Internal stress fields from dislocations and other defects such as precipitates are spatially varying, that is, they are functions of the location r of the migrat­ing point defect. However, the stresses o;y(r) may be viewed as continuous functions except at material interfaces such as grain boundaries and free surfaces, and their differences between adjacent lattice sites may be approximated by і R where a summation over the repeated index k is implied. The saddle point energy as given in eqn [81] is now replaced by

= ES — °ер1 spq(r)-0

When this is inserted into eqn [76], a new diffusion tensor is given obtained that, to linear order in the stress gradients, is given by

Xjk

R, m,v

exp [-bELM + b/(r)] [86]

Here, the first term is just the diffusion tensor obtained for a uniform stress field, but now with the stress replaced by the local stress field at r. The second term is a correction linear in the stress gradient. How­ever, this term vanishes for the following reasons. We have seen that for fcc crystals there are pairs of oppo­site jump directions that have identical transformation strain values ep^, but they differ only in the signs of the jump vector components Xk. Hence, these opposite pairs of jump directions cancel each other’s contribu­tion to the sum in eqn [86]. As a result, there is no correction to the diffusion tensor that is linear in stress gradients.

Spatially varying stress field, however, produce a drift force. Inserting eqn [84] into the formula [77] leads to

-bEL(r) + bEf (r)

The factor with the sum has a remarkable resem­blance with the expression for the diffusion tensor. Indeed, it is straightforward to show that

d @

F‘(r) = — dXkD, k(r) — Dik(r)bOfdXkspq(r) [88]

Since the average energy of the point defect in its stable configuration in the presence of a stress field is given by

EF (r) = Ef — Of spq (r) [89]

we may also write eqn [88] as

F (r) = — exp [b£f (r)] dL {exp [-b£f (r)] D, k(r)} [90]

In this last expression, the product of the two func­tions in the curly brackets depends now only on the saddle point energies, and this shows that it is the spatial dependence of the saddle point energy only that gives rise to a drift force, while the variation of the defect formation energy is not contributing to the drift force.

The general formulae [88] or [90] reveal that stress — induced diffusion anisotropy affects the direction of the drift force such that it is in general not collinear with the stress-induced interaction force.

A wide range of defect cluster morphologies can be created by particle irradiation.8,21,22 The thermody­namic stability of these defect cluster geometries is dependent on the host material and defect cluster size as well as the potential presence of impurities. There are four common geometric configurations for clusters of vacancies and self-interstitial atoms (SIAs): two planar dislocation loop configurations (faulted and perfect loops) that occur for both vacancies and SIAs, and two three-dimensional configurations that occur only for vacancy clusters (the stacking fault tetrahedron, SFT, and cavities).

The faulted loop (also called Frank loop) is most easily visualized as either insertion or removal of a layer of atoms, creating a corresponding extrinsic or intrinsic stacking fault associated with condensation of a planar monolayer of vacancies and SIAs, respec­tively. The faulted loop generally forms on close packed planes, i. e., {111} habit planes with a Burgers vector of b = 1/3(111) for face-centered cubic (fcc) materials, {110} habit planes with b = 1/2(110) for body-centered cubic (bcc) metals, and {1010} habit planes with b = a/2 (1010) for hexagonal close packed (HCP) metals.23 Faulted loops with b = a/2 [0001] on the (0001) basal plane are also observed in many irradiated HCP materials. All of these faulted loops are immobile (sessile). The high stacking fault energy of bcc metals inhibits faulted loop nucle — ation and growth, and favors formation of perfect loops. There have been several observations of faulted loops consisting of multiple atomic layers.8,21

