There are eight major property changes that may occur in irradiated materials due to a variety of micro­structural changes. Listed in order of increasing temper­ature where the effects are typically dominant, these
phenomena are radiation-induced amorphization, radi­ation hardening (often accompanied by loss of tensile elongation and reduction in fracture toughness), decrease in thermal and electrical conductivity, mechanical property or corrosion degradation due to radiation-induced segregation and precipitation, dimensional instabilities due to three distinct phenom­ena (anisotropic irradiation growth, irradiation creep, void swelling), and high temperature embrittlement of grain boundaries due to helium accumulation. The microstructural origins associated with these eight deg­radation processes are summarized in the following sections, and more detailed descriptions ofthe property degradations in metals and nonmetals are given in accompanying chapters in this Comprehensive. The radiation doses at which these phenomena emerge to become of practical engineering significance are gener­ally dependent on irradiation temperature, PKA energy, and material.

Next, we consider 1/4 (110) {110} loops in spinel. Spinel {110} planes alternate in composition, (AlO2)~-(MgAlO2)+…, such that each layer is a mixed cation/anion layer. To insert a charge-neutral interstitial slab along (110) in spinel requires that we insert a {110} double-layer block, (AlO2)~-(MgAlO2)+, that is, a stoichiometric MgAl2O4 unit. The thickness of this slab is a/4 (110), where a is the spinel cubic lattice parameter. Along the (110) direction normal to the traces of the {110} planes, the registry of the {110} planes varies between adjacent planes, analogous to the registry shifts that occur between adjacent {111} planes in spinel (discussed earlier). The O atom patterns are identical in all {110} planes, but the registry of the O atom patterns between adjacent {110} planes alternates every other layer, analogous to the BCBC… stacking described earlier. The Mg atom patterns are identical in each (MgAlO2)+ layer, while the registry of the Mg atom patterns alternates every other (MgAlO2)+ layer. We denote the Mg stacking sequence by aj a2 aj a2 …. There are two Al atom patterns along (110): (1) the first occurs in each (AlO2)_ layer with no change in registry between layers (we denote this Al pattern by p0); and (2) the second occurs in each (MgAlO2)+ layer, and the registry of these Al atom patterns alternates every other (MgAlO2)+ layer (we denote this Al stacking sequence by p1 p2 p1 p2 .. .). Combining all these considerations, we can write the {110} planar stacking sequence in spinel as fol­lows: (p0B) (a1p1C) (p0B) (a2p2C).

Now, as with the spinel {111} case described earlier, when an extra 1/4 (110) two-layer block, (AlO2)~-(MgAlO2)+, is inserted into the spinel {110} stacking sequence, (p0 B) (a1p1C) (p0B) (a2p 2C), astack — ing fault occurs as follows:

(p0B) («1p1 C) (p0B) («2p2c) (p0B) (a^C)

(p0B) (a2p2C) (before)

(p0B) («1p1 C) (p0B) («2p2C) (p0B) («1p1C)

(p0B) («1p1 C) (p0B) (a2p2 C) (after)

(p0B) («1p1 C) (p0B) («2p2C) (p0B) («1p1C) I (p0B) 1 (a1p1C) (p0B) (a2p2C) (after, showing stacking fault positions) [6]

Notice in eqn [6] that after block insertion, the anion sublattice is not faulted (BCBC… layer stacking is preserved), whereas the cation sublattice is faulted, specifically at the positions of the red vertical lines in the last sequence (the left-hand red line corresponds to the cation fault position for cation planar registries moving from right to left; likewise, the right-hand red line corresponds to the cation fault position for cation planar registries moving from left to right). Thus, the dislocation loop formed by 1/4 (110) two — layer block insertion in spinel is an extrinsic, cation — faulted, sessile interstitial Frank loop.

