Consider then a small defect with a radius p situated at the center of an elastically isotropic sphere of radius R. The defect is modeled in four different

ways:

As a center of dilatation (CD): the defect is created by displacing the inner radius r of the surrounding matrix by an amount

u(P) — cp [A2]

Here, the parameter c is referred to as the strength of the defect.

As a cavity (CA): the inner surface is loaded by a pressure p, giving rise to a radial stress component in the surrounding medium, which has the boundary value
partial differentiation. Hooke’s law then connects the stresses with the elastic part of the strains, which for the isotropic case reads as

Oij m(uij A Uj;i) A 1dijuk, k 2pej [A5]

Here, repeated indices imply a summation and a comma before an index, say j, means a partial deriva­tive with regard to xj. The mechanical equilibrium equations are Oj;j — 0, or after substitutions of Hooke’s law

(m A 1)uy; ji A pui; jj — 0 [A6]

The pressure p can also represent radial forces that the defect exerts on its nearest neighbor atoms of the surrounding matrix, and they are then referred to as Kanzaki forces.

As an inclusion (INC): the material within the defect region is subject to a transformation strain ej as if it had endured a phase transformation. We shall treat here only the simple case of an isotropic transformation strain ej — pd, j where dj is the Kronecker matrix.

As an inhomogeneity (IHG): In addition to a transformation strain, the defect region has also acquired different elastic constants from the sur­rounding matrix.

The boundary conditions at the interface between the defect and its surrounding matrix are that the radial displacement and the radial stress component must be continuous.

These different models of defects correspond in essence to the three possible boundary conditions on the defect-matrix interface, namely the Dirichlet boundary condition for the CD, the von Neuman condition for the CA, and the mixed one for the INC and IHG.

One could also consider these three different boundary conditions for the external surface. However, prescribing a value for the displacement of the external surface or a mixed boundary condition implies the presence of yet another surrounding medium, in which the defect-containing sphere would be embed­ded. Here, these cases will not be considered.

We then restrict the following treatment to the boundary condition of a free external surface for which

Or(R) = 0 [A4]

The solution of the elasticity problems associated with all defect models is rather elementary and can be found in many textbooks. However, as a way to intro­duce our notation, we sketch the procedure.

The deformation is described by a displacement field «i(r), from which the strain tensor is obtained by

We assumed here that the transformation strains eij are constant within the defect region.

For a spherically symmetric problem, the dis­placement field possesses only one radial component u, which satisfies the differential equation 1 d / 2 s

7dT(r u)

The solution to this equation is

u(r) — At A B/r2 [A8]

with two unknown constants A and B for each region, for the matrix that surrounds the defect and for the defect region itself. However, for the defect region B — 0. The remaining constants are to be determined by the boundary conditions. For later use we give the radial stress component

sr — (2m a 31)(a — m) — 4mB/r3

— 3K (a — m) — 4mB/r3 [A9]

for the case of an isotropically transformed inclusion, that is, when ej — mdj. Furthermore, instead of the Lame’s constants m and l, we introduced the bulk modulus K and the shear modulus m. The following ratio involving the two elastic constants will appear often, and we reserve the symbol

4m _ 2(1 — 2v) 3K — 1 A v

for it, where v is Poison’s ratio and gE is the Eshelby factor.

The hydrostatic stress is related to the elastic dilatation or the lattice parameter change by the equation

1 . Da

Oh — — ff„ — 3K (a — m) — 3^ [A11]

3 a

We see that for a defect in the center of an isotropic sphere the lattice parameter changes are uniform throughout the matrix.

The density of strain energy, on the other hand, is strongly peaked near the defect, as can be seen from the formula

f (r) = [1] [2] GijUij = 2K(A — rj)[3] + 6pB2/r6 [A12]

Integrating this function over the entire sphere gives the strain energy associated with the defect,

tR

(r)r2dr [A13]

0

Finally, let us state the formulae for volume changes. The external volume change is computed from the equation

A V = 4pR2u(R) = 4pR[4] [5](A + B/R3)

or AV = 3(A + B/R3) [A14]

In a similar fashion, one defines the change of the defect volume due to the constraints imposed by the surrounding matrix, as

AU = 3(A + B/p3) [A15]

A last point must be made regarding the measure­ment of the lattice parameter change.