The perfect loop in fcc materials is typically created from initially formed faulted loops by nucleation of an a/6(112) Shockley partial dislocation that sweeps across the surface of the faulted loop and thereby restores perfect stacking order by this atomic shear of one layer of atoms. The resultant Burgers vector in fcc materials is a/2 (110), maintaining the {111} loop habit planes. After unfaulting, rotation on the glide cylinder gradually changes the habit plane of the fcc perfect loop from {111} to {110} to create a pure edge loop geometry. After the loop rotates to the {110} habit plane, the perfect loop is glissile. Experimental studies of irradiated fcc materials typically observe perfect loops on either {111} or {110} habit planes (or both), depending on the stage of the glide cylinder rotation process. The glissile perfect loop configurations for bcc materials consist of b = a/2 (111) loops on {111} habit planes and b = a( 100) loops on {100} habit planes. The typical corresponding HCP perfect loop configura — tionis b = a/3 (1120) on {1120} prismatic habit planes.

SFTs are only observed in close-packed cubic structures (i. e. fcc materials). The classic Silcox — Hirsch24 mechanism for SFT formation is based on dissociation of b = 1/3(111) faulted loops into a/6 (110) stair rod and a/6(121) Shockley partial dis­locations on the acute intersecting {111} planes. Interaction between the climbing Shockley partials creates a/6 (011) stair rod dislocations along the

tetrahedron edges. The Silcox-Hirsch mechanism has been verified during in situ transmission electron microscope (TEM) observation of vacancy loops in quenched gold.25 Evidence from molecular dynamics (MD) simulations26-29 and TEM observations12’19’30-32 during in situ or postirradiation studies indicate that SFT formation can occur directly within the vacancy — rich cascade core during the ‘thermal spike’ phase of energetic displacement cascades.

There is an important distinction between the defi­nitions for the terms void’ bubble’ and cavity’ all of which describe a three-dimensional vacancy cluster that is roughly spherical in shape. Void refers to an object whose stability is not dependent on the presence of internal pressurization from a gaseous species such as helium. Bubbles are defined as pressurized cavities. The term cavity can be used to refer to either voids or bubbles and is often used as a generic term for both cases. In many cases, voids exhibit facets (e. g. truncated octahedron for fcc metals) that correspond with close — packed planes of the host lattice’ whereas bubbles are generally spherical in shape. However, the absence of facets cannot be used as conclusive evidence to dis­criminate between a void and a bubble.

Figure 1 shows the calculated energy for different vacancy geometries in pure fcc copper.22 The SFT is calculated to be the most energetically favorable configuration in copper for small sizes (up to about 4 nm edge lengths). Faulted loops are calculated to be stable at intermediate sizes, and perfect loops are calculated to be most stable at larger sizes. In practice,
many metastable defect cluster geometries may occur. For example, it is well established that the transition from faulted to perfect loops is typically triggered by localized stress such as physical impin­gement of adjoining loops, and not simply by loop energies; the activation energy barrier for unfaulting may be on the order of 1 eV atom-1.8 Similarly, large activation energy barriers exist for the conversion between planar loops and voids.33

This discussion has used neutron irradiations for illustration purposes. Reactors provide an effective instrument for achieving high neutron exposures under conditions relevant for most nuclear applica­tions. However, reactor irradiations suffer from many difficult-to-control and, sometimes, uncontrolled variables. The neutron energy spectrum is responsi­ble for large differences in irradiation effects between different reactors.

The mechanism of atomic displacement is well understood.15 With a known neutron energy spec­trum, neutron atomic displacements can be calcu­lated as a function of fluence for a given reactor. Transmutation of elements in the material under study, which is a strong function ofneutron spectrum, results in wide variation in some mechanical proper­ties. This is of particular importance in applying fission reactor results to fusion. In a fusion device, helium and hydrogen will be generated through (n, a) and (n, p) reactions in nearly all common structural materials. Hydrogen has a very high diffusivity in metals so that an equilibrium concentration will be
established at a level that is believed to be benign.16 By contrast, helium is insoluble in metals, segregating at grain boundaries and other internal surfaces and discontinuities.

