A g’ B (after, showing stacking fault positions) [4]

Once again, both the anion and cation sublattices are faulted, and we obtain an extrinsic, cation+anion faulted, sessile interstitial Frank loop.

Multiscale modeling hierarchically links various computational techniques over widely differing length and time scales.159 Multiscale modeling of He trans­port, fate, and consequences requires linking ab initio electronic structure, MD, kinetic Monte Carlo/Lattice Monte Carlo (KMC/KLMC), and mean field RT simulations to predict microstructural evolutions as an ultimate basis for modeling mechanical behavior. (see Chapter 1.08, Ab Initio Electronic Structure Calculations for Nuclear Materials; Chapter 1.09, Molecular Dynamics; Chapter 1.14, Kinetic Monte Carlo Simulations of Irradiation Effects; Chapter 1.15, Phase Field Methods and Chapter 1.16,

Dislocation Dynamics) In this section, we emphasize electronic-atomistic models of He behavior in Fe lattices.

If the elastic distortions associated with self­interstitials could be adequately treated with linear elasticity theory, and if the repulsive interactions between the dumbbell atoms with their nearest neigh­bors were like that between hard spheres, then the volume change of a solid upon insertion of an intersti­tial atom would be equal to the volume change AVas derived above. This follows from the analysis of the inclusion in the center of a sphere given in Appendix A. From the results listed in Table A2, under column INC, we see that the volume change of the solid with a concentration S of inclusions is simply given by where 3m is the volume dilatation per inclusion as if it were not confined by the surrounding matrix. This remarkable result has been proven by Eshelby21 to be valid for any shape of the solid and any location of the inclusion within it, provided the inclusion and the solid can be treated as one linear elastic material. In other words, the elastic strains within the inclusion and within the matrix must be small.

However, this is not the case for the elastic strains produced by self-interstitials. Here, the elas­tic strains are quite large. For example, the volume of the confined inclusion, also listed in Table A2 under column INC, is given by

Au 3m 3m 1 + v

u 1 + o gE 3^ 3(1 — v)

and so it is reduced to about 62% of the uncon­strained volume for a Poisson’s ratio of v = 0.3.

This amounts to an elastic compression of 42% of the ‘volume’ of the self-interstitial in fcc materials, and 25% for the self-interstitial in bcc materials. Clearly, nonlinear elastic effects must be taken into account.

Zener22 has found an elegant way to include the effects of nonlinear elasticity on volume changes pro­duced by crystal defects such as self-interstitials and dislocations. If Urepresents the elastic strain energy of such defects evaluated within linear elasticity theory, and if one then considers the elastic constants in the formula for U to be in fact dependent on the pressure, then the additional volume change dV produced by the defects can be derived from the simple expression
found by Schoeck.23 Its application to the strain energy of self-interstitials leads to the following result.

dv = rBV/m+MTK — i 1 3K + 4m к

12 (K m — K m) 1 U [37]

(3K + 4m)(9K + 8m) m к 1

Here, m and K’ are the pressure derivatives of the shear and bulk modulus, respectively. The first term arising from the dilatational part of the strain energy was derived and evaluated earlier by Wolfer.19 It is the dominant term for the additional volume change for self-interstitials in fcc metals. Here, we evaluate both terms using the compilation of Guinan and Steinberg24 for the pressure derivatives of the elastic constants, and as listed in Table 7.

The calculated relaxation volumes for self­interstitials,

Vrel = AV + dv [38]

are given in the eighth column of Table 7, and they can be compared with the available experimental values also listed.

We shall see that the relaxation volume of self­interstitials is of fundamental importance to explain and quantify void swelling in metals exposed to fast neutron and charged particle irradiations.

So far, we have ignored possible geometric prefer­ences between the constituent defects of a defect cluster. Of course, for oppositely charged defects, electrostatic considerations would drive the defects to sit as close as possible to one another, which would be described as a nearest neighbor configuration. However, as we saw in the previous section, defects can cause considerable lattice strain. Consequently, the most stable defect configuration will be dictated by a balance between electrostatic and strain effects.