Figure 3 shows an example of 1/4 (110) interstitial dislocation loops in spinel, produced by neutron irra — diation.12 The alternating black-white fringe contrast within the loops is an indication of the presence of a stacking fault within the perimeter of each loop. The character of the {110} loops was determined by Hobbs and Clinard using the TEM imaging methods of Groves and Kelly,14,15 with attention to the pre­cautions outlined by Maher and Eyre.16 These loops were determined to be extrinsic, faulted 1/4 (110) {110} interstitial dislocation loops. It is evident in Figure 3 that the extrinsic fault associated with these loops is not removed by internal shear, even when the loops grow to significant sizes (> 1 pm diameter). This is the subject of our next topic of discussion, namely, the unfaulting of faulted Frank loops.

Interactions with defects, like dislocations and inter­faces, play a dominant role in the fate of He in complex multiconstituent, multiphase structural alloys. MD, MS, and ab initio calculations have been carried out to characterize the interactions of He with Fe SIA and their clusters.159,270,271 These simula­tions revealed various reactions-interactions involving (a) a spontaneous recombination-replacement reac­tion, where a single SIA replaces a Hes atom leaving a Hei; and, (b) SIA cluster trapping-detrapping reac­tions with single Hes and Hei atoms, as well as with small He clusters. The simulations also showed that small SIA clusters are strongly bound by a single Hei and Hei clusters with high Eb from 1.3 to 4.4 eV. Such trapping significantly retards the primarily one dimensional motion of SIA clusters, which oth­erwise are highly mobile with a migration energy of less than 0.1 eV in pure Fe.

Interactions of He with microstructural features such as dislocations, GBs, and nanometer-scale pre­cipitates have also been modeled.136 The Dimer method272 was used to efficiently identify saddle point activation energies that were then used in MD simulations to observe interaction mechanisms and reaction paths. Energy landscapes for He around dis­locations were modeled for (a/2)[111] and [—1-12] edge dislocations as well as (a/2)[111] screw disloca — tions.273,274 Similar calculations have also been carried out for four symmetric tilt boundaries with a common (101) axis (£3(112), £11(323), £9(114), and S3 {111}) using a two-part rectangular cell.275 The interactions of He with coherent nanometer-scale Cu precipitates have also been modeled.136

Figure 39(a) shows the Hes Eb and excess volume per unit area (Vex) for edge and screw dislocations as a function of the distance from the core. The maximum Eb is much larger for edge («0.5 eV) compared with screw dislocations («0.15 eV). The Eb closely corre­late with the Vex. As shown in Figure 39(b), a similar relationship is found for different GBs. The Eb of substitutional and interstitial He varies from «0.2 to «0.8 eV and «0.6 to «2.7 for screw and edge disloca­tions, respectively, increasing linearly with increasing Vex. Figure 39(c) shows the Eb for both a single vacancy and a Hes atom as a function of distance from a 2 nm coherent Cu precipitate.136 Both the Hes and a single vacancy have very similar energy — distance relations, with the maximum Eb « 0.6 eV at the precipitate surface. Table 4 summarizes the results for these extended defect models. The qualitative implications of these results are that Hei is strongly trapped at various common microstructural features, while Hes is more weakly trapped. Thus, it is likely that detrapping of Hei involves a Hei + V! Hes reaction.

1.01.5.1 The Misfit or Size Interaction

Many different sources of strain fields may exist in real solids, and they can be superimposed linearly if they satisfy linear elasticity theory. If this is the case, we need to consider here only the interaction between one particular defect located at rd and an extraneous displacement field u0(r) that originates from some other source than the defect itself. In particular, it may be the field associated with external forces or deformations applied to the solid, or it may be the field generated by another defect in the solid.

