A number of different point defects can form in all ceramics, but their concentration and distributions are interrelated. In the event of the production of a vacancy by the displacement of a lattice atom, this released atom can be either contained within the crystal lattice as an interstitial species (forming a Frenkel pair), or it can migrate to the surface to form part of a new crystal layer (resulting in a Schottky reaction). Figure 1 represents a Frenkel pair: both cations and anions can undergo this type of disor­der reaction, resulting in cation Frenkel and anion Frenkel pairs, respectively. In ceramic materials, both the vacancy and interstitial defects are usually charged, but the overall reaction is charge-neutral. The energy necessary for this reaction to proceed is the energy to create one vacancy by removing an ion from the crystal to infinity plus the energy to create one interstitial ion by taking an ion from infinity and placing it into the crystal. The implication of remov­ing and taking ions from infinity implies that the two species are infinitely separated in the crystal (unlike the two species shown in Figure 1). As separated species, these are defects at infinite dilution, a

well-defined thermodynamic limit. As the two defects are charged, they will interact if not infinitely sepa­rated, a point we will return to later.

As with the Frenkel reaction, the Schottky disorder reaction must be charge-neutral. Here, only vacancies are created, but in a stoichiometric ratio. Thus, for a material of stoichiometry AB, one A vacancy and one B vacancy are created. The displaced ions are removed to create a new piece of lattice. It is impor­tant to realize that we are dealing with an equilibrium process for the whole crystal. Thus, as the tem­perature changes, many thousands of vacancies are created/destroyed and new material containing many thousands of ions is formed. Thus, it is not simply that one new molecule is formed, but there is also an increase in the volume of the crystal, which is why the lattice energy is part of the Schottky reaction.

The energy for the Schottky disorder reaction to proceed in an AB material is the energy to create one A-site vacancy by removing an ion from the crystal to infinity plus the energy to create one B-site vacancy by removing an ion from the crystal to infinity, plus the lattice energy associated with one unit of the AB compound. For example in Al2O3, the energy would be that associated with the sum of two Al vacancies plus three O vacancies plus the lattice energy of one Al2O3 formula unit. Again, the vacancy species are assumed to be effectively infinitely separated.

In a crystalline material with more than one type of atom, each species usually occupies its own sub­lattice. If two different species are swapped, this pro­duces an antisite pair (see Figure 2). For example, in an AB compound, one A atom is swapped with the B atom. While this would be of high energy for an AO compound, where A and O are of opposite charge (e. g., Mg2+ and O2—), in an ABO3 material where A and B may have similar or even identical positive charges, antisite energies can be

chemistry terms, a lattice

site means a position in the crystal that an ion will usually occupy in that crystal structure), the number of ways, OA, of arranging n A-site vacancies is

N!

n!(N — n)!

As we have n B-site vacancies to distribute over N, B-lattice sites, the total number of configurations is the product of OA and OA:

N! ‘

n!(N — n)!

( N! ‘

n!(N — n)!

where N and n are large, as they are when dealing with crystals, we can invoke Stirling’s approxima­tion, which states that ln(MZ) = M ln(M) — M. Thus,

ASc = 2k[N ln(N) — (N — n)ln(N — n) — nln(n)]

Therefore,

AG=nAgf — 2kT [N ln(N) — (N—n)ln(N—n) — nln(n)]

=nAgf — 2kT

To find the equilibrium number of defects, we need to find the minimum of AG with respect to n (see Figure 3). That is

0=Agf — 2kT ln

Assuming that the number of defects is small in comparison to the number of available lattice sites, then N n N:

n Ag Ah As

N =exp — Ш =exp — rn exp 2k

Usually, we assume that the energy associated with the change in vibrational entropy is negligible so that the concentration of defects (n/N) is dominated by the enthalpy of reaction:

M=N=“p(-2T)

However, this is not always a valid assumption and care must be taken. When defect concentrations are measured experimentally, they are presented on an Arrhenius plot of ln(concentration) versus 1/ T which yields straight lines with slopes that are pro­portional to the disorder enthalpy (see Figure 4).