If the atoms in the defect region contribute to the diffraction pattern just as the matrix atoms, we must calculate the appropriate average lattice parameter change over the entire volume V. We denote this
average by < > brackets. However, if the defect region consists of a very different crystal structure or atoms with much lower form factors, then the defect region does not contribute to the diffraction pattern, and the lattice parameter change is due to the matrix only. When the defect region is excluded, the lattice parameter change is indicated by ( ) parentheses.

We obtain the coefficients A and B by solving the equations for the boundary conditions. For example, for the case of an inhomogeneity, the radial stress components vanish on the external surface,

sr(R) = 3KA — 4pB/R3 = 0 [A16]

On the interface between the inclusion region and the matrix, that is, at r= p, the displacements and the radial stress components must be continuous, or

AiP = Ap + B/p2 [A17]

and

3Ki(Ai — r) = 3KA — 4pB/p3 [A18]

Here, the subscript ‘I’ designates a parameter for the inclusion region, while parameters for the matrix are without a subscript. Solution of the three eqns [A17]-[A19] determines the three parameters A, B, and AI listed in the first three rows and the last column of Table A1. We note that the stress within the inhomogeneous inclusion is purely hydrostatic (because BI = 0), and the results do not depend on

Table A1 The solutions for the integration constants A and B, and volume changes, lattice parameter changes, and defect strain energies that derive from them

IHG

the shear modulus of the inhomogeneity, but only on the ratio

к = K/KI [A19]

For the inclusion (INC), its bulk modulus is the same as for the matrix. Therefore, its parameters follow from those for the IHG by setting к = 1.

With the parameters A and B determined, it is straightforward to evaluate volume changes, lattice parameter changes, and the defect strain energy U0. Table A1 contains a tabulation of all these para­meters for the four different defect types, and for all possible values of the defect volume fraction S. For application to defect concentrations in irradiated materials, it may be assumed that S<<1, and the expressions in Table A1 can be simplified to those shown in Table A2.

Solute atoms of importance include elements origi­nally added to the material during fabrication and species produced by nuclear transmutation reactions (e. g., He and H, and a range of other elements). Solute atoms may exhibit preferential coupling with point defects created during irradiation, leading to either enhancement or depletion of solutes at point defect sink structures such as dislocations, grain boundaries, preexisting precipitates, and voids.193-198 The solute-defect coupling can modify the kinetics for point defect diffusion, and the resultant solute enrichment or depletion may sufficiently modify the local composition to induce the formation of new phases. There are three general categories of precip­itation associated with radiation-induced segregation processes10 , : radiation-induced (phases that form due to irradiation-induced nonequilibrium solute segregation and dissolve during postirradiation annealing), radiation-enhanced (precipitate forma­tion accelerated or occurring at lower temperatures due to irradiation, but are thermally stable after formation), and radiation-modified (different chemi­cal composition of precipitates compared to thermo­dynamically stable composition). In some materials,
radiation-retarded precipitation (phase formation shifted to higher temperatures or longer exposure times) has been reported.2

A phenomenon that is uniquely associated with ion irradiation is the potential for the ions from the irra­diating beam to modify the microstructural evolution by perturbing the relative balance of SIAs compared to vacancies flowing to defect sinks. The injected ions act as a source of additional interstitial atoms and can significantly suppress void nucleation and growth.149,154,201,202 The peak concentration of the injected ions occurs near the displacement damage peak for ion irradiation, and therefore considerable care must be exercised when evaluating the void swelling data obtained near the peak damage region in ion-irradiated materials.15 ,201,202 Figure 22 shows an example ofthe dramatic changes in microstructure that can occur in the injected ion region.203 In this example, void formation in ion-irradiated nickel at 400 °C is completely suppressed in the regions with the injected interstitials and the void microstructure is replaced with an aligned array of small interstitial — type dislocation loops.

Numerous studies have observed that the precipita­tion behavior during irradiation can strongly influence microstructural evolution, for example, the swelling behavior of austenitic stainless steels.103,106,204-206

_i_________________ і________________ і_________________ і———————— >————————- 1——————— — j— ———————- 1—

0 1 2 3 a) Voids

Depth (pm) b) Random loops

and voids
c) Ordered loops

Figure 22 Cross-section TEM microstructure of nickel irradiated at 400 °C with 14 MeVCu ionstoafluenceof 5 x 1020 ions m~2 which produced a peak damage level of about 55 dpa at a depth near 2 pm. Void formation is completely suppressed in the injected interstitial regime (~1.3-2.8 pm) and the void microstructure is replaced with an array of small interstitial-type dislocation loops aligned along {100} planes. Reproduced from Whitley, J. B. Depth dependent damage in heavy ion irradiated nickel. University of Wisconsin, Madison, 1978.