To find the interaction energy, we assume that the defect under consideration is modeled by applying a set of Kanzaki26 forces f(“)(R(“)), a = 1, 2, …, z, at z atomic positions R(a as described in greater detail

in Appendix B. We imagine that once applied, the extraneous source is switched on, whereupon the atoms are displaced by the field u0(r), and work is done by the Kanzaki forces of the amount

z

W = -‘ f(a) • u0 rd + R(“> [39]

a=1 ‘ ‘

If the displacement field varies slowly from one atom position to the next, we may employ the Taylor expansion for each displacement component,

rd + R(“^ ~ «0>(rd) + Uij(rd)R(°^ + ••• [40]

and obtain -4ЫЕfia)-мЕгґ]Ra)— [41

The first term vanishes because the sum of Kanzaki forces is zero, as pointed out in eqn [B4], and the second term can be expressed in terms of the dipole tensor defined in [B6]. Furthermore, since the dipole tensor is symmetric, we obtain finally for the interaction energy

W1 ~-P„ 4(rd) [42]

where є0 (rd) is the extraneous strain field at the location of the defect.

Point defects can also be modeled as inclusions, as we have seen, and they can be characterized by a transformation strain tensor, ekl. In Appendix B it is shown, with eqn [B25], that the dipole tensor is then given by

Pij = OCijki eu [43]

and the interaction energy can be written as

W = — OCiju eu 4 = — П eu ff° [44]

Finally, for the simplest model of a point defect as a misfitting spherical inclusion, as treated in Appendix A,

ekl = ro dkl

and

w = —.Vrel s“k [45]

In this last form, we used the notation for the relaxation volume for the misfit or transformation volume, that is, AV = Vrel, to emphasize the fact that the interaction energy of vacancies and self-interstitials does not depend on their formation volumes, but on their relax­ation volumes. This dependence on the misfit volume or defect size gave this energy the name of misfit or size interaction.

Under equilibrium conditions, the number of elec­tronic defects of energy E is given by, ,

n(E) = N(E) • F(E)

where N(E) is the volume density of electronic levels that have energy E (known as the density of states) and F(E) is the probability that a given level is occu­pied, called the Fermi-Dirac distribution function.

N(E) is a function of energy. It is the maximum density of electrons of energy E allowed (per unit volume of crystal) by the Pauli exclusion principle. For a semiconductor, this has an approximately para­bolic behavior close to the band edges (i. e., N(E) « E1/2, refer to Figure 11).

where m* is the effective mass of an electron in the conduction band. Similarly, the effective valence band density of states is given by2,3

2ят*kT 3=2

‘{-t-

where m* is the effective mass of a hole in the valence band. Note that m* and mh are between two and ten times greater than the mass of a free electron. Also, per volume, these densities are approximately four orders of magnitude less than the typical atom den­sity in a solid.

The Fermi-Dirac distribution function is given

by2,3

F (E)

At 0 K, this implies that all energy levels are occupied up to Ef, the Fermi energy. This is a step function. The Fermi-Dirac function with respect to the energy is represented in Figure 12. At the Fermi energy, F(E) is 1=2. Above 0 K, some energy levels above Ef are occupied. This implies that some levels below Ef are empty.

Intrinsic

semiconductor

Figure 13 Characteristic electron energy band levels for a metal, an intrinsic semiconductor and an insulator, where Ec is the bottom of the conduction band, Ev is the top of the valence band, Eg is the band gap, and Ef is the Fermi level.

Radiation-induced amorphization can proceed by sev­eral different mechanisms, including direct impact amorphization and gradual accumulation of lattice defects and chemical disorder that eventually causes destabilization of the crystalline matrix.213 Figure 36 shows an example of the microstructure near the crystalline to amorphous transition dose in ion-irradiated SiC, where the amorphization is induced by gradual buildup of radiation defects.209

At intermediate doses, amorphous islands gradu­ally emerge from the initially crystalline matrix in SiC irradiated at low temperatures. Direct amorphization within individual displacement cas­cades has been observed in several intermetallic,334 semiconductor,12,335 and ceramic insulator15,336,337 materials. In many other materials, extensive chemi­cal disordering from displacement cascades or point defects precedes amorphization.3,76,87 The chemical disordering can be monitored either on the nanoscale dimensions (e. g., due to individual displacement cas­cades) by techniques such as transmission electron microscopy,12,338 or an integrated average value by various techniques including X-ray diffraction, TEM, and Rutherford backscattering spectrometry.76,339 As previously noted in Section 1.03.3.8, the intense ionization associated with swift heavy irradiation can lead to amorphization either directly within ion tracks, or by a cumulative process involving chemical disordering before amorphization due to multiple overlapping ion tracks.