Irradiation growth (due to anisotropic nucleation and growth of dislocation loops on different habit planes) can be of significant practical concern at intermediate temperatures in anisotropic materials such as Zr alloys, Be, BeO, Al2O3, uranium, and graphite.40’126’259-261 Anisotropic growth in individual grains in polycrys­talline materials can produce large grain boundary stresses, leading to loss of strength and grain bound­ary fracture in some materials. Figure 27 shows the large anisotropy in measured lattice parameter change in the basal and prism planes for BeO irradiated near room temperature.262 For neutron fluences above 2 x 1020 cm-2 (~0.2 dpa) with a c-axis expansion >0.5% and an a-axis expansion near 0.1%, a rapid decrease in flexural strength was observed.262,263 In materials with highly textured grains, unacceptable anisotropic growth at the mac­roscopic level can occur. One engineering solution is to use processing techniques to produce randomly aligned, small grain-sized materials.

Irradiation creep occurs in the presence ofapplied stress, due to biased absorption of point defects at cavities and along specific dislocation orientations relative to the applied stress.264 Irradiation creep produces dimensional expansion that acts in addition to normal thermal creep mechanisms and is most

prominent at temperatures from recovery Stage III up to temperatures where thermal creep deformation becomes rapid (typically above 0.5 TM). The magni­tude ofsteady-state irradiation creep is proportional to the applied stress level and dose, and consists of a creep compliance term and a void swelling term. The magnitude of typical irradiation creep compli­ance coefficients260,265,266 for fcc and bcc metals

is 0.5-1 x 10-12 Pa-1 dpa-1. The irradiation creep compliance for ferritic/martensitic steels appears to be about one-half of that for austenitic steels.109 Accelerated irradiation creep due to differential absorption of point defects at low temperatures (e. g. below recovery Stage V) or at low doses can produce creep deformation rates that are up to 10-100 times larger than the steady-state irradiation

267,268

creep rates.

Volumetric swelling from void formation occurs at temperatures above recovery Stage V in fcc and HCP materials (and above Stage III for bcc materi­als), and typically exhibits a linear increase with dose after an initial transient regime. As summarized in Figure 28 the dose-dependent swelling in fast fission reactor-irradiated austenitic stainless steel progresses

at a swelling rate of ^1%/dpa without evidence for saturation up to swelling levels approaching 100%.1 Similar high swelling levels without evidence of sat­uration have been observed in pure copper108 and some simple bcc alloys.131 Volumetric swelling levels in structural materials in excess of ^5% are difficult to accommodate by engineering design,269 and addi­tional embrittlement mechanisms may appear in aus­tenitic stainless steel for swelling levels above 10% including void channeling and loss of ductility.270,271 Therefore, there is strong motivation to design struc­tural materials that are resistant to void swelling by introducing a high matrix density of point defect sinks or other techniques. In general, the amount of void swelling is lower in bcc materials compared to

fcc materials.50,92,109 For example, the observed void

swelling in many ferritic/martensitic steels is <2% after fission neutron damage levels of 50 dpa or higher, whereas the void swelling in simple austenitic stainless steels may be 30% or higher.109 The superior swelling resistance in ferritic/martensitic steels is largely due to a higher transient dose before onset of steady-state swelling, along with a lower steady-state swelling rate. For many HCP materials, the amount of void swelling is relatively small compared to fcc materials due to anisotropic point defect migration that tends to pro­mote defect recombination.128 However, the potential for anisotropic swelling associated with cavity formation in HCP materials may induce large stresses and poten­tial cracking at grain boundaries.263,272,273 Figure 29 shows an example of aligned cavity formation and grain boundary separation in Al2O3 following fast fis­sion reactor irradiation.272

Having examined the crystallography of unfaulting reactions in alumina and spinel, it is interesting now to compare and contrast what is experimentally observed so far as dislocation loop evolution in irra­diated Al2O3 versus MgAl2O4 is concerned. First, it is observed that the 1/3 [0001] (0001) basal loops and 1/3(1010){1010} prismatic loops readily unfault under irradiation, by the reactions shown in eqns [7,8] and [9,10], respectively.6 These reactions occur when the loops reach ^50 nm diameter,10 and each reaction produces an unfaulted loop with a 1/3(1011) perfect Burgers vector. Once formed, these unfaulted loops grow without bound until they intersect other growing dislocation loops, ultimately forming a dis­location network. Such a dislocation network in neu­tron irradiated Al2O3 is shown in Figure 6.