0.4dpa/0.2 appm He/675 0C 109dpa/2000 appm He/675 0C

Figure 23 Comparison of the cavity microstructures for a pure Fe-13Cr-15Ni austenitic alloy (left panel) and the same alloy with P, Si, Ti, and C additions that produced dense radiation-induced phosphide precipitation (center and right panels) following dual beam Ni + He irradiation at 675 °C. The irradiation conditions were 0.4 dpa and 0.2 appm He for the left panel (70 dpa and 35 appm for the inset figure), and 109 dpa and 2000 appm He for the other two figures. Reproduced from Mansur, L. K.; Lee, E. H. J. Nucl. Mater. 1991, 179-181, 105-110.

In extreme cases, large-scale phase transformations can occur such as the g (austenite, fcc) to a (ferritic, bcc) transformation in austenitic stainless steel following high dose neutron irradiation.106,207 Depending on the type of precipitation, either enhanced or suppressed swelling can occur. Void swelling enhancement has generally been attributed to a point defect collector mechanism and typically occurs for moderate densities of relative coarse precipitates such as G phase in austenitic stainless steels, whereas void swelling suppression is generally observed for high densities of finely dispersed precipi­tates and is usually attributed to high sink strength effects.103,1 1,208 Figure 22 shows an example of the strong void swelling suppression associated with forma­tion of radiation-induced Si — and Ti-rich phosphide precipitates compared to a simple Fe-Cr-Ni ternary austenitic alloy.208 Similarly, the He/dpa ratio can influence the types and magnitude ofpoint defect clus­ters and precipitation due to modifications in the point defect evolution under irradiation (Figure 23).106

So far we have considered Coulombic charge and fault­ing for 1/3 [0001] (0001) loops in alumina and 1/6 (111) {111} loops in spinel. Now, we must repeat these considerations for 1/3(1011){1010} prismatic loops in alumina and 1/4 (110) {110} loops in spinel. We begin with alumina prismatic loops. Alumina {1010} prism planes contain both Al and O in the ratio 2:3, that is, identical to the Al2O3 compound stoichiometry. Along the (1010) direction normal to the traces of the {1010} planes, the registry of the {1010} planes varies between adjacent planes, analo­gous to the registry shifts that occur between adjacent (0001) basal planes in alumina (discussed earlier). However, the patterns of Al atoms in all {1010} planes are identical. Similarly, the O atom patterns are identi­cal in all {1010} planes. The registry of the O atom patterns between adjacent {1010} planes alternates every other layer, analogous to the BCBC… stacking of oxygen basal planes (Table 1, eqns [2-4]). On the other hand, the registry of the Al cation patterns is distinct from the O pattern registries (B and C), and the registry ofthe Al patterns only repeats every fourth layer. In other words, the stacking sequence of {1010} plane Al atom patterns can be described using the same nomenclature as in Table 1 and eqn [2] for (0001) alumina planes, that is, ax a2 a3 ax a2 a3…. Putting the anion and cations together, we can write the {1010} stacking sequence in alumina as (a1B) (a2C) (a3B) (axC) (a2B) (a3C).

Now, as with the basal plane story described earlier, when an extra 1/3(1010) two-layer block, (Al2O3)x-(Al2O3)x, is inserted into the stacking sequence, (ajB) (a2C) (a3B) (axC) (a2B) (a3C), a stack­ing fault occurs as follows:

(a1B) (a2C) (a3B) (a1 C) (a2B) (a3C) (a1B)

(a2C) (a3 B) (a1 C) (a2B) (a3 C) (before)

(a1B) (a2C) (a3B) (a1 C) (a2B) (a3C) (a1B) (a2C)

(a1B) (a2 C) (a3B) (a1 C) (a2 B) (a3C) (after)

(a1B) (a2C) (a3B) (a1 C) (a2B) (a3C) (a1B) (a2C) | (a1B) (a2 C) (a3B) (a1 C) (a2 B) (a3C)

(after, showing stacking fault position) [5]

Notice in eqn [5] that after block insertion, the anion sublattice is not faulted (BCBC… layer stacking is preserved), whereas the cation sublattice is faulted, specifically at the position of the red vertical line in the last sequence. Similar to the case of basal plane interstitial loop formation in alumina (discussed ear­lier), the dislocation loop formed by 1/3(1010) block insertion in alumina is an intrinsic, cation-faulted, sessile interstitial Frank loop.