To illustrate cluster geometry preference, we will consider simple defect pairs in the fluorite lattice, specifically in cubic ZrO2. These are formed between a trivalent ion, M3+, that has substituted for a tetra — valent lattice ion (i. e.,M’Zr) and its partially charge — compensating oxygen vacancy (i. e., VO*). This doping process produces a technologically important fast ion-conducting system, with oxygen ion transport via oxygen vacancy migration.2,

The lowest energy solution reaction that gives rise to the constituent isolated defects14 is

M2O3 + 2ZrXr! 2M’Zr + VO* + 2ZrO2 with the pair cluster formation following:

MZr + VO*! {M’zr + VO*}*

Figure 9 shows the options for the pair cluster geom­etry, in which, if we fix the trivalent substitutional ion at the bottom left-hand corner, the associated oxygen vacancy can occupy the first near neighbor, the sec­ond (or next) near neighbor, or the third near neigh­bor position.

Defect energy calculations have been used to pre­dict the binding energy of the pair cluster as a func­tion of the ionic radius15 of the trivalent substitutional

Figure 9 First, second, and third neighbor oxygen ion sites with respect to a substitutional ion (M3+).

ion.14 These suggest (see Figure 10) that there is a change in preference from the near neighbor configu­ration to the second neighbor configuration as the ionic radius of the substitutional ion increases. The change occurs close to the Sc3+ ion. Furthermore, the binding energy of the near neighbor cluster falls as a function of radius; conversely, the binding energy of the second neighbor cluster increases. Consequently, the change in preference occurs at a minimum in binding energy. The third neighbor cluster is largely independent of ionic radius. Interestingly, the minimum coincides with a maximum in the ionic conductivity, perhaps because the trapping of the oxygen vacancies as they move through the lattice is at a minimum.14

The change in preference for the oxygen vacancy to reside in a first or second neighbor site is a conse­quence of the balance of two factors: first, the Cou — lombic attraction between the vacancy and the dopant substitutional ion, which always favors the first neighbor site, and is largely independent of ionic radius, and second, the relaxation of the lattice, a crystallographic effect that always favors the second neighbor position. This is because, in the second neighbor configuration, the Zr4+ ion adjacent to the oxygen vacancy can relax away from the effectively positive vacancy without moving away from the effectively negative substitutional ion. Nevertheless, lattice relaxation in the first neighbor configuration contributes an important energy term. However, in the first neighbor configuration, the relaxation of oxygen ions is greatly hindered by the presence of larger trivalent cations, while small trivalent ions provide more space for relaxation. Thus, the relaxa­tion preference for the second neighbor site increases in magnitude as the ionic radius increases and conse­quently, the second neighbor configuration becomes more stable compared to the first.14

This example shows that even in a simple system such as a fluorite, which has a simple defect cluster, the factors that are involved in determining the clus­ter geometry become highly complex. Even so, we have so far only considered structural defects. Next, we investigate the properties of electronic defects.

Irradiation of fcc metals under energetic displace­ment cascade conditions induces the formation of stacking fault tetrahedra. Figure 34 shows an exam­ple of the formation of small dislocation loops and SFTs (triangle-shaped projected images) in copper due to irradiation with 750 MeV protons (2.5 MeV average PKA energy) at -90 °C to -0.7 dpa.302

The SFTs are thermally stable up to recovery Stage V. SFTs have been observed in numerous irradiated fcc metals, including aluminum,305

copper,12’53’146’302’306’307 nickel,304’307-309 silver,306’307 gold,307’310’311 palladium,310,312 and austenitic stain­less steels.53’96’307’312’313 Evidence from thin film and low-dose irradiation studies using ion beams or other energetic displacement cascade conditions suggests that SFTs can be formed directly in displacement cascades when the PKA energy exceeds a threshold value of ~5—10 keV, in agreement with molecular dynamics simulations.26,29 There are also several observations of SFT formation in some fcc metals due to point defect nucleation and growth during electron irradiation.299,305 The results from irradiations performed under energetic displacement cascade con­ditions at temperatures near recovery Stage I suggest that SFTs are not visible, perhaps due to insufficient rearrangement of the vacancy-rich core within the rapidly quenched displacement cascade.74,305

Graphite (C) is a very important material for nuclear energy applications. Graphite is a moderator used to thermalize neutrons in thermal gas and water-cooled reactors in the United Kingdom and the Soviet Union, respectively.45 Pyrolitic graphite is one of the barrier coating materials used in TRISO coated fuel particles.33 Graphite and carbon composites are also used as plasma-facing materials in fusion reactors.46 Numerous radiation effects studies have been performed on graphite. Nevertheless, the behav­ior of graphite in a radiation damage environment remains poorly understood. This is due primarily to the fact that graphite comes in so many forms and is produced in so many different ways, that in fact, the structure and chemistry of graphite used in nuclear applications is not a well-defined constant. Neverthe­less, there are some aspects of the crystal structure of graphite and the changes in this structure induced by irradiation that are somewhat analogous to the discussion of Al2O3 versus MgAl2O4, presented earlier in this chapter.