Once the dislocation network in irradiated alu­mina is formed, it has been demonstrated that the product dislocations within the network are free to climb6 The continuous climb of network disloca­tions in Al2O3 provides unsaturable sinks for Al and O interstitials arriving in stoichiometric proportions. All the conditions for a substantial supersaturation of vacancies are now in place. Al and O interstitials are readily absorbed at network dislocations, leaving behind numerous unpaired Al and O vacancies in the lattice. These unpaired vacancies inevitably con­dense to form voids. Under these conditions, void swelling must be the anticipated radiation response of the material.

Contrast the evolution described above for alumina to the observed microstructural evolution in spinel. The predominant 1/4 (110) {110}

dislocations in spinel do not unfault under most experimental conditions tested to date.6 (Kinoshita eta/.51 observed unfaulted 1/2 (110) {110} perfect loops in MgAl2O4 single crystals following neutron irradiations in the JOYO fast breeder test reactor (fast neutron fluences up to 6.5 x 1025nm~2 (equivalent to 6.5 dpa), and temperatures between 673 and 873 K). Kinoshita eta/.51 also proposed a growth pro­cess of loops in spinel as follows: 1/6 [111] (111) 1/4 [110] (111) 1/4 [110] (101) 1/4 [110] (110) 1/2 [110] (110). Notice that this sequence ends in an unfaulted, perfect interstitial loop. This final loop configuration should be a good sink for interstitials, thus promoting a supersaturation of vacancies in the lattice. However, in neutron irradiations of MgAl2O4 single crystals in the fast flux test facility (FFTF), no evidence for 1/2 [110] (110) perfect dislocations was found, for neutron fluences ranging from 2.2 x 1026 to 2.17 x 1027nm~2 (equivalent to 22-217dpa) in the temperature range 658-1023 K.51 Therefore, the proposed progression of spinel interstitial loop characteristics described above has, to date, been confirmed only under the JOYO irradiation condi­tions reported by Kinoshita et a/.51) According to Clinard eta/.:6

Persistence of the 1/4 (110) {110} stacking fault amounts to a failure of a 1 /4(112) partial dislocation to nucleate, sweep across the loop plane, and so remove the fault.

The reason for this failure is paradoxical. Apparently, stacking fault energy cannot be the reason. The stack­ing fault energy estimate for 1/4 (110) {110} stacking faults in spinel, 180 mJ m~2,18 is similar to the energy estimates for 1/3 [0001] (0001) and 1/3(10І0){10І0} stacking faults in alumina (320 and 750 mJ m~2, respectively).10 Therefore, there seems to be a rea­sonable ‘driving force’ available to favor unfaulting of 1/4 (110) {110} stacking faults in spinel. Perhaps the explanation is simply that the magnitude of the par­tial shear vector required to unfault the faulted loops is prohibitively large. In spinel, the magnitude of the unfaulting 1/4(112) vector is ~5A, compared with the 1/3 [0001] (4.32 A) and 1/3(1010) (2.74 A) unfaulting vectors in alumina.

Whatever the reason, spinel 1/4 (110) {110} stacking faults do not unfault, and this leads to void swelling resistance and impressive inherent radiation tolerance in spinel compared alumina. Hobbs and Clinard summarize the situation as follows:

The absence of void swelling ( in spinel ) can be attributed to the failure of the loops to unfault and develop into dislocation networks; they therefore remain less than perfect interstitial sinks since the energy per added interstitial never drops below the fault energy. Vacancy-interstitial recombination thus remains the dominant mode ofdefect accommo­dation, and saturating defect kinetics inevitably ensue.