The results of atomistic modeling of He-defect inter­action energies are presented in this section for both ab initio and MD and molecular statics (MS) meth­ods. The ab initio results provide interaction energies 0 K that can be used in Monte Carlo simulations that are generally restricted to very small, nonequilib­rium, high-pressure (solid) He-vacancy (V) clusters. This contrasts with the models described in Section 1.06.3 that treat He as a continuum phase within the capillary approximation. In this latter case at suffi­ciently large bubble sizes and high temperatures, and in the absence of irradiation, the pressure in a relaxed strain-free bubble is given by the capillary approximation as P = 2g/ту, where the gas pressure is balanced by the average surface tension. Note that this approximation ignores the faceted shape of small bub­bles.260 This is a lower energy cluster configuration than those at higher He pressures that are balanced by a contribution from an elastically strained matrix, P = (2g + mb)/rb, where m is the shear modulus and b is the Burgers vector. MD can simulate gas bubbles at finite temperatures.260-263 A recent MD study at tem­peratures up to 700 K shows that the He pressure is very high in 2 nm clusters but saturates at «25GPa for a He/vacancy ratio of 3, due to spontaneous SIA emission at the theoretical strength of the Fe lattice.260 The MD simulations show that the equilibrium bubble He/vacancy ratio (m/n) is <1 and decreases with
increasing temperature and bubble size as expected. However, the MD studies suggest that the pressure in small bubbles predicted by MD is a factor of ~2 times lower than that predicted by the simple capillary approximation for Van der Waals and hard sphere equations of state.263 The reasons for this discrepancy are not yet understood.

Although quantitative details differ, the most important results of the atomistic models are that (a) both substitutional and interstitial He have very high formation energies; (b) there are large positive binding energies for He and vacancies in HemVn clusters even at m/n > 1; (c) at high m/n, the clusters relax and increase n by emitting SIA, or punching SIA loops; clusters with m/n < 1 are very stable.

Ab initio calculations in the framework of density functional theory (DFT) have been used to obtain formation and binding energies (Eb) of small clusters typically containing up to combinations of four He atoms and vacancies.82,134 The results are sum­marized in Table 3. Here, the Eb of the He in the cluster is referenced to tetrahedral Hei, with an energy that is slightly lower than that for other interstitial configurations. The ab initio calculations show that Hei clusters are bound to even absent vacancies. However, the Eb for He-V clusters is much larger. The Eb for He monotonically decreases, while the vacancy Eb monotonically increases, with larger helium to vacancy ratios, m/n, except for the case HemV, where the He Eb do not change much for m < 4. These trends reflect a high level of overpres­surization in the small clusters relative to a relaxed bubble with n > m. The He Eb also increases with cluster size, from 2.3 eV at m = n = 1 to 3.05 eV at m = n = 4. The most important implications of

these results are that, except at very high tem­peratures, m > «2 and n > «2 clusters are likely the stable nucleation sites for He-vacancy cluster formation and subsequent bubble evolution, as assumed in the models described in Section 1.06.3. Once formed and equilibrated, even small He-vacancy clusters are extremely stable up to very high temperatures.

MD simulations of larger HemVn (m < 20, n < 20) clusters predict vacancy Eb trends that are very similar to those found in the ab initio calculations for small clusters.265 However, the Eb for He derived from these MD simulations are consistently «1e V higher than those from ab initio calculations. The MD simulations used an embedded atom method (EAM) potential for Fe-Fe266 and pair potentials for Fe-He267 and He-He.268 While this set of potentials has been widely used, they predict significantly larger differ­ences between Hes and Hei formation energies than the ab initio results.269 Improved EAM potentials have been developed and a three-body potential for Fe-He-Fe264,269 predicts cluster formation and binding energies that are generally closer to the ab initio results.

