Void swelling is only one component of microstruc­tural and microchemical evolutions that take place in alloys under irradiation. In addition to loops and network dislocations, other coevolutions include sol­ute segregation and irradiation-enhanced-induced — altered precipitation. In the mid-1980s, CBM and RT models of dislocation loop and network evolution were self-consistently integrated in the computer code MicroEv, which also included a parametric treatment of precipitate bubble-void nucleation sites.133,144 Later work in the 1990s further developed and refined this code.163 A major objective of much of this research was to develop models to make quanti­tative predictions of the effect of the He/dpa ratios on void swelling for fusion reactor conditions.

CBMs have been used to parametrically evaluate the effects of many irradiation variables and ma­terial parameters15,114,118,128,129,140,149,150 as well as

to model swelling as a function of temperature, dpa and dpa rates, and the He/dpa ratio (see both Stoller and Odette references). The CBMs have also been both informed by and compared with data from experiments in both fast and mixed thermal-fast spectrum test reactors, including EBR-II (fast), FFTF (fast), and HFIR (mixed),16,119 complemented by exten-

26,124,125,128,129,157,164a,164-171

sive dual ion CPI results.

The semiempirical CBM models and concepts ratio­nalize a wide range of seemingly complex and some­times disparate observations, including the following:

• Void nucleation on bubbles

• The general trends in the temperature, dpa, and He/dpa dependence of the number densities of bubbles and voids

• Incubation dpa and postincubation swelling rates, including the effects of temperature and stress

• The occurrence of bimodal cavity size distribu­tions of small He bubbles and larger voids

• Bubble nucleation on dislocations and precipitate interfaces

• Swelling that is increased, decreased, or unaltered by increasing GHe, depending on the combination of other irradiation and material variables

• Suppression of void swelling by a very high num­ber of densities of bubbles

• Highly coupled concurrent evolutions of all the microstructural features, resulting in weaker trend toward refinement of precipitate and loop struc­tures at higher GHe and, in the limit of very high Nb, suppression of loops and precipitation

• Strong effects of the schedule and temperature history of He implantation in CPI

• Effects of alloying elements on swelling incubation associated with corresponding influence on pre­cipitation, solute segregation, and the self-diffusion coefficient

• Swelling resistance of AuSS that have stable fine — scale precipitates that trap He in small interface bubbles

• The much higher swelling resistance of bcc FMS compared with fcc AuSS

The concept of trapping He in a high number den­sity of bubbles to enhance the swelling and HTHE resistance (and creep properties in general) was imple­mented in the development of AuSS containing fine-scale carbide and phosphide phases. Figure 19 shows the compared cavity microstructures resulting in «6% void swelling in a conventional AuSS (Figure 19(a)) to an alloy modified with Ti and heat treated to produce a high density of fine-scale TiC (Figure 19(b)) phases with less than 0.2% bubble swelling following irradiation to 45 dpa and 2500appm He at 600°C.172 There are many other examples of swelling-resistant AuSS that were suc­cessful in delaying the onset of swelling to much

higher dpa than in conventional AuSS. However, as illustrated in Figure 7, these steels also eventually swell. This has largely been attributed to thermal — irradiation instability and coarsening of the fine-scale precipitates that provide the swelling resistance.172

FMS are much more resistant to swelling than advanced AuSS.15102,104116128,129,162,169,174,175 The swelling resistance of FMS, compared with AuSS, can be attributed to a combination of their (a) lower dislocation bias; (b) higher sink densities for parti­tioning He into a finer distribution of bubbles, thus increasing m*; (c) low void to dislocation sink ratios; (d) a higher self-diffusion coefficient that increases m*; and (e) lower He/dpa ratios.15,176 However, void swelling does occur in FMS, as well as in unalloyed Fe,177 and is clearly promoted by higher He/dpa ratios. Higher He can decrease incubation times for void formation and increase Zv/Zd ratios closer to 1, resulting in higher swelling rates.52,157,168-171 Recent models predict significant swelling in FMS,178 and the potential for high postincubation swelling rates in these alloys remains to be assessed. Swelling in FMS clearly poses a significant life-limiting chal­lenge in fusion first wall environments in the temper­ature range between 400 and 600 °C.