Therefore, in conclusion, the significant swelling of Al2O3 alumina at high temperatures is attributable to the unfaulting of interstitial dislocation loops and the subsequent formation of dislocation networks, which serve as efficient sinks for the absorption of intersti­tial atoms. This leaves behind a supersaturation of lattice vacancies, that is, an excess of unpaired vacan­cies in the bulk of the Al2O3.

In irradiated MgAl2O4, only high-energy faulted loops are available as sinks for interstitials. Therefore, in this case, interstitial-vacancy (i-v) recombination is the dominant mechanism for defect accommoda­tion, and negligible swelling results.

Developing fusion as a large-scale energy source and high-energy proton accelerator-based technologies are the primary motivations for studying He effects in structural alloys. These environments produce copious quantities of He (and H) by transmutation reactions. High levels of He, coupled with displacement (dpa) radiation damage, lead to a wide variety of property degradation phenomena over a wide range of irradia­tion temperatures, including both severe embrittle­ment and dimensional instabilities of various types.

Indeed, there is growing evidence that He-dpa syner — gisms will severely limit the operating window for leading candidate FMS in fusion first wall structures.

Ultimate resolution of He effect issues in fusion applications will require a dedicated high-energy neutron source to develop an information base to construct and verify rigorous physically based pre­dictive models of the effects of the fusion environ­ment on performance-sustaining properties. These models must account for the interactions of a large number of variables, characterizing both the irra­diation service environment and alloy of interest. However, in the interim, there are a number of irradiation techniques that can be used to simulta­neously introduce high levels of He and dpa into materials. The most notable irradiation techniques include multibeam charged particle irradiations (CPI), ISHI in mixed spectrum fission reactors, and spallation neutron sources. Each of these methods has limitations, but significant progress in understanding He effects has been, and will continue to be, achieved by closely integrating all of these irradiation tools with advanced physical models.

A primary objective of such modeling, and asso­ciated experiments, is to understand and predict the transport, fate, and consequences of He, and its inter­actions with displacement damage. There is a very large historical literature on irradiation effects in AuSS, including the key role played by He. Helium is critical to void swelling and high-temperature embrittlement (HTHE), where the latter is mani­fested as severe reductions in creep rupture times and strains. Standard AuSS provide an excellent basis
for developing an understanding of these phenomena, since they are (a) sensitive to many manifestations of irradiation damage and especially He effects; and (b) generate high levels of He from two-step Ni thermal neutron reactions in mixed spectrum fission reactors (typically of the order 50appm He/dpa) and much lower, but significant, amounts of He (<1 appm He/dpa) in fast reactors. Both void formation and HTHE are characterized by a significant incubation period prior to forming growing cavities followed by rapid swelling or creep rupture.

Helium bubbles are typically the formation sites for both voids and grain boundary creep cavities. RT-based thermodynamic-kinetic models rationalize many important trends in these phenomena. In par­ticular, the critical bubble concept relates the combi­nation irradiation variables, of temperature and dpa rates, along with a number of material-defect vari­ables and parameters, to the critical size and helium content of bubbles that convert to voids, due to dislo­cation bias for SIA or grain boundary creep cavities due to stress. The critical bubble concept rationalizes and provides a basis to quantitatively model the incu­bation periods for both of these phenomena. RT can also be used to model post-incubation swelling rates and creep rupture times and strains. These models predict a number of important observations like bimodal bubble-void cavity size distributions and precipitate-associated bubbles and voids.