Graphite is a hexagonal crystalline material, with an ABAB. .. layer stacking arrangement of carbon sheets. These carbon layers have obvious hexagonal atom patterns in them. However, they are not fully dense triangular atom nets, as would be the case in a close-packed structure. They are so-called graphene sheets, in which the atom pattern is a honeycomb pattern, identical to the cation layer patterns in Al2O3 (see Section 1.05.2.3.1). Each C atom is sur­rounded by three nearest-neighbor C atoms, and the bonding linking each C atom with its neighbors is characterized by sp2 hybridization. The bonding that links adjacent graphene layers is weak, Van der Waals-type bonding.

The interstitial dislocation loops that form in irra­diated graphite, by the condensation of freely migrat­ing interstitial point defects, form (not surprisingly) on (0001) basal planes, between adjacent graphene layers. In some of the earliest work on radiation effects in graphite, this was described as follows47:

When subjected to bombardment with fission neu­trons, primary collisions displace carbon atoms from their normal sites in the layers, driving them to sites between planes (interstitial or interlamellar positions).

This loop nucleation is analogous to the (0001) basal interstitial loops that form in Al2O3 during the initial stages of irradiation (Section 1.05.2.3.1). However, the basal loops in graphite do not grow to any signifi­cant size. Instead, the graphene layers adjacent to interlamellar loop nuclei buckle, which causes a net increase in the c-dimension of the hexagonal material and a concomitant decrease in the a-dimension.48 This buckling is believed to be due to sp3 bond formation between C interstitials and C atoms in the graphene planes.49,50 The overall macroscopic effect of c-axis expansion and a-axis shrinkage is dimensional changes of crystallites within the graphite. Macroscopic radia­tion damage effects in graphite are discussed in detail in Chapter 4.10, Radiation Effects in Graphite.

For dislocation loops, the equilibrium vacancy con­centration can be found by a similar analysis. How­ever, as point defects are absorbed or emitted by the loop, it shrinks or expands, thereby changing its strain energy.

Let us first consider an interstitial loop in a crystal subject to a stress Sj. The force exerted on the loop plane is ti = , where n is a normal vector to the

loop plane. Upon forming the loop in the crystal, the atomic planes adjacent to the loop platelet are dis­placed by the Burgers vector b. Therefore, the work done by the external stresses is pR1biacjjnj, where pR1 is the area of the loop assumed to be circular in shape. Since the separation of the two atomic planes next to the loop platelet is n • b = nfa, the volume of the inserted platelet is pR2n, b, = NO, where N is the number of atoms in the loop. The work done by the external forces can then be written as b, Gjnj O/ (n, b,)

Note, when a vacancy loop is formed, the crystal contracts and the work performed by external forces is just the opposite, namely

sVLO = — b, ajnj О/(п, Ь,) [98]

Let us return to the case of an interstitial loop and consider the change in loop energy when a vacancy is emitted. This change is £Loop(N + 1) — ELoop (N) ffi dELoop/dN, where the number N of intersti­tials is related to the loop radius as pR2n, b, = NO. This change may be considered as a vacancy chemi­cal potential associated with the prismatic interstitial loop of radius R, and it may be computed according to the equation

dELoop O dELoop

dN 2p (n, b, )R dR

Following our procedure now, we consider the total change in Gibbs free energy when a vacancy is cre­ated in the crystal lattice next to the dislocation core and the extra atom is attached to the loop platelet. This change is

AG = — 4 + Tf — Ei(rc,’) + O^L + VvRs°H

— AmVoop — kT indL [ioo]

and it must be equal to zero when local thermody­namic equilibrium prevails. This condition then determines the local vacancy concentration as

EI(rC;’) OsIL+ VV AmV P

‘ kT kT kT

Loop

AmV p

‘ kT

The local equilibrium vacancy concentration at an interstitial-type loop differs from the value at a
straight edge dislocation mainly by an exponential factor containing the vacancy chemical potential change. In fact, it is lower than for a straight edge dislocation, as will be shown below.