NFA, which are dispersion strengthened by a high density of nanometer-scale Y-Ti-O-enriched features, are even more resistant to swelling and other mani­festations of radiation damage than FMS.22,23,51,179,180 Irradiation-tolerant alloys will be discussed in Section 1.06.6.

There are essentially three primary energy sources for the billions of people living on the earth’s surface: the sun, radioactivity, and gravitation. The sun, an enormous nuclear fusion reactor, has transmitted energy to the earth for billions of years, sustaining photosynthesis, which in turn produces wood and other combustible resources (biomass), and the fossil fuels like coal, oil, and natural gas. The sun also provides the energy that steers the climate, the atmospheric circulations, and thus ‘fuelling’ wind mills, and it is at the origin of photovoltaic processes used to produce electricity. Radioactive decay of primarily uranium and thorium heats the earth underneath us and is the origin of geothermal energy. Hot springs have been used as a source of energy from the early days of humanity, although it took until the twentieth century for the potential of radioactivity by fission to be discovered. Gravitation, a non-nuclear source, has been long used to generate energy, primarily in hydropower and tidal power applications.

Although nuclear processes are thus omnipresent, nuclear technology is relatively young. But from the moment scientists unraveled the secrets of the atom and its nucleus during the twentieth century, aided by developments in quantum mechanics, and obtained a fundamental understanding of nuclear fission and fusion, humanity has considered these nuclear processes as sources of almost unlimited (peaceful) energy. The first fission reactor was designed and constructed by Enrico Fermi in 1942 in Chicago, the CP1, based on the fission of uranium by neutron capture. After World War II, a rapid exploration of fission technology took place in the United States and the Union of Soviet Socialist Republics, and after the Atoms for Peace speech by Eisenhower at the United Nations Congress in 1954, also in Europe andJapan. Avariety of nuclear fission reactors were explored for electricity generation and with them the fuel cycle. Moreover, the possibility of controlled fusion reactions has gained interest as a technology for producing energy from one of the most abundant elements on earth, hydrogen.

The environment to which materials in nuclear reactors are exposed is one of extremes with respect to temperature and radiation. Fuel pins for nuclear reactors operate at temperatures above 1000 °C in the center of the pellets, in fast reactor oxide fuels even above 2000 °C, whereas the effects of the radiation (neutrons, alpha particles, recoil atoms, fission fragments) continuously damage the material. The cladding of the fuel and the structural and functional materials in the fission reactor core also operate in a strong radiation field, often in a dynamic corrosive environment of the coolant at elevated temperatures. Materials in fusion reactors are exposed to the fusion plasma and the highly energetic particles escaping from it. Furthermore, in this technology, the reactor core structures operate at high temperatures. Materials science for nuclear systems has, therefore, been strongly focussed on the development of radiation tolerant materials that can operate in a wide range of temperatures and in different chemical environments such as aqueous solutions, liquid metals, molten salts, or gases.

The lifetime of the plant components is critical in many respects and thus strongly affects the safety as well as the economics of the technologies. With the need for efficiency and competitiveness in modern society, there is a strong incentive to improve reactor components or to deploy advanced materials that are continuously developed for improved performance. There are many examples of excellent achievements in this respect. For example, with the increase of the burnup of the fuel for fission reactors, motivated by improved economics and a more efficient use of resources, the Zircaloy cladding (a Zr-Sn alloy) of the fuel pins showed increased susceptibility to coolant corrosion, but within a relatively short period, a different zirconium-based alloy was developed, tested, qualified, and employed, which allowed reliable operation in the high burnup range.

Nuclear technologies also produce waste. It is the moral obligation of the generations consuming the energy to implement an acceptable waste treatment and disposal strategy. The inherent complication of radioactivity, the decay that can span hundreds of thousands of years, amplifies the importance of extreme time periods in the issue of corrosion and radiation stability. The search for storage concepts that can guarantee the safe storage and isolation of radioactive waste is, therefore, another challenging task for materials science, requiring a close examination of natural (geological) materials and processes.