Critical bubble-cavity growth models highlight the critical roles played by the overall He bubble microstructures on irradiation effects. Bubbles are necessary for the formation of voids and large
numbers of creep cavities. However, high concentra­tions of bubbles, as generally associated with high He generation rates, can prolong incubation times by increasing the number of He partitioning-trapping sites in small highly subcritical bubbles; and, in the case of swelling and other manifestations of matrix radiation displacement damage. High bubble densi­ties reduce the excess vacancy supersaturations and excess defect fluxes by acting as dominant defect sinks. These insights provide important guidance to developing irradiation-tolerant alloys. Enhanced damage tolerance can be achieved by creating fine — scale and stable microstructural features that can form small, harmless bubbles that sequester high levels of He suppressing swelling and protecting GBs from HTHE and grain boundary decohesion, leading to enormous DBTT shifts. Swelling-resistant advanced stainless steels, with extended incubation times, have used alloy carbide and phosphide phases to manage He in this manner. FMS, with high disloca­tion densities and fine lath structures, are intrinsically more damage resistant than standard austenitic alloys for a variety of reasons. These reasons include low He generation rates in fission reactors. However, the rates of He generation are much higher in D-T fusion spectra, and the irradiation damage tolerance of FMS may be significantly degraded in this case.

SPNI have been the primary source of recent insight into the effects of He in structural alloys, especially mechanical property effects. Perhaps the most significant result of the SPNI studies is that high He-hardening synergisms can lead to enormous shifts in the DBTT shifts and IG fracture in FMS. The mechanisms responsible for such synergistic embrittlement, which can lead to transition tempera­ture elevations of 600 °C or more, are He-induced weakening of GBs as well as enhanced hardening at higher temperatures and higher dpa levels. Thus, there is concern that severe embrittlement and void swelling may close the window for useful application of these alloys in fusion environments.

A parallel set of activities is needed to develop predictive models of He effects. Integrated master models, based on a multiscale-multiphysics paradigm, are being developed to predict He transport, fate, and consequences in realistic alloy microstructures. These master models contain a number ofparameters and must be both mechanistically and microstructu­rally informed. First-principles electronic structure theory and atomistic simulations can provide required model parameters and mechanistic insights. These tools include DFT, embedded atom-based

MD and MS, various Monte Carlo methods, and RT. For example, recent first principles and atomistic research has provided important information on He-vacancy cluster energetics, He interactions with a range of microstructural features, He diffusion mechanisms, and rates in the matrix, along dislocations and in GBs. These models show high He-vacancy binding energies even at the smallest cluster sizes, strong interactions between He and dislocations and dislocation jogs, and both homo — and heterophase interfaces. MD methods have also been used to char­acterize phenomena such as resolutioning of He from bubbles, showing this to be a minor mechanism; and determining the dislocation interaction strength of cavities, ranging from under to overpressurized con­ditions, showing that near-equilibrium bubbles are the strongest obstacles. Recent MD studies have also sug­gested that capillary models may overpredict He gas pressures in equilibrium bubbles.

As noted above, the concepts and models described above can be used to guide the development of irradiation-tolerant alloys. The most promising class of such materials have been dubbed NFA, which contain an ultrahigh density of remarkably stable, nanometer-scale Ti-Y-O NF. The NF are now believed to be primarily a complex oxide Y2Ti2O7 pyrochlore phase. The NF provide remarkable high — temperature creep strength, so that NFA can operate above the displacement damage regime, where recov­ery processes are much faster than defect accumula­tion rates. More significantly the NF trap helium in a very high density of very fine-scale bubbles. In prin­ciple, the bubbles suppress or mitigate essentially all manifestation of radiation damage and property deg­radation. Limited proof in principle validation of the ability of NFA to manage He has been provided by ISHI irradiations that were carried out in the HFIR. For example, an irradiation of NFA MA957 to 9 dpa and 380 appm He produced a very high density of nanometer-scale bubbles on the NFs and dislocations. The same irradiation of the FMS F82H produced roughly an order of magnitude lower density of cav­ities, composed of a bimodal distribution of bubbles and larger voids. Bubbles primarily form on disloca­tions in this case. Further, limited data showed rela­tively bubble-free interfaces in MA957, in contrast to interfaces in F82H that are highly decorated with small bubbles.

The results ofthe ISHI experiments compare very favorably with a He transport and fate master model that is under development. The master model is both microstructurally informed and incorporates parameters derived from the atomistic/electronic models cited above. Preliminary RT models have also been used to extrapolate the ISHI data to predict void swelling at higher dpa and He levels. These models suggest that FMS may experience significant swelling at damage levels greater than 50 dpa, while the NFA will remain void free. Indeed the combination of the experimental and modeling results suggest that He can be transformed from a liability to an asset in NFA.