For a vacancy-type loop, the sign in this exponen­tial factor is positive, and the local equilibrium vacancy concentration is enhanced compared to the straight edge dislocation according to

C ffi c? exp(Ammr!) [101]

To quantify how different these local vacancy con­centrations can be from the true equilibrium concen­tration, we consider the case of perfect prismatic loops. The Burgers vector b in this case is parallel to the normal vector n, and n, b, = b. The energy is given by36,37

; R{K(1 —rC)— E(1— rC)} + pr1 tsf

— + pRl»sf I103]

where K and E are the complete elliptic integrals. For large loop radii R much larger than the core radius rc, the approximation given in the second part of the above equation can be used. The second term in addition to the strain energy represents the energy of the stacking fault. Note that the shear modulus m is not to be confused with the vacancy chemical potential AmV.

Taking the derivative of the above equation with respect to N gives the vacancy chemical potential difference as

AmVoop(R)=—

+ OgSF [104]

Inserting this result into eqns [101] and [101] gives the ratio of the vacancy concentration near the dislo­cation core of prismatic dislocation loops relative to the equilibrium concentration [93]. As an example, these ratios are evaluated for Ni and at a temperature of 773 K and shown in Figure 22. Both the exact expression for the loop energy and the logarithmic one in eqn [103] are employed to demonstrate that the latter provides an excellent approximation. How­ever, for small prismatic loops, both expressions for the loop energy become questionable, and atomistic calculations are necessary to obtain energies ofsmall, plate-like clusters of self-interstitials or of vacancies.

Irradiation temperature typically invokes a very large influence on the microstructural evolution of irra­diated materials. There are several major tempera­ture regimes delineated by the onset of migration of point defects. Early experimental studies used iso­chronal annealing electrical resistivity measurements on metals irradiated near absolute zero temperature to identify five major defect recovery stages.61-64 Figure 6 shows the five major defect recovery stages for copper irradiated with electrons at 4 K.65 The quantitative magnitude of the defect recovery in each of the stages generally depends on material, purity, PKA spectrum, and dose. Based on the cur­rently accepted one-interstitial model, Stage I corre­sponds to the onset of long-range SIA migration. Stage I often consists of several visible substages that have been associated with close-pair (correlated) recombi­nation of Frenkel defects from the same displacement event and long range uncorrelated recombination of defects from different primary displacement events. Stage II involves migration of small SIA clusters and SIA-impurity complexes. Stage III corresponds to the onset of vacancy motion. Stage IV involves migration of vacancy-impurity clusters, and Stage V corresponds to thermal dissociation of sessile vacancy clusters. It should be noted that the specific recovery stage tem­perature depends on the annealing time (typically 10 or 15 min in the resistivity studies), and therefore needs to be adjusted to lower values when considering the onset temperatures for defect migration in typical

Figure 5 Comparison of the molecular dynamics simulations of 1-50 keV PKA displacement cascades in iron. PKA energies of 1 (red), 10 (green), and 50 (blue) keV for times corresponding to the transient peak number of displaced atoms are shown. The length of the Z (horizontal) dimension of the simulation box is 170 lattice parameters (49 nm). Adapted from Stoller, R. E., Oak Ridge National Lab, Private communication, 2010.

Source: Eyre, B. L. J. Phys. F1973, 3(2), 422-470.

Zinkle, S. J.; Kinoshita, C. J. Nucl. Mater. 1997, 251,200-217.

Schilling, W.; Ehrhart, P.; Sonnenberg, K. In Fundamental Aspects of Radiation Damage in Metals, CONF-751006-P1; Robinson, M. T.; Young, F. W., Jr., Eds. National Tech. Inform. Service: Springfield, VA, 1975; Vol. I, pp 470-492.

Hautojarvi, P.; Pollanen, L.; Vehanen, A.; Yli-Kauppila, J. J. Nucl. Mater. 1983, 114(2-3), 250-259.

Lefevre, J.; Costantini, J. M.; Esnouf, S.; Petite, G. J. Appl. Phys. 2009, 106(8), 083509.

Schultz, H. Mater. Sci. Eng. A 1991, 141, 149-167.

Xu, Q.; Yoshiie, T.; Mori, H. J. Nucl. Mater. 2002, 307-311(2), 886-890.

Young, F. W., Jr. J. Nucl. Mater. 1978, 69/70, 310.

Hoffmann, A.; Willmeroth, A.; Vianden, R. Z. Phys. B 1986 62, 335.