The more than 50 years of research and development of fission and fusion reactors have undoubtedly demonstrated that the statement ‘technologies are enabled by materials’ is particularly true for nuclear technology. Although the nuclear field is typically known for its incremental progress, the challenges posed by the next generation of fission reactors (Generation IV) as well as the demonstration of fusion reactors will need breakthroughs to achieve their ambitious goals. This is being accompanied by an important change in materials science, with a shift of discovery through experiments to discovery through simulation. The progress in numerical simulation of the material evolution on a scientific and engineering scale is growing rapidly. Simulation techniques at the atomistic or meso scale (e. g., electronic structure calculations, molecular dynam­ics, kinetic Monte Carlo) are increasingly helping to unravel the complex processes occurring in materials under extreme conditions and to provide an insight into the causes and thus helping to design remedies.

In this context, Comprehensive Nuclear Materials aims to provide fundamental information on the vast variety of materials employed in the broad field of nuclear technology. But to do justice to the comprehensiveness of the work, fundamental issues are also addressed in detail, as well as the basics of the emerging numerical simulation techniques.

R. J.M. Konings European Commission, Joint Research Centre, Institute for Transuranium Elements, Karlsruhe, Germany

T. R. Allen

Department ofEngineering Physics, Wisconsin University, Madison, WI, USA

R. Stoller

Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA

S. Yamanaka

Division of Sustainable Energy and Environmental Engineering, Graduate School of Engineering, Osaka University, Osaka, Japan

As we have shown in Appendix A, lattice imperfec­tions, such as a self-interstitials, a small precipitate, or small dislocation loops can also be modeled as an inclusion. These can be created by transforming a region O in the perfect crystal to a different crystal structure. If the transformation strain is emn(R), then the displacement field outside this region is given by

d3 RG, y.k'(r, r’ + R)emn(R) [B20] where G is now the elastic Green’s function.

Again, far from the defect region where |r — r’| >> |R|, we may employ a Taylor series expansion for the Green’s function, and we end up with the multipole expansion of eqn [B9]. The mul­tipole tensors are now given by

Pjk Cjkmn d R emn(R)

Pjkl Cjkmn d RR1 emn(R)

d3RR/ Rp emn(R) tensors then encompasses their description by either Kanzaki forces or by transformation strains. These tensors, in particular the dipole tensor Pjk, serve as more general parameter for their properties.

At very low temperatures where motion of SIAs or SIA clusters is limited, a crystalline to amor­phous phase transition can be induced. The phase transition usually produces large swelling (5-30%) and decreases in elastic moduli.15,91,182,209 This phase transition typically occurs for damage levels of ~0.1—1 dpa at low temperatures and has been attrib­uted to several mechanisms including direct amorphi — zation within collision cascades, and an increase in

the crystalline free energy due to point defect accu­mulation and disordering processes.86’147’210-213 The dose dependence for accumulation of the amorphous volume fraction is significantly different for the direct impact mechanism compared to point defect accumu­lation or multiple overlap mechanisms.213 As the irra­diation temperature is raised to values where long range SIA and SIA cluster migration occurs, point defect diffusion to reduce the increase in free energy occurs and the dose to induce amorphization typi­cally increases rapidly with increasing tempera­ture until a temperature is reached where it is not possible to induce amorphization. In many cases, the critical temperature for amorphization increases with increasing PKA energy. Figure 24 compares the effect of PKA energy on the temperature- dependent dose for complete amorphization for an intermetallic alloy214 and a ceramic139 material. In both materials, for all types of irradiating particles, the critical dose for amorphization increases rapidly when the irradiation temperature exceeds a critical value. The critical temperature for amorphization is significantly higher for heavy ion irradiation condi­tions compared to electron irradiation conditions.

In principle, faulted interstitial Frank loops can unfault by dislocation shear reactions. This should occur at a critical stage in interstitial loop growth, when the energy of the faulted dislocation loop, with a relatively small Burgers vector, becomes equal to an equivalently sized, unfaulted dislocation loop, with a larger Burgers vector. (In the absence of a stacking fault, the energy of a dislocation scales as b2, where b is the magnitude of the Burgers vector.) From this critical point on, the energy cost to incre­mentally grow the size of a dislocation loop favors the unfaulted loop, since there is no cost in energy due to a stacking fault within the loop perimeter. We examine first the unfaulting of 1/3 [0001] (0001) loops in alumina.