Note that, while we are not aware of specific simulation results, overpressurized bubbles would be expected to be biased sinks for vacancies and against SIA. Taken together, these results further reinforce the assumption that, in vacancy-rich envir­onments produced by displacement damage, stable He-V clusters nucleate at very small sizes (m — n = 2) and grow along a near-equilibrium bubble path. Thus, a critical issue is how He is transported and interacts with other microstructural features.

The dumbbell configuration of a self-interstitial gives it a certain orientation, namely the dumbbell axis, and upon migration this axis orientation may change. This is indeed the case for self-interstitials in fcc metals, as illustrated in Figure 15.

Suppose that the initial location of the self­interstitial is as shown on the left, and its axis is along [001]. A migration jump occurs by one atom of the dumbbell (here the purple one) pairing up with one nearest neighbor, while its former partner

(the blue atom) occupies the available lattice site. Computer simulations of this migration process have shown25 that the orientation of the self-interstitial has rotated to a [010] orientation, and that this combined migration and rotation requires the least amount of thermal activation.

Similar analysis for the migration of self­interstitials in bcc metals has revealed that a rotation may or may not accompany the migration, and these two diffusion mechanisms are depicted in Figure 16. Which of these two possesses the lower activation energy depends on the metal, or on the interatomic potential employed for determining it.

In general, however, the activation energies for self-interstitial migration are very low compared to the vacancy migration energy, and they can rarely be measured with any accuracy. Instead, in most cases only the Stage I annealing temperatures have been measured. In the associated experiments, specimens for a given metal are irradiated at such low tempera­tures that the Frenkel pairs are retained. Their con­centration is correlated with the increase of the electrical resistivity. Subsequent annealing in stages then reveals when the resistivity declines again upon reaching a certain annealing temperature. The first annealing, Stage I, occurs when self-interstitials become mobile and in the process recombine with
vacancies, form clusters of self-interstitials, or are trapped at impurities. Table 8 lists the Stage I temperature,7 Tjm, for pure metals as well as two alloys that represent ferritic and austenitic steels. For a few cases, an associated activation energy Em is known, and in even fewer cases, a preexponential factor, D0, has been estimated.

1.02.4.1 Formation

Electronic defects are formed when single or small groups of atoms in a crystal have their electronic structure changed (e. g., electrons removed, added, or excited). In particular, they are formed when an electron is excited from its ground state configuration into a higher energy state. Most often this involves a valence electron, although electrons from inner orbits can also be excited if sufficient energy is available. In either case, the state left by this transition, which is no longer occupied by an electron, is usually termed a hole. These defects can be generated thermally, opti­cally, by radiation or through ion beam damage. The excited electron component may be localized on a single atomic site and if the electron is transferred to another center, it is represented as a change in the ionization state of the ion or atom to which it is localized. This is sometimes described as a small polaron or trapped electron. Such electronic defects might migrate through the lattice via an activated hopping process. An example of a small polaron electron is a Ce3+ ion in CeO2_x.16 Alternatively, the excited electron may be delocalized so that it moves freely through the crystal. In this case, the electron occupies a conduction band state, which is formed by the superposition of atomic wave functions from many atoms. This is the case with most semi­conductor materials. Similarly, the hole may also be localized to one atomic center and be represented as a change in the ionization state of the ion or atom. Holes may also move via an activated hopping pro­cess. An example is a Co3+ ion in Co1_xO. Similarly, the hole may also be delocalized. Intermediate situa­tions may occur with the hole or electron being localized to a small number of atoms or ions (known as a large polaron) or a specific type of hole state associated with a particular chemical bond.

The relationship between doping and its influence on electronic defects is of great technological impor­tance in the field of semiconducting materials. For example, doping silicon with defect concentrations in the order of parts per million is sufficient for most microelectronic applications. Incorporation of a phosphorous atom in silicon results in a shallow state below the conduction band that will easily donate an electron to the conduction band. The remaining four valence electrons of the phosphorous dopant will form sp3 hybrid bonds with the four neighboring tetrahedral silicon atoms. Recently, it has been suggested that the state from which the electron is removed is associated with the dopant species and the four silicon atoms surrounding it; in other words, it is associated with a cluster.17