Takamura, S.; Kobiyama, M. Rad. Eff. Def. Sol. 1980, 49(4), 247.

Kobiyama, M.; Takamura, S. Rad. Eff. Def. Sol. 1985, 84(3&4), 161.

neutron irradiation experiments that may occur over time scales of months or years. Table 1 provides a summary of defect recovery stage temperatures for several fcc, bcc, and HCP materials.8,18,63,66-69 Although there is a general correlation of the recovery temperatures with melting temperature, Table 1 shows there are several significant exceptions. For example, Pt has one of the lowest Stage I temperatures among fcc metals despite having a very high melting temperature. Similarly, Cr has a much higher Stage III temperature than V or Nb that have higher melting points. As illustrated later in this chapter, the micro­structures of different materials with the same crystal structure and irradiated within the same recovery stage temperature regime are generally qualitatively similar.

Several analytic kinetic rate theory models have been developed to express the dose dependence of defect cluster accumulation in materials at different temper­ature regimes.6,70-72 In the following, summaries are provided on the experimental microstructural obser­vations for five key irradiation temperature regimes.

1.04.7.1 Introduction

The class of ferritic-martensitic alloys with chro­mium concentrations in the range of 9-12% has attracted interest in the fast reactor programs because of its radiation resistance, in particular, very low swelling and low irradiation creep. Alloys such as Sandvik HT-9 (12Cr1Mo.6Mn.1Si.5W.3V)) and other alloys of this class were irradiated in the EBR-II,24 in research reactors25 and with heavy ions.26 The quantitative results from the ion irradia­tions in this class of alloys and the low neutron absorption cross section led to inclusion of ferritic alloys into the fast reactor alloy development pro­grams, in particular in the United States in the mid-1970s. The radiation resistance has been con­firmed to displacement levels of 70 dpa.12,14

Further interest in this class of alloys was initiated by the fusion reactor programs in Europe and the United States when the necessity for low neutron activation structural materials was realized. Further
research on martensitic alloys by fusion programs in Europe, the United States, and Japan led to the devel­opment of low-activation alloys by replacing elements that result in long-term activation products. Molyb­denum and niobium, both of which result in long — lived activation products, were replaced by tungsten and tantalum. This research led to radiation-resistant alloys with a fracture toughness superior to that of the commercial alloys even in the unirradiated condi­tion.27 The compositions of representative members of this class of alloys referred to in this chapter are presented in Table 1. An excellent review of irradia­tion behavior of this class of alloys has been published by Klueh and Harries.27 Details of the metallurgy of martensitic alloys appears in Chapter 4.03, Ferritic Steels and Advanced Ferritic-Martensitic Steels.

The previous section included examples of void swelling. Voids result from the clustering of vacancies produced by displacement damage, as characterized by the number of dpa. Atomic displacements produce equal numbers of vacancy and SIA defects. As noted previously, descriptions of swelling mechanisms, including the role of He, can be found in excellent reviews.113-116 Early RT models showed that swelling is due to an excess flux of vacancies to voids, which is a consequence of a corresponding excess flux of SIA to biased dislocation sinks.106,107 Typical dis­placement rates (Gdpa) in high-flux reactors (HFR) are «10-6-10-7dpas—1 Hence, an irradiation time of 108s («3 years) produces up to «100 dpa. Only about 30% of the primary defects survive short­time cascade recombination.137 The residual defects undergo long-range migration and almost all either recombine with each other or annihilate at sinks. However, a small fraction of SIA and vacancies cluster to form dislocation loops and cavities, respectively. Ultimate survival of only 0.1% of the dpa in the form of clustered vacancies leads to 10% swelling at 100 dpa.