To unfault a 1/3 [0001] (0001) dislocation loop in alumina, we must propagate a 1/3[1010] partial shear dislocation across the loop plane.6 This is described by the following dislocation reaction:

3[0001] + |[10T0] ! 1[10І1] (basal) [7

faulted loop partial shear unfaulted loop

1/3 [1010]

This reaction is shown graphically in Figure 4. Note that the magnitude of 1 / 3 [1010] is approximately the Al—Al (and O—O) first nearest-neighbor spacing in Al2O3. When we pass a 1/3[1010] shear through a 1/3 [0001] (0001) dislocation loop, the cation planes beneath the loop assume new registries such that in eqn [2], ab a2, and a3 commute as follows: a1 ! a3 ! a2 ! a1. The anion layers beneath the loop are left unchanged (B ! B, C! C). Taking the faulted (0001) stacking sequence in eqn [2] and assuming that the planes to the right are above the ones on the left, we perform the 1/3[1010] partial shear operation as follows:

a1 B аг C аз B а1 C аг B аз C a1 B аг |C a1 B аг C аз B a1 C аг B аз C (faulted)

C аз B a1 C аг B аз C a1 B аг C

a1 B аг C аз B a1 C аг B аз C a1 B аг C аз B a1 C аг B аз C a1 B аг C (unfaulted)

[8]

After propagating the partial shear through the loop, we are left with an unfaulted layer stacking sequence. The Burgers vector of the resultant dislocation loop, 1 / 3 [1011], is a perfect lattice vector; therefore, the newly formed dislocation is a perfect dislocation. The resultant 1/3 [1011] (0001) dislocation is a mixed dis­location, in the sense that the Burgers vector is canted (not normal) relative to the plane of the loop.

Migration of He in Fe has been studied using

ab initio134 and EAM MD/MS methods.159,160 Inter­stitial Hei diffusion is almost athermal, with very low migration energy, Em « 0.06 eV134 to 0.08 eV.159,160 Hes diffuses by either (a) a classical vacancy exchange (Hes-V) mechanism; or (b) by dissociation mecha­nism that involves a Hes! Hei + V reaction, fol­lowed by the diffusion of Hei. The activation energy (Ea) for the dissociation mechanism is estimated

1.0

(b) VgbA (nm)

=——— 1 8 fl*—8S—8—■———————— ■——— =

0 2 4 6 8 10 12

(c) Distance from cluster (A)

Figure 39 (a) Substitutional He atom binding energies to edge and screw dislocations along with the excess volume as a function of the distance from the core. (b) Maximum binding energy of interstitial and substitution He atoms as a function of the excess volume per unit GB area. (c) Binding energies of vacancies and substitutional He to a 2-nm coherent Cu precipitate as a function of the distance.

Table 4 Binding energies of He atoms to various micro­structural features

Feature Maximum

binding energy (eV)

Hei Hes

to be «2.4 to 3.70 eV based on and EAM MD/MS methods, respectively.134’159 The vacancy exchange mechanism is similar to that for any substitutional solute, except that there is an unusually high binding energy for HesV complex. In bcc crystals Hes dif­fuses by a sequence of an initial Hes-V exchange, followed by a jump of the vacancy from the first nearest neighbor (NN) to a second NN position. Thus, Hes diffusion then requires that a (the same or another) vacancy jump back to a different NN position than the one involved in the initial exchange. The rates of exchange, including jumps to a third NN position, are needed to model the Hes diffusion coefficient,

DHes. Indeed, a minimum of five jump frequencies must be considered in modeling any substitutional solute diffusion coefficient in a bcc lattice, in this case Hes (DHes).276 The analytical five-frequency model provides both the correlation coefficient and net migration energy (Em) for Hes diffusion expressed in terms of the individual vacancy exchange activa­tion energies. The activation energies for the various exchanges have been evaluated by both MD159,160 simulations and ab initio calculations.134 These activa­tion energies have been used in the five-frequency model to estimate DHe as well as in direct KLMC simulations. The KLMC model yields:

DHes = 2.8 x 10~4exp(—2.35/kT)(m2s~1) [20]