Mechanical deformation of metals and alloys after irradiation at temperatures below recovery Stage V produces deformation microstructures that typically evolve from predominantly dislocation cell micro­structures in the unirradiated and low-dose irradiated conditions to a variety of localized deformation microstructures above a threshold damage level including twinning, planar dislocation deformation, and formation of dislocation channels.314-316 Forma­tion of cleared dislocation channels has been sug­gested to be the cause of low uniform elongations observed in tensile tests of metals and alloys irradiated at temperatures below recovery Stage V,221,317 and dislocation channeling is frequently observed follow­ing deformation of irradiated materials that exhibit low uniform elongation.95,96,100,312,316’318-321 An alter­native mechanism for the low uniform elongations in irradiated materials, based on a material-specific threshold stress for plastic instability, has also been proposed.216,322-324 The spacing between dislocation channels is typically on the order of 1 pm, and the width of the individual channels ranges from ^20 to 200 nm. Localized deformation visible as surface slip steps in irradiated copper following tensile straining has been directly correlated with cleared dislocation channels.325 The matrix regions between the cleared channels do not exhibit evidence of substantial dislo­cation activity, suggesting that all of the dislocation motion associated with deformation is restricted to the dislocation channel regions. Figure 35 shows an example of cleared dislocation channels observed in austenitic stainless steel following fission neutron

irradiation to 0.78 dpa near 80 °C and subsequent uniaxial tensile deformation to 32% strain.326

The mechanisms responsible for annihilation of SFTs by gliding dislocations within the dislocation channel include stress-induced collapse to triangle loops, multiple shear, partial annihilation with a rem­nant apex, collapse to a triangle loop or complete annihilation with multiple super jogs, and complete annihilation by screw dislocations followed by cross slip.327-329 Computer simulations of dislocation loop interactions with gliding dislocations suggest multi­ple potential mechanisms that could lead to defect — cleared dislocation channels, including absorption, unfaulting, and shear of the loops.330-333 Detailed experimental confirmation of these annihilation mechanisms is still needed.

Numerous additional ceramics have been either used or proposed for nuclear reactor materials applica­tions. These include graphite (discussed in other chapters in this volume) as well as carbides and nitrides, such as ZrC and ZrN, which have higher thermal conductivities than their sister oxide com­pound, ZrO2. Research into the radiation damage properties of these materials is in its infancy, and therefore, these compounds are not described in further detail here.

1.05.2. Summary

The response of ceramic materials to radiation is espe­cially complex because ceramics (with the exception of graphite) are made up of anions and cations (some­times several different cations) such that the atomic defects that initiate radiation damage are different in their size, chemistry, charge, mobility, and so on. Thus, it is difficult to predict how the microstructure of a ceramic will evolve under irradiation and, in turn, how properties such as structural stability will change in response to the radiation-induced microstructural alterations. Nevertheless, we present a case study (described below) wherein researchers have succeeded in explaining the extraordinary differences between the radiation responses of two important engineering ceramics.

We devoted much of this chapter to comparing and contrasting the high-temperature radiation dam­age response of two quite similar refractory, dielectric ceramics: a-alumina (Al2O3) and magnesio-aluminate spinel (MgAl2O4). Al2O3 is highly susceptible to radiation-induced swelling, whereas MgAl2O4 is not. The swelling of Al2O3 is due to excessive void forma­tion in the crystal lattice. We considered in detail in this chapter the atomic and microstructural mechan­isms that help to explain why voids nucleate and grow in Al2O3 to a very significant degree, whereas in MgAl2O4, this problem is much less pronounced. We showed that the reasons for the great differences between the radiation damage behavior of Al2O3 and MgAl2O4 have mainly to do with differences in the way interstitial loops nucleate and grow in these two oxides. The hope is that by understanding these dif­ferences, we will by analogy be able to understand the radiation damage behavior of other ceramic materials.

In this chapter, we also examined two different phenomena that lead to degradation in the mechani­cal properties of ceramics: (1) nucleation and growth of interstitial dislocation loops and voids and (2) crystal-to-amorphous phase transformations. Both these phenomena cause macroscopic swelling of materials. This ultimately leads to the failure of materials because of unacceptable dimensional changes, microcracking, excessive increases in hard­ness (or alternatively, softening in the case of amor- phization), and so on.

We concluded this chapter with brief discussions of a few ceramics additionally important for nuclear energy applications, namely silicon carbide (SiC), uranium dioxide (UO2), and graphite (C).