Y. Dai

Paul Scherrer Institut, Villegen PSI, Switzerland

G. R. Odette and T. Yamamoto

University of California, Santa Barbara, CA, USA © Introduction and Overview

This chapter reviews the profound effects of He on the bulk microstructures and mechanical properties of alloys used in nuclear fission and fusion energy systems. Helium is produced in these service envir­onments by transmutation reactions in amounts rang­ing from less than one to thousands of atomic parts per million (appm), depending on the neutron spec­trum, fluence, and alloy composition. Even higher amounts of H are produced by corresponding n, p reactions. In the case of direct transmutations, the amount of He and H are simply given by the content weighted sum of the total neutron spectrum averaged energy dependent n, a and n, p cross-sections for all the alloy isotopes ((sn, a)) times the total fluence (ft). The spectral averaged cross-sections for a specified neutron spectrum can be obtained from nuclear data­base compilations such as SPECTER,1 LAHET,2 and MCNPX3 codes. He and H are also produced in copious amounts by very high-energy protons and neutrons in spallation targets of accelerator-based nuclear systems (hereafter referred to as spallation proton-neutron (SPN) irradiations, SPNI).4, The D-T fusion first wall spectrum includes 14MeV neutrons («20%), along with a lower energy spec­trum («80%). The 14MeV neutron energy is far above the threshold for n, a («5 MeV) and n, p («1 MeV) reactions in Fe.6 Note that some impor­tant transmutations also take place by multistep nuclear reactions. For example, thermal neutrons (nth) generate large amounts of He in Ni-bearing alloys by a 58Ni(nth, g)59Ni(nth, a) reaction sequence. These various irradiation environments also produce a range of solid transmutation products.

High-energy neutrons also produce radiation- induced displacement damage in the form of vacancy and self-interstitial atom (SIA) defects. Vacancies and SIA are the result of a neutron reaction and scattering — induced spectrum of energetic primary recoiling atoms with energies ranging from less than 1 keV, in neutron irradiations, up to several MeV in SPN irra­diations.7 The high-energy primary recoils create cascades of secondary displacements of atoms from their crystal lattice positions, measured in a calcu­lated dose unit of displacements per atom (dpa). As in the case of n, a transmutations, dpa production can also be evaluated using spectral averaged dis­placement cross-sections8 that are calculated using the codes and nuclear database compilations cited above.

Typical operating conditions of various fission, fusion, and spallation facilities are summarized in Table 1. Notably, He (and H) generation in fast fission (He/dpa << 1), fusion (He/dpa « 10), and spallation proton-neutron (He/dpa up to 100) environments differs greatly and this is likely to have significant effects on the corresponding microstructural and mechanical property evolutions.

The primary characteristic of He, which makes it significant to a wide range of irradiation damage phenomena, is that it is essentially insoluble in solids. Hence, in the temperature range where it is mobile, He diffuses in the matrix and precipitates to initially form bubbles, typically at various microstructural trapping sites. The bubbles can serve as nucleation sites of growing voids in the matrix and creep cavities on grain boundaries (GBs), driven by displacement damage and stress, respectively. While He effects are primarily manifested as variations in the cavities, all microstructural processes taking place under irradia­tion are intrinsically coupled; hence, difference in the He generation rate can also affect precipitate, dislo­cation loop, and network dislocation evolutions as well (see Section 1.06.3).