Classical models138,139 demonstrated that for the low Gdpa in neutron irradiations, homogeneous void nucleation rates are very low at temperatures in the peak swelling regime for AuSS between about 500 and 600 °C. However, heterogeneous void nucleation on He bubbles is much more rapid than homogeneous nucleation.109 Indeed, nucleation is not required when the He bubbles reach a critical size (r*) and He content (m*). The CBM concept has provided a great deal of insight into the effects of He on swelling.15,109- 112,114,118,130-133,140-151 In particular, the CBM rationa­lized the extended incubation dpa in fast reactor irradiations prior to the onset of rapid swelling. As previously shown in Figure 2(d) and 2(e), here we clearly distinguish between bubbles, which shrink or grow only by the addition of He, from larger voids, which grow unstably by the continuous accumulation of vacancies. In the case of bubbles, the gas pressure (p) plus a chemical stress due to irradiation (see Section 1.06.3.4) just balances the negative capillary stress 2g/ rb, where g is the surface energy and r> the bubble radius. By definition dr,/dt = 0 for bubbles, while the growth rate is positive and negative for cavities that are slightly smaller and larger than rb, respectively. In the case of voids (v), drv/dt is positive at all rv greater than the critical radius. Voids are typically underpres­surized with p<< 2g/rv More generally, cavities include both bubbles and voids and can contain an arbitrary number of vacancies (n) and He atoms (m).

The evolution of the number of discrete vacancy (n)-He (m) cavities, N(n, m), in a two-dimensional n—m space can be numerically modeled using cluster dynamics (CD) master equations. In the simplest case of growth or shrinkage by the absorption or emission of the monomer diffusing species (He, vacancies, and SIA), an ordinary differential equation (ODE) for each n, m cluster, dN(n, m)/dt, tracks the transitions from and to all adjacent cluster classes (n ± 1 and m ± 1), as characterized by He, vacancy, and SIA rates of being absorbed (bHe, v,0 and the vacancy emission (av) rate, as dN (n, m)/dt = bne (n, m — 1)N (n, m — 1)

+ bv(n — 1, m)N (n — 1, m) — av(n + 1, m)N (n + 1, m)

+ bi(n + 1, m)N (n + 1, m) — [bHe(n, m) — bv(m, n)

— bi(n, m)]N (n, m) [1

Note that thermal SIA and He emission rates are low and need not be included in eqn [1]. However,

He may be dynamically resolutioned by displace­ment cascades.152’153 There are a total of nmax x mmax such coupled ODEs. The rate coefficients, a and b, are typically computed from solutions to the diffu­sion equation, to obtain cavity sink strengths,107’113-116 along with the concentrations of the various spe­cies in the matrix and vacancies in local thermody­namic equilibrium with the cavity surface. The local vacancy concentrations are controlled by the surface energy of the void, g, via the Gibbs Thomson effect, and the He gas pressure.109,139,141 Conservation equa­tions are used to track the matrix concentrations of the mobile He, vacancies, and SIA based on their rates of generation, clustering, loss to all the sinks present, and, for the point defects, vacancy-SIA

recombination.144

Similar RT CD methods can also be used to simultaneously model SIA clustering to form dislo­cation loops, as well as climb driven by the excess flux of SIA to network dislocations.111,144 In AuSS, loop unfaulting produces network dislocations, and net­work climb results in both production and annihila­tion of the network segments with opposite signs. Thus, dislocation structures evolve along with the cavities.

However, the a and b rate coefficients depend on a number of defect and material parameters that were not well known during the period of intense research on swelling in the 1970s and 1980s, and integrating a very large number of nmax x mmax coupled ODEs was computationally prohibitive at the time these models were first proposed. One simplified approach, based on analytically calculating the rate of void nucleation on an evolving distribution of He bubbles, coupled to a void growth model provided consider­able insight into the role of He in void swelling.109,111 These early models, which also included parametric treatments of void and bubble densities,1 — 2 led to the correct, albeit seemingly counterintuitive, predictions that higher He may decrease, or even totally suppress, swelling in some cases, while in other cases swelling is enhanced, or remains unaf­fected. These early models also predicted the forma­tion of bimodal cavity size distributions, as confirmed by subsequent modeling studies and many experi­mental observations.111’112’114’118’131’133’134’148’151

Most aspects of void formation and swelling incubation can be approximately modeled based on the CBM concept. A critical bubble is one that has grown to a radius (r*) and He content (m*), such that, upon the addition of a single He atom or vacancy, it immediately transforms into an unstably growing void (see Figure 2(d) and 2(e)) without the need for statistical nucleation. Note that while a range of n and m clusters are energeti­cally highly favorable compared with equal numbers of He atoms and vacancies in solution, bubbles rep­resent the lowest free energy configuration in the vacancy-rich environments, characteristic of mate­rials experiencing displacement damage. That is, in systems that can swell due to the presence of sink bias mechanisms that segregate excess fractions SIA and vacancies to different sinks and at low reactor relevant damage rates, cavities primarily evolve along a bubble path that can ultimately end in a conversion to voids.