Although helium is produced in all nuclear reac­tors, the thermal spectrum is responsible for the highest concentrations. The largest contributors to helium in a thermal reactor are boron and nickel by the following reactions:

10B(n, a)7Li

58Ni(n, g)59Ni 59Ni(n, a)56Fe

Boron is present as a trace element in most alloy­ing elements but only at ppm levels. Nickel is a major constituent of many alloys and a minor con­stituent of still others. The two nickel reactions constitute a two-step generation process for helium, which starts slowly and accelerates as 59Ni builds up in the alloy, limited only by the supply of 58Ni, which for practical purposes is often unlimited. In austenitic alloys, the high flux isotope reactor (HFIR) has generated over 4000 appm He in austenitic stainless steels. The generation rate is so high that multistep absorber experiments have been conducted to reduce the helium generation rate to that characteristic of fusion reactors, 12 appm He per dpa in austenitic stainless steels.17 (see Chapter 1.06, The Effects of Helium in Irradiated Structural Alloys).

Other transmutation products may also compli­cate reactor irradiation studies. Examples are the transmutation of manganese to iron by the follow­ing reaction: 55Mn (n, g) 56Mn! 56Fe and the trans­mutation of chromium to vanadium by 50Cr (n, g) 51Cr! 51V. The first reaction leads to loss of an alloy constituent, and the second leads to doping with an extraneous element. However, neither of these reactions has been shown to significantly affect mechanical properties of steels.18

Helium remains the most studied transmutation product, and it can have profound effects on tensile properties, especially at high temperatures. Experi­ments have been conducted in various reactors throughout the world to assess the effects of helium on mechanical properties of alloys.19 An interesting result is that helium has little effect on strength. This is illustrated in Figure 5 where a comparison has been made between austenitic steels irradiated in Rapsodie, a fast spectrum reactor, and steels irradiated in HFIR, a mixed-spectrum reactor with a very high thermal flux. The saturation yield strength of all alloys remains within a single scatter band.20,21

The tramp impurity elements sulfur and phos­phorus have significantly high (n, a) cross sections at high energies, as shown in Figure 6. Although the cross section for phosphorus is large only at energies characteristic of fusion, a boiling water reactor pro­duces 500 appm He from sulfur and 40 appm He from phosphorus in eight years of operation. An Liquid Metal Fast Breeder Reactor (LMFBR) can produce 100 times these concentrations. All these elements are expected to enhance embrittlement when seg­regated to grain boundaries, but it remains to be determined which is more detrimental, helium, sul­fur, or phosphorus.

The primary techniques used to characterize the behavior of He and He bubbles in materials include TEM, small-angle neutron scattering (SANS), posi­tron annihilation spectroscopy (PAS), and thermal desorption spectroscopy (TDS). All of these techni­ques, and their numerous variants, have individual limitations. Complete and accurate characterization of He transport and fate requires a combination of these methods; however, such complementary tools are seldom employed in practice. Note that there are also a variety of other methods of studying helium in solids that cannot be discussed due to space limitations.

TEM, with a practical resolution limit of about 1 nm, is the primary method for characterizing He bubbles. Bubbles and voids are most frequently observed by bright field (BF) ‘through-focus’ imaging in thin regions of a foil. The Fresnel fringe contrast changes from white (under) to black (over) as a func­tion of the focusing condition. The bubble size is often taken as the mid diameter of the dark under focus fringe. Two critical issues in such studies are artifacts introduced by sample preparation, which produce similar images and determine the actual size, especially below 2 nm.66-68 Electron energy — loss spectroscopy (EELS) can be used to estimate the He pressure in bubbles.69,70