Figure 1, adopted from Molvik et al.,9 schemati­cally illustrates the effects of high He as a function of lifetime-temperature limits in a fusion first wall structure for various irradiation-induced degrada­tion phenomena. At high temperatures, lifetimes (green curve) are primarily dictated by chemical compatibility, fatigue, thermal creep, creep rupture, and creep-fatigue limits. In this regime, He can fur­ther degrade the tensile ductility and the other high — temperature properties, primarily by enhancing grain boundary cavitation, in some cases severely. In aus­tenitic stainless steels (AuSS), high-temperature He embrittlement (HTHE) has been observed at concentrations as low as 1 appm.10,11 In contrast, 9Cr ferritic-martensitic steels (FMS), which are cur­rently the prime candidate alloy for fusion structures, are much more resistant to HTHE.12,13

Figure 1 Illustration of the materials design window for the fusion energy environment, as a function of temperature. Reproduced from Molvik, A.; Ivanov, A.; Kulcinski, G. L.; et al. Fusion Sci. Technol. 2010, 57, 369-394.

At intermediate temperatures (blue curve), growing voids form on He bubbles, and He accumulation largely controls the incubation time prior to the onset of rapid swelling (see Section 1.06.3). FMS are also much more resistant to swelling than stan­dard austenitic alloys,14,15 although the microstruc­tures of the latter can be tailored to be more resistant to void formation by He management schemes.16 High He concentrations can also extend irradiation hardening and fast fracture embrittlement to inter­mediate temperatures.17

At lower temperatures (red curve), where irradiation hardening and loss of tensile uniform ductility are severe, high He concentrations enhance large positive shifts in the ductile-to-brittle transition temperature (DBTT) in bcc (body-centered cubic) alloys.18-20 This low-temperature fast fracture embrittlement phenomenon is believed to be primarily the result of

He-induced grain boundary weakening, manifested by a very brittle intergranular (IG) fracture path, inter­acting synergistically with irradiation hardening. , High concentrations also increase the irradiation hardening at dpa levels that would experience satu­ration in the absence of significant amounts of He.17 A significant concern for fusion is that the dpa- temperature window may narrow, or even close, for a practical fusion reactor operating regime.

What is sketched above is only a very broad-brush, qualitative description of some of the important He effects. The quantitative effects of He, displacement damage, temperature and stress, and their interactions, which control the actual positions of the schematic curves shown in Figure 1, depend on the combination of all the irradiation variables, as well as details of the alloy type, composition, and starting microstructure (material variables). The effects of a large number of interacting variables, the complex interactions of a plethora of physical mechanisms, and the implica­tions to the wide range of properties of concern are not well understood; and even if they were, such
complexity would beg easy description. Therefore, a first priority is to develop a good understanding of and models for the transport and fate of He at the point when it is effectively immobilized in bubbles and voids, often at various microstructural sites. Such insight provides a basis for developing microstruc­tures that can manage He and thus mitigate its dele­terious effects. To this end we next briefly outline key radiation damage processes, including the role ofHe.

Figure 2 schematically illustrates the combined effects of He and displacement damage on irradiation- induced microstructural evolutions.2 Figure 2(a) shows a molecular dynamics simulation of primary displacement damage produced in displacement cas­cades. Most of the initially displaced atoms return to a lattice site (self-heal). Residual cascade defects include single and small clusters of vacancies and SIA. In the temperature range of interest, vacancies (red cir­cles) and SIA (green dumbbells) are mobile. SIA clusters, in the form of dislocation loops, are also believed to be mobile in some cases, undergoing one-dimensional diffusion on their glide prisms.

However, the cascade loops may also be trapped by interactions with solutes. Small cascade vacancy clus­ters may coarsen in the cascade region by Ostwald ripening and diffusion coalescence mechanisms. Both isolated and clustered defects interact with alloy solutes forming cascade complexes. The cascade vacancy clusters dissolve over a time associated with cascade aging, which depends strongly on tempera­ture. The concentration of cascade vacancy clusters, which act as sinks (or recombination centers) for migrating vacancies and SIA, scales directly with the damage rate. Thus, the overall defect production microstructures can be viewed as being composed of steady-state concentrations of diffusing defects, small loops, and cascade vacancy clusters; the latter are important if the irradiation time is much less than the cluster annealing time. Vacancy-SIA recombina­tion at clusters, in the matrix and at vacancy trapping sites, can give rise to important damage rate, or flux, effects.