He atoms also diffuse quasi one dimensionally along a dislocation core. The detailed mechanisms and acti­vation energies have been studied by MD and Dimer method.136 For example, interstitial Hei trapped on an a/2(111){110}edge dislocation in a-Fe is in a (111) crowdion configuration. Thus, He atoms can migrate along the dislocation line by jumping as a crowdion to an adjacent close-packed row with the migration energy of «0.4-0.5 eV.273 Hei also migrates along a a/2 (111) screw dislocation within or near the core with a similar migration energy of «0.4 eV, in this case via exchanges between octahedral interstitial sites. Hes migrates near the core of the screw dislocations by vacancy mechanism. The migration energy of 1.1 eV is associated with vacancy jumps from NN to second NN positions.274 Thus, the Em for Hei is higher on dislocations than in the matrix and lower for Hes.

Diffusion of He atoms on two symmetric tilt GBs, S3{112}, S 11{323}, was also studied using MD and Dimer methods.277-279 Hei diffusion was found to be one to three dimensional depending on the boundary characteristics. Hei diffuses one dimensionally along <1-13> direction in the S11{323} GBs at tempera­tures from 600 to 1200 K. In the S3{112} GBs, Hei diffuses two or three dimensionally at lower and higher temperatures, respectively.278 The mean square displacement in a long-term MD simulation indicated in Em = 0.28 (eV) for interstitial He migra­tion on S3{112} boundary and Em = 0.34 (eV) for migration on S11{323} boundary. Both of these GBs Em are higher than the value 0.087 eV in a-Fe lattice. The preexponential coefficient was found to be 4.35 x 10-8 (m2s—*) in both cases.

Dimer saddle point searches of possible migration paths of the Hei yield somewhat different Em but rationalize the slightly different results for the two

GBs. Possible migration paths of vacancies and Hes — vacancy complexes have also been studied using Dimer method. An important observation is that both tend to migrate one dimensionally, especially at low temperatures.279 These results depend on the EAM potentials and cannot be considered to be quantitative. However, the trends provide considerable insight, and it is notable that the estimated grain boundary vacancy and Hes-vacancy Em are lower in the GBs than bulk Fe matrix, while the Hei Em is higher. These Em are summarized in Table 5.

The definition of the dipole tensor and of multipole tensors with Kanzaki forces assumes that they are applied to atoms in a perfect crystal and selected such that they produce a strain field that is identical to the actual strain field in a crystal with the defect present. In particular, the dipole tensor reproduces the long-range part of this real strain field, and it can be determined from the Huang scattering measure­ments of crystals containing the particular defects. The actual specification of the exact Kanzaki forces is therefore not necessary. However, if the crystal is
under the influence of external loads, the Kanzaki forces may be different in the deformed reference crystal. Consider for example the case of a crystal with a vacancy and under external pressure. In the absence of pressure, the vacancy relaxation volume has a certain value. However, under pressure, the volume of the vacancy may change by a different amount than the average volume per atom, and therefore, the Kanzaki forces necessary to reproduce this additional change will have to change from their values in the crystal under no pressure. The change of the Kanzaki forces induced by the extraneous strain field may then also be viewed as a change in the dipole tensor by dPj. Assuming that this change is to first-order linear in the strains,

dpij = aijki [46]

The tensor a. jjki has been named diaelastic polariz­ability by Kroner27 based on the analogy with dia­magnetic materials.

When the change of the dipole tensor is included in the derivation of the interaction energy per­formed in the previous section, an additional contri­bution arises, namely

W2 = -2 a4u ej e0 [47]

The factor of 1/2 appears here because when the extraneous strain field is switched on for the purpose of computing the work, the induced Kanzaki forces are also switched on. This additional contribution W2, the diaelastic interaction energy, is quadratic in the strains in contrast to the size interaction, eqn [44], which is linear in the strain field.

A crystalline sample that contains an atomic frac­tion n of well-separated defects and is subject to external deformation will have an enthalpy of

H(c) = 2(Cijki — O j4e0i + O(V — pij4) N

per unit volume.