SANS provides bulk measures of He bubble microstructures. In ferromagnetic steels, both nuc­lear and magnetic scattering cross-sections can be measured by applying a saturating magnetic field («2T) perpendicular to the neutron beam. The coherent scattering cross-section variations with the scattering vector are fit to derive the bubble size distribution, with a potential subnanometer resolu­tion limit.71 The magnitude of the scattering cross­section is proportional to the square of the scattering length density contrast factor between the matrix and the bubble times the total bubble volume fraction. Since the magnetic scattering factor contrast is known (He is not magnetic), the bubble volume fraction, and corresponding number densities, can be directly determined by SANS. The nuclear scattering cross­section provides a measure of the He density in the bubbles. Thus, the variation in the ratio of the nuclear (He dependent) to magnetic (He independent) scat­tering cross-sections with the scattering vector can be used to estimate the He pressure (density) as a func­tion of the bubble size.71,72 Some studies have shown that SANS bubble size distributions are in good agreement with TEM observations,73’74 while others show considerable differences for small (<^2 nm) bubbles.72 Limitations of SANS include distin­guishing the bubble scattering from the contributions of other features; note that, in many cases, these features may be associated with the bubbles. Other practical issues include measurements over a suffi­cient range of scattering vectors and handling of radioactive specimens. Note that small-angle X-ray scattering studies can also be used to characterize He bubbles, and this technique is highly complementary to SANS measurements.

PAS is a powerful method for detecting cavities that are smaller than the resolution limits of TEM and SANS. Indeed, positrons are very sensitive to vacancy type defects, and even single vacancies can be readily measured in PAS studies.75,76 PAS can also be used to estimate the He density, or He/vacancy ratio, in bubbles.77 In the case of He-free cavities, the positron lifetime increases with increasing the nano­void size, saturating at several tens of vacancies. How­ever, in the case of bubbles, the lifetime decreases with increasing He density. In principle, positron orbital electron momentum spectra (OEMS) can also provide element-specific information about the anni­hilation site.78 Thus, for example, OEMS might detect the association of a bubble with another microstruc­tural feature. Limitations of positron methods include that they generally do not provide quantitative and unique information about the cavity parameters. The application of PAS to studying He in steels has been very limited to date.

TDS measures He release from a sample as a function of temperature during heating or as a function of time during isothermal annealing. The time-temperature kinetics of release provides indi­rect information about He transport and trapping/ detrapping processes. For example, isothermal anneal­ing experiments on low-dose (<2appm) a-implanted thin Fe and V foils showed that substitutional helium atoms migrate by a dissociative mechanism, with dissociation energies of about 1.4 eV, and that dihe­lium clusters are stable up to 637 K in Fe and up to 773 K in V.79 At higher concentrations in irradiated alloys, He can be deeply trapped in cavities (bubbles and voids); in this case, He is significantly released only close to melting temperatures.80, 1 Given the complexity and multitude of processes encountered in many studies, it is important to closely couple TDS with detailed physical models.82,83 Techniques that can quantify He concentrations at small levels

used in TDS can also be used to measure the total He contents in samples that are melted.81

In summary, a variety of complementary techni­ques can be used to characterize He and He bubbles in structural materials. A good general reference for these techniques and He behavior in solids can be found in Donnelly and Evans.84 TEM and SANS can measure the number densities, size distributions, and volume fractions of bubbles, subject to resolution limits and complicating factors. The corresponding density of He in bubbles can be estimated by TEM-EELS, SANS, and PAS. TDS can provide insight into the He diffusion and trapping/detrapping processes. Unfortunately, there have been very lim­ited applications in which various methods have been applied in a systematic and complementary manner. Major challenges include characterizing subnan­ometer bubbles in complex structural alloys, includ­ing their association with various microstructural features.

In elastically isotropic materials, the interaction of vacancies as well as interstitials with spherical cav­ities depends only on the radial distance r to the

 

Ei(-rC=ra) — Ei(—rc/R)