Figure 2(b) shows that SIA can recombine with diffusing and trapped vacancies, in this case one trapped on a precipitate interface. Figure 2(b) also shows that both bubbles (blue part circle) and voids (orange part circle) often form on precipitates. Figure 2(c) shows that dislocation loops (green hexagon) nucleate and grow due to preferential absorption of SIA (bias). Preferential accumulation of SIA also takes place at network dislocation segments (inverted green T), causing climb. Loop growth and dislocation climb can lead to creation (loops and Herring-Nabarro sources) and annihilation (of oppositely signed network segments) of disloca­tions, ultimately leading to quasi-steady-state densi­ties, as is observed in the case of AuSS.

Figure 2(d) shows that He precipitates to form bubbles (larger blue circles) at various sites, in this case in the matrix. Small bubbles are stable since they absorb and emit vacancies in net numbers that exactly equal the number of SIA that they absorb; thus bubbles grow only by the addition of diffusing He atoms (small blue circle). However, Figure 2(e) shows that when bubbles reach a critical size they convert to unstably growing underpressurized voids (large orange circle containing blue He atoms) due to an excess flux of vacancies over SIA arising from the dislocation bias for the latter defect. Figure 2(e) shows the corresponding growing creep cavities transformed from critical He bubbles on stressed GBs. Designs of microstructures that mitigate, or even fully suppress, these various coupled evolutions are described in Section 1.06.6 and discussed in references.2 ,

Therefore, a master overarching framework for measuring, modeling, and managing He effects must be based on developing and understanding the domi­nant mechanisms controlling its generation, trans­port, fate, and consequences, as mediated by the irradiation conditions and the detailed alloy micro­structure. Figure 3 illustrates such a framework for He generation, transport, and fate. In this framework, experiments and models can be integrated to estab­lish how He is transported to various microstructural trapping (-detrapping) features and how He locally clusters to form bubbles at these sites, as well as in the matrix. The master models must incorporate para­meters that describe He diffusion coefficients under irradiation, binding energies for trapping at the vari­ous sites and He-vacancy cluster and other interac­tion energies.

Given the length and comprehensive character of this chapter, it is useful to provide the reader a guide to what follows. Notably, we have tried to develop useful semi-standalone sections.

Section 1.06.2 describes the various experimental approaches to studying He effects in structural alloys including both neutrons and various types of charged-particle irradiations (CPI).

Section 1.06.3 reviews the historical knowledge base on He effects, which has been developed over the past 40 years, with emphasis on bubble evolution, void swelling, and HTHE processes. While less of current interest, the examples included here pri­marily pertain to standard AuSS, discussions of experiment and modeling are closely integrated to emphasize the insight that can be derived from such coupling. Particular attention is paid to the critical bubble mechanism for the formation of growing voids and grain boundary cavities and the corresponding consequences to swelling and creep rupture. The implications of the coupled models and experimental observations to designing irradiation-tolerant alloys that can manage He are discussed in some detail.

Section 1.06.4 focuses on a much more recent body of observations on He effects in SPNI. The emphasis here is on descriptions of defect and cavity microstruc­tures in both FMS and AuSS irradiated at low to intermediate temperatures and the corresponding effects on their strength, ductility, and fast fracture resistance. Similarities and differences between the SPNI effects and those observed for fission irradia­tions are drawn where possible.

Section 1.06.5 summarizes some key examples of atomistic modeling of He behavior, which has been the focus of most recent modeling efforts. Insight into mechanisms and critical parameters provided by these models will form the underpinning of the com­prehensive master models of He transport, fate, and consequences.