Although these conclusions represent an optimis­tic view of the status of research on measuring, mod­eling, and managing He-dpa effects in structural alloys, especially for fusion applications, there are a number of outstanding issues that require additional science-based research to resolve. Space does not permit a complete listing, but these issues and research needs can generally be classified into broad categories related to He effects per se and those related to developing, optimizing, and qualifying alloys that can manage He in a way that may provide near immunity to radiation damage. He-related issues and research needs include the following:

• Extension of experimental observations of He transport, fate, and consequences to more alloys for a wide range of microstructures and to higher damage (He and dpa) levels and higher temperatures over a range of He/dpa, using all the irradiation techniques: CPI, ISHI, and SPNI. Characterization of the management of He in NFA at temperatures up to 750 °C, or more, is particularly critical.

• Intercomparisons of accelerated high-rate CPI and lower reactor relevant rate ISHI and SPNI data and development of experimental-modeling approaches that will permit reliable predictions of He-dpa effects at very high damage levels, up to several hundred dpa, that cannot be accessed prac­tically in neutron ISHI and SPNI.

• Very detailed characterization of He and He bub­ble distributions, along with the balance of irra­diation microstructures, with particular emphasis on quantitative evaluations, like the distribution of He and bubbles on interfaces, using a suite of advanced characterization methods tailored to this application.

• Focused mechanism studies that can provide both parameters and insights into key mechanisms that govern the transport, fate, and consequences of He in various complex alloy and model systems.

• Continued development and refinement of multi­scale master models of He transport fate and con­sequences, along with the narrower first principles and atomistic studies (models and experiments) needed to better parameterize and mechanistically inform the master models. One of many examples of modeling needs is the resolution of differ­ences between continuum capillary versus atomis­tic MD-based models of helium-vacancy cluster and bubble properties.

• Developing models and basic experiments that relate the microstructural consequences of the fate of He to the degradation of performance — sustaining mechanical properties. One of many examples is the detailed model of the behavior of He on GBs and the corresponding reduction in the grain boundary fracture strength.

However, it is clear that He must be managed as well as understood. Thus, issues and research needs also include development of practical NFA or alloys with similar attributes. These include the following items:

• Identify alloy composition-synthesis designs and thermal mechanical processing paths that optimize the NF, and the balance of NFA microstructures, so as to provide a balanced suite ofoutstanding and isotropic properties.

• Resolve issues of low fracture toughness and aniso­tropic properties in extruded NFA product forms.

• Demonstrate and understand the thermal and irra­diation stability of far-from-equilibrium NF and NFA microstructures.

• Develop practical fabrication and joining methods, which preserve optimal NFA microstructures and yield defect-free components.

• Reduce costs, improve alloy homogeneity and reproducibility, and establish industrial-scale sup­ply sources.

• Qualify new alloys for nuclear service for extended lifetimes.

We believe that there is a science-based framework to resolve these and other critical issues related to the effects and management of He in practical alloy sys­tems. (see Chapter 1.03, Radiation-Induced Effects on Microstructure and Chapter 1.09, Molecular Dynamics)

To evaluate the diffusion tensor and the drift force, we consider here the diffusion of self-interstitials and vacancies in fcc crystals, and begin first with the ideal case of a crystal free of any stresses other than those produced by the migrating defect itself. The jump vectors for both self-interstitials and vacancies coin­cide with the n = 12 nearest neighbor locations of the defect in its stable configuration. Table 12 lists the components Xi of these jump vectors, and they are divided into three groups according to the orienta­tions of the dumbbell axes before and after the jump. These are indicated in the first row of Table 12 by the indices 1, 2, or 3 when the dumbbell axis is aligned with the xb x2, or x3 crystal coordinate direc­tion, respectively.

In the absence of stress, all saddle point energies are equal, say ES, and ES — E = Em is just the defect migration energy. It is then a simple matter to