Table 15 Capture radii and bias factors of edge dislocations with size and modulus interactions Element rc, eff/b SIA rc, eff/b vacancy zd zd ZV Net bias = Zd/Zd — 1 Al 20.38 3.55 1.70 1.21 0.405 Cu 18.30 2.89 1.66 1.18 0.408 Ni 20.09 2.98 1.69 1.18 0.434 Cr 10.47 2.69 1.46 1.17 0.256 Fe 10.75 3.12 1.47 1.19 0.240 Mo 12.73 3.27 1.53 1.20 0.276
cavity center. Denoting by WS(r) this interaction energy at the saddle point configurations, the radial defect flux that includes the drift can be written as	Next, the modulus interaction with the strain field of the cavity is60
3aG 8 m2
2g (R)
W2(R/r)	[153]
jr = — exp[—fiWS(r)]—{DC(r)exp[bWS(r)]} [150] dr The defect current J through each concentric sphere around the cavity is a constant, that is, 4яг2/р = J Integration of [150] then leads to DC (r)exp[b WS(r)] =DC(R)exp[bW S(R)]
The strain field around the cavity is caused by both a pressure p when gas resides in the cavity, and by the surface stress g(R). Note that this surface stress is not equal to the surface energy or a constant as usually assumed. As shown in Section 1.01.7.4.3, it changes with the cavity radius and the gas pressure p. For the case of voids, when p = 0, the surface tension is found to be
exp[b W S(r)]
2 me 1 + (1 — 2v)(R/ b)
Matching this solution to the boundary conditions at r! 1, where C(r) ! C, and at r = R, where C(R) = C0 is the defect concentration in local equi­librium with the cavity, gives the defect current = 4pR(DC — DC0) J0R=(R+h) exp [b WS(R/r)] d(R/r) = 4pRZ0(DC — DC0) [151] The integration extends up to the distance h(R) from the cavity surface where the strain energy of the defect becomes zero, as discussed in Section 1.01.5.3 and shown in Figure 17. The interaction energy WS(R/r) consists of two parts, the image interaction and a modulus interac­tion. The former has been discussed in Section 1.01.5.3, and it has been given by Moon and Pao32 in the form
where e* is the residual surface strain of a planar surface. It is shown in Section 1.01.3.1 , that this residual surface strain parameter can be related to the relaxation volume of the vacancy, and according to eqn [11] 2 e* = —(1 — v)VVel/O [155] With eqns [154] and [155], the modulus interaction then takes the final form
2
[156]
1 [157]
Z0(R)
(1+v)2m(vrel )2 36p(1 — v)R3
Uim (r, R)
requires numerical integration. As an example, the bias factors for self-interstitials and vacancies are com­puted for Ni, at a temperature of 773 K, and using the defect parameters in their stable configurations as given in Tables 2, 7, and 11. The results are presented in Figure 27. The solid curves are obtained when both the image and the modulus interactions are included, while the dashed curves neglect the contribution of
n(n — 1)(2n — 1)(2n + 1) R 2n+2 П 2 n2 + (1 — 2v)n + 1 — v r
[152]
The series converges slowly as r approaches the cavity radius R. However, when R/r < 0.99, no more than about 1000 terms are required to obtain accurate results.

the modulus interaction to the void bias factors. It is seen that the modulus interaction contributes little to the void bias for vacancies, while it enhances the void bias for interstitials significantly.

Atomic bonding (i. e., metallic, ionic, covalent, and polar covalent) is a potential factor to consider when comparing the microstructural evolution between metals and nonmetals, or between different nonmetal­lic materials that may have varying amounts of direc­tional covalent or ionic bonds. For example, several authors have proposed an empirical atomic bonding

500nm

Figure 17 Comparison of the microstructure of Type 316 LN austenitic stainless steel and 9%Cr-2%WVTa ferritic/ martensitic steel after dual beam ion irradiation at 650°C to 50 dpa and 260 appm He. Reproduced from Kim, I.-S.; Hunn, J. D.; Hashimoto, N.; Larson, D. L.; Maziasz, P. J.; Miyahara, K.; Lee, E. H. J. Nucl. Mater. 2000, 280(3), 264-274.

criterion to correlate the amorphization susceptibility of nonmetallic materials.136’137 Materials with ionicity parameters above 0.5 appear to have enhanced resis­tance to irradiation-induced amorphization. However, there are numerous materials which do not follow this correlation,86’138’139 and a variety of alternative mech­anisms have been proposed to explain resis­

tance to amorphization. Atomic bonding can directly or indirectly influence point defect migration and annihilation mechanisms (e. g., introduction of recom­bination barriers), and thereby influence the overall microstructural evolution.