It follows from this formula that the presence of defects changes the effective elastic constants of the sample by

n

DCijki о aijki [49]

Such changes have been measured in single crystal specimens of only a few metals that were irradiated at cryogenic temperatures by thermal neutrons or elec­trons. Significant reductions of the shear moduli C44 and C’ = (C11-C12)/2 are observed from which the corresponding diaelastic shear polarizabilities listed in Table 9 are derived.25,28 These values are per Frenkel pair, and hence each one is the sum of the shear polarizabilities of a self-interstitial and a vacancy. By annealing these samples and observing the recovery of the elastic constants to their original values, one can conclude that the shear polarizabil­ities of vacancies are small and that the overwhelming contribution to the values listed in Table 9 comes from isolated, single self-interstitials.

The softening of the elastic region around the self­interstitial to shear deformation is not intuitively obvious. However, the theoretical investigations by Dederichs and associates29 on the vibrational proper­ties of point defects have provided a rather convinc­ing series of results, both analytical and by computer simulations. According to these results, the self­interstitial dumbbell axis is highly compressed, up to 0.6 of the normal interatomic distance between neighboring atoms. Therefore, the dumbbell axis can be easily tilted by shear of the surrounding lattice and thereby release some of this axial compression. The weak restoring forces associated with this tilt introduce low-frequency vibrational modes that are also responsible for the low migration energy of self­interstitials in pure metals.

Computer simulations carried out by Dederichs et ai. with a Morse potential for Cu gave the results presented in Table 10.

While the shear polarizabilities compare favor­ably with the experimental results for Cu listed in Table 9, the bulk polarizability in the last column of Table 10 is too large, and most likely of the wrong sign for the self-interstitial. The experimental results for Cu indicate that the bulk polarizability for the Frenkel pair is close to zero. Atomistic simulations

have also been reported by Ackland31 using an effec­tive many-body potential. The predicted diaelastic polarizabilies all turn out to be of the opposite sign than those reported by Dederichs and those obtained from the experimental measurements. Furthermore, Ackland also reports that the simulation results are dependent on the size of the simulation cell, that is, on the number of atoms. Evidently, the predictions depend very sensitively on the type and the particu­lar features of the interatomic potential as well as on the boundary conditions imposed by the periodicity of the simulation cell.

The model of the inhomogeneous inclusion pio­neered by Eshelby may be instructive to explain the diaelastic polarizabilities of vacancies and self­interstitials. A defect is viewed as a region with elastic constants different from the surrounding elastic con­tinuum. We suppose that this region occupies a spherical volume of NO, has isotropic elastic con­stants K and G, and is embedded in a medium with elastic constants K and G. Here, O is the volume per atom, N the number of atoms in the defect region, and Kand G the bulk and shear modulus, respectively. As Eshelby has shown, when external loads are applied to this medium and they produce a strain field Є0 in the absence of the spherical inhomogene­ity, then an interaction is induced upon forming it that is given by

W = —N Q (KA e0 4 + 2 GB [50]

31

aG = NQGB к a44 + (a11 — a12) [55]

The approximation in the last equation is based on Voigt’s averaging of the shear moduli of cubic mate­rials to obtain an isotropic value.

Let us first apply the formulae [52] to [55] to a vacancy. It seems plausible to select N = 1 and assume that K* G* 0. Then 15(1 — v) 7 — 5v

With these expressions, it is easy to compute the bulk and shear polarizabilities for vacancies, and their values are listed in the second and third columns of Table 11.

Next, we consider the bulk polarizability of self­interstitials. The two atoms that form the dumbbell are under compression, and the local bulk modulus that controls their separation distance may be esti­mated as follows:

K*» K + dK d. DU

dp dV

Here, Au’ is the volume compression of the dumbbell exerted on it by the surrounding material, and as shown in Section 1.01.4, it is given by

Au’ = Au — AV =(1/gE — 1)A V [59]

surrounds the dumbbell becomes significantly dilated due to nonlinear elastic effects. This additional dila­tation, 8V/O, has to be added to A V/O to obtain relaxation volumes that agree with experimental values. We repeat the values for 8V/O in the last column of Table 11 as a reminder. As a result of this additional dilatation, the atomic structure adja­cent to the dumbbell is more like that in the liquid phase, as it lost its rigidity with regard to shear. For this dilated region, consisting of Np atoms that include the two dumbbell atoms, we assume that its shear modulus G = 0. Then