Section 1.06.6 builds on the discussion in Section 1.06.3 regarding managing He by trapping it in a population of small stable bubbles. A specific example comparing FMS to a new class of high — temperature, irradiation-tolerant nanostructured fer­ritic alloys (NFA) irradiated in a High-Flux Isotope Reactor (HFIR) at 500 °C to 9 dpa and 380 appm He is described. The results of this study offer proof in principle of the enormous potential for developing irradiation-tolerant NFA that could turn He from a liability to an asset. Section 1.06.6 again couples these experimental observations with a master multi­scale model of the transport and fate of He in both

FMS and NFA. The predictions of the master model, that is both microstructurally informed and parame­terized by atomistic submodels, are favorably com­pared to the HFIR data.

Section 1.06.7 briefly summarizes the status of understanding of He effects in structural alloys and concludes with some outstanding issues. Reading this summary first may be helpful to general readers who then can access the more detailed information at their own discretion.

Surface areas in solids can be changed by elastic deformation, and this leads to the concept of surface stresses, gab, and surface strains, eS^. Here the lower indices designate two orthogonal coordinate direc­tions, xl and x2, which are tangential to the surface. Greek indices are used here for surface quantities, and they assume the values of 1 or 2. In contrast, Latin indices are employed for vector — and tensor-valued quantities within the bulk material, and they assume values of 1, 2, or 3.

The definition of the surface stress in a Lagrang — ian reference frame is according to Cahn38

dg

gab = [111]

@eab

Here, the specific surface free energy g is assumed to depend now on the elastic strain components tangen­tial to the surface. We note that eqn [Ill] differs from

the Shuttleworth39 equation. The latter is inconsis­tent with continuum mechanics as has been argued by Gutman.40 In contrast, the surface elasticity theory of Gurtin and Murdoch41,42 is compatible with the elasticity theory of bulk solids, and the two are connected by appropriate boundary conditions as described below. The theory of Gurtin and Murdoch requires that the surface energy be a function of the surface strains.

As found from recent ab initio calculations and from atomistic calculations with empirical interatomic potentials, the dependence of the specific surface energy g on surface strains can be written as

g(eSb) = g(°)+ 2r «byd4eeS<5 + °(fi3) [112]

for small surface strains. Here, Gapyg is the surface elastic modulus tensor, and summation is implied over repeated indices. The surface elastic constants have a dimension of N m — while bulk elastic con­stants have the dimension of N m-2. Since the surface is attached to a bulk solid, surface strains eS^ and the bulk strains ejj are connected. Bulk strains are defined relative to a stress-free reference configuration of the bulk solid. However, a stress-free reference state for its surfaces may not coincide with the stress-free reference configuration for the bulk solid. Ab initio calculations on slabs43-47 of metals have in fact revealed that surface layers on stress-free bulk mate­rials possess in general a positive residual strain, e^. The surfaces ofmetals are naturally stretched and will contract if they can deform the underlying solid.

When the solid is elastically strained, the total surface strains become

eSb = <8 + eab [113]

where the strains eab are identical to the bulk strains ejj at the surface.

The surface stresses relative to the stress-free ref­erence configuration of the bulk are now defined as

EaP T Gap-/SeyS

The first term gives the residual or intrinsic surface stresses, and they are connected to the residual sur­face strains by

gtf = G«pyse;s [115]

This mechanical theory of surface stresses, as out­lined above, has been developed previously in its full mathematical rigor by Gurtin and Murdoch for gen­eral finite strains.41 The above linearized version for
small strains has also been presented by these authors, and they have shown42 that for isotropic materials, the residual surface strains are

e*;8 = e;dab [116]

and the surface elastic modulus tensor

Ga[]yS mS(dayd8d T dadg8y) T 8gyd [117]

contains two surface elastic constants m-S and lS. Typ­ically, e* is between 0.01 and 0.1, and the surface stretch modulus is on the order of 2mMd/(1—2nM), where mM and nM are the elastic constants of the bulk material and d is the thickness of the surface layer, about one interatomic distance.