15(1 — v)

7 — 5v and

ap = NpQGBI

If the dilated region extends out to the first, sec­ond, or third nearest neighbors, then Np = 14, 20, or 44, respectively, for fcc crystals, and Np = 10, 16, or 28 for bcc crystals. From these numbers we shall select those that enable us to predict a value for ap that comes closest to the experimental value. Matching it for fcc Cu indicates that the dilated region reaches out to third nearest neighbors, and hence Np = 44. However, the best match for bcc Mo is obtained with Np = 10, a region that only includes the dumbbell and its first nearest neighbors. These respective values for Np are also adopted for the other fcc and bcc elements in Table 11 , and the shear polarizabilities so obtained are listed in the fifth column.

To compare these estimates with experimental results, the approximation given in eqn [55] is used with the data in Table 9 for the shear polarizabilities of Frenkel pairs. These isotropic averages are listed in the sixth column of Table 11, and they are to be compared with (ap + ap). It is seen that the
inhomogeneity model is quite successful in explain­ing the experimental results, in spite of its simplicity and lack of atomistic details.

Materials can be classified based on the occupancy of the energy bands (Figure 13). In an insulator or a semiconductor, an energy band gap, Eg, is between the filled valence band, Ev, and the unoccupied (at 0 K) conduction band. In metals, the conduction band is partially filled (refer to Figure 13 ). Typical semicon­ductors have band gaps up to 1.5 eV; when the band gap exceeds 3.5 eV, the material is considered to be an insulator. Table 1 reports the band gaps of some important semiconductors (Ge, Si, GaAs, and SiC) and insulators (UO2, MgO, MgAl2O4, and Al2O3).

1.02.4.2 Excited States

The definition of an electronic defect is effectively ‘a deviation from the ground state electronic

3s —— —|-

configuration.’ The defects discussed in Section 1.02.4.2. were holes and electrons. Here, we consider defects in which the excited species is localized around the atom by which it was excited.

If an electron is excited into a higher lying orbital, there must be a difference between the angular momentum of the ground state and the excited state to accommodate the angular momentum of the photon that has been absorbed during the excitation process (conservation of angular momentum). For example, if the ground state is a singlet, then the excited state may be a triplet. A simple example would be 2p! 3s excitation of an oxygen ion in MgO (Figure 14).

Notice how the energy levels in Figure 14 alter their energies between the ground state and excited states. Therefore, in this case, it is not correct to estimate the energy difference between the ground state and excited states based on the knowledge of only the ground state energy configuration.

If the excitation energy is calculated based on the ground state ion positions, it is known as the Franck — Condon vertical transition. When a photon is ab­sorbed, the energy can be equal to this transition. However, the electron in the higher orbital will cause the forces between the ions to be altered. Con­sequently, the ions in the lattice will change their positions slightly, that is, relaxation will occur. Such relaxation processes are known as nonradiative, that is light is not emitted. Notice that the total energy of the system in the excited state decreases. However, if the triplet excited state now decays back to the singlet ground state (a process known as luminescence,18 see Figure 15), locally the ions are no longer in their optimum positions for the ground state. That is, the relaxed system in the ground state has become higher. The difference between the excitation energy

2p-H — +ь +ь

Ground state Excited state

Figure 14 The 2p! 3s excitation of an oxygen ion in MgO.

and the luminescence energy is known as the Stokes shift.18

Figure 16 represents an example of an excited state electron in MgO, known as a self-trapped exci — ton.19 The model uses the idea that an exciton is composed of a hole species and an excite electron. Notice that the excited electron has an orbit that is between the hole and its nearest neighboring cations. Thus, the hole is shielded from the cations. This means that the cations do not relax to the extent

they would if there was a bare hole (the small relaxa­tions are indicated by the arrows). Experimentally, the excitation energy in MgO is 7.65 eV, and the luminescence is 6.95 eV, which yields a small Stokes shift of only 0.7 eV.20

In comparison, a model for the exciton in alkali halides is shown in Figure 17. In this case, the exciton is composed of a Vk center (a hole shared between two halide ions) and an excited electron (the so-called Vk + e model). However, it is to be noted that the two halide ions that comprise the Vk center are not displaced equally from their original lattice positions. In fact, one of the halide ions is essentially still on its lattice site, while the other is almost in an interstitial site. As calculations suggest that the hole is about 80% localized on this intersti­tial halide ion, it is almost an interstitial atom known as an H-center. Also, the electron is shifted away from the hole center and is sited almost completely in the empty halide site (called an F-center). As such, the model is almost a Frenkel pair plus an electron localized at a halide vacancy (the so-called F—H pair model).

Whichever model is nearest to reality, Vk + e or F—H pair, it is clear that there is considerable lattice relaxation. This is reflected in the large Stokes shift. In LiCl, the optical excitation energy is 8.67 eV and the я-luminescence energy is only 4.18 eV, leading to a Stokes shift of 4.49 eV.21

As previously outlined in Sections 1.03.3.9 and

1. 03.4, phase changes associated with irradiation can be manifested in a variety of geometries, including randomly distributed matrix or grain boundary preci­pitates, continuous grain boundary films, precipitate — free zones near grain boundaries or other point defect sinks, spatially ordered arrays of precipitates, large-scale (>100nm) phase transformations, and

dissolution or growth of thermally stable precipitates. Preferential coupling of solute atoms to point defect fluxes can lead to modifications in the chemistry of precipitates as well as nucleation ofphases that would not be stable under thermal equilibrium conditions. Figure 37 shows an example of radiation-induced platelet precipitates observed in the grain interiors of V—4Cr—4Ti following neutron irradiation to 0.1 dpa at 505 °C.224 A precipitate-free zone is observed adjacent to the grain boundary in this figure.

1.06.2.1 Single, Dual, and Triple-Beam CPI

Single (He), dual (typically heavy ions to produce dpa and He), and triple (typically heavy ions, He and H) beam CPI have been extensively used to study He effects for a wide variety of materials and conditions. The number of facilities worldwide, both current and historically, and the large resulting liter­ature cannot be fully cited and summarized in this chapter, but some examples are given in Section

1.06.1 . A more complete overview of these facilities can be found in a recent Livermore National Labo­ratory Report.24 Extensive high-energy He implan­tation studies of creep properties were carried out at Forschungszentrum Julich using a 28 MeV He cyclo­tron.2 Major dual — and triple-beam studies were previously carried out at Oak Ridge National Labo­ratory (180 keV H, 360 keV He, 3.5 MeV Fe)26 and many other facilities around the world.24 The new JANNUS facility at Saclay couples a 3 MV Pelletron with a multicharged ion source and a 2.5 MV single Van de Graaff and a 2.25 MeV tandem accelerator.27 Another multibeam facility at Orsay couples a 2 MV couple, a tandem accelerator, and a 190 kV ion im — planter to a 200 kV transmission electron microscope (TEM) to allow simultaneous co-irradiation and observation.27

The advantages of He implantation and multibeam ion irradiations include the following: (a) conditions can be well controlled and in many cases selectively and widely varied; (b) high dpa, He, and H levels can be achieved in short times; (c) the specimens are often not, or only minimally, acti­vated; and (d) in situ TEM observations are possible in some cases. The disadvantages include the follow­ing: (a) highly accelerated damage rates compared with neutron irradiations; and in the case of
multibeam ion irradiations, (b) shallow damage depths and the proximity of free surfaces; (c) non­uniform damage production and the deposition of foreign ions; and (d) inability to measure bulk prop­erties. High-energy He implantation can be used on bulk specimens tested, either in situ or post­implantation, to measure tensile, creep, and creep rupture properties. The corresponding disadvantages are that He implantation results in high He/dpa ratios («6000 appm He/dpa).2 The differences be­tween CPI and neutron irradiation can significantly affect microstructural evolution.

Thus, it must be emphasized that He implantation and multibeam CPI do not simulate neutron irradia­tions. Although it has been argued that CPI reveal general trends and that corrections, like temperature adjustments, allow extrapolations to neutron-irradiation conditions, both assertions are problematic. The proper role of He implantation and multibeam CPI is to help inform and calibrate models and to identify and quantify key processes based on carefully designed mechanism experiments.