With the above equations, the specific surface energy expansion, eqn [112], can be written as

y(eab) y0 T 2 (mS T As )e eaa T AS eafteaft

T 2^S eaae88

for isotropic materials. The energy associated with the residual surface stresses g^ is included in the term y0. It represents the surface energy of a planar surface on a stress-free solid.

events becomes important.6,70-72,90 The critical dose

for this kinetic transition is dependent on the con­centration of other defect sinks in the lattice (disloca­tions, grain boundaries, precipitates, etc.). The high sink strength associated with the immobile vacancies limits the growth rate (i. e., size) of the SIA loops for doses above ^0.1 dpa, and the observable defect clus­ter size and density typically approach a constant value at higher doses. Figure 8 shows an example of the microstructure of AlN following ion irradiation at 80 K (mobile SIAs, immobile vacancies) to a damage level of about 5 dpa.91 The microstructure consists of small (<5 nm diameter) interstitial dislocation loops.

Helium is conveniently introduced into nickel­bearing alloys through thermal neutron irradiation. Although helium is usually detrimental, especially if the material is to be subsequently welded,63 it offers a method to simulate the production of helium expected in the very hard spectrum of a fusion reac­tor. In the case of vanadium and other refractory metal alloys, the effect of helium has been studied using two primary methods of introduction of helium. One method is implantation of a-particles with an accelerator; the other is the use of the decay of tritium. Tritium rapidly diffuses into group V refractory metals at elevated temperatures. The elevated temperature serves more to dissolve the protective oxide layer than to accelerate the kinetics of dissolution. The tritium thus introduced is permit­ted to decay, by p-decay with a residual nucleus of 3He. Helium-doped specimens have subsequently been neutron irradiated to study the synergistic effects of helium and atomic displacement damage.

A limited number of experiments have used tech­niques to simultaneously implant He and produce atomic displacements through an irradiation environ­ment of Li. The concept of introducing tritium into an irradiation capsule with the specimens in contact with lithium has been investigated to study vanadium

alloys with the He/dpa ratio characteristic of a fusion environment. The tritium charge, the production of tritium from lithium, and the production of tritium from 3He are some of the important considerations in the design of the experiment.64 Although conceptually valid, the desired results have not yet been obtained with experiments of this type.

Cyclotron-implanted helium has been used, also to study the effects of the fusion irradiation environ­ment. Tanaka showed severe embrittlement with the introduction of 90 and 200appm He at 700 °C in V-20Ti.65 Grossbeck and Horak showed that a level of 80 at. ppm He implanted as part of the same experiment had no significant effect on elongation in V-15Cr-5Ti at 700 °C.52 Braski also observed no sig­nificant effect on ductility in V-15Cr-5Ti at 600 °C with similar levels of He introduced from decay of tritium.66 The alloys, Vanstar-7 and V—3Ti—1Si, were also investigated, in some cases with an improvement in ductility upon introduction of helium.66 Follow­ing irradiation, severe embrittlement was observed in V-15Cr-5Ti at 600 °C in tritium trick samples by Braski66 and at 625 °C in cyclotron-implanted samples by Grossbeck and Horak.52 Irradiation experiments with refractory metals, unless using a Li environment, frequently subject the specimens to contamination by interstitial impurities, also leading

to embrittlement.52,67

Unalloyed molybdenum, Mo-0.5Ti, and Mo-50Re were irradiated in EBR-II by Wiffen at exposures of

Test temperature fC)

3.5—6.1 n cm~2 (E > 0.1 MeV) (18-32 dpa).68 Although Mo alloys are known to exhibit increased ductility with increasing temperature in the unirradiated con­dition, at temperatures above 400-550 °C, all three materials suffered plastic instability with uniform elongations below about 0.5%. This effect is shown in Mo in Figure 2 968 where irradiation temperature is shown to be the critical parameter and where speci­mens irradiated at 455 and 1136 °C were embrittled even in room temperature tests.

This class of alloys is discussed further in Chapter

4.6, Radiation Effects in Refractory Metals and Alloys.