In addition to spinel and alumina layer stacking sequences, Table 1 also shows the layer ‘blocks’ that have been found to comprise {111} and (0001) inter­stitial dislocation loops in spinel and alumina, respec­tively. An interstitial loop in spinel is composed of four layers such that the magnitude of the Burgers vector, b, along (111) is 1/6 (111). The composition of each of these blocks has stoichiometry M3O4, where M represents a cation (either Mg or Al) and O is an oxygen anion. The upper 1/6 (111) block in Table 1 has an actual composition of Al3O4, while the lower 1/6 (111) block has a composition of Mg2AlO4. If Mg and Al cations assume their formal valences (2+ and 3+, respectively), and O anions are 2—, then the blocks described here are charged: (Al3O4)1+and (Mg2AlO4)1-. This may result in an untenable situation of excess Coulombic energy, as each molecular unit in the block possesses an elec­trostatic charge of 1 esu. It has been proposed that this charge imbalance is overcome by partial inver­sion of the cation layers in the 1/6 (111 ) blocks.12 (Inversion in spinel refers to exchanging Mg and Al lattice positions such that some Mg cations reside on o sites, while a similar number of Al cations move to t sites.) If a random cation distribution is inserted into either the upper or lower 1/6 (111) block shown in Table 1 , then the block becomes charge neutral, that is, (MgAl2O4)x.

Table 1 Layer stacking of {111} planes along (111) in cubic spinel and (0001) planes along [0001] in hexagonal alumina

Layer # Layer height Spinel (MgAi2O4) {111}-layer stacking along (111) direction Alumina (а-АІ20з) (0001)-layer stacking along [0001] direction

Frank loop Burgers vectors

O = oxygen t = tetrahedral interstices o = octahedral interstices Layer registry (ABCABC-type O-stacking) Layer Frank loop composition Burgers vectors O=oxygen t = tetrahedral interstices o = octahedral interstices Layer registry (BCBC-type O-stacking) Layer composition 24 23/24 t C — t — 23 22/24 (11/12) O B O4 O C O3 22 21/24 (7/8) t A Mg1 t — 21 20/24 (5/6) o C AI1 o a3 Al2 20 19/24 t B Mg1 t — 19 18/24 (3/4) O A O4 г O B O3 18 17/24 t C — 1/6<111> t — 17 16/24 (2/3) o B AI3 =4 layers o a2 Al2 16 15/24 (5/8) t A — (Al3O4)1 + t — 15 14/24 (7/12) O C O4 L O C O3 14 13/24 t B Mg1 t — 13 12/24 (1/2) o A Al1 o a1 Al2 12 11/24 t C Mg1 t — 11 10/24 (5/12) O B O4 O B O3 10 9/24 t A — t — 9 8/24 (1/3) o C Al3 o a3 Al2 8 7/24 t B — t — 7 6/24 (1/4) O A O4 r O C O3 6 5/24 t C Mg1 1/6<111> t — 5 4/24 (1/6) o B Al1 =4 layers o a2 Al2 4 3/24 (1/8) t A Mg1 (Mg2AlO4)1- t — 3 2/24 (1/12) O C O4 L O B O3 2 1/24 t B — t — 1 0/24 o A Al3 o a1 Al2 0 -1/24 t C — t —

1/3 [0001] = 4 layers (excluding empty tetrahedral layers) (Al2O3)x

Table 1 indicates that an (0001) interstitial dislo­cation loop in alumina consists of a four-layer block (excluding the empty t layers) such that the magni­tude of the Burgers vector, b, along [0001] is 1/3 [0001]. The composition of each of these blocks is Al2O3, which is charge neutral, that is, (Al2O3)x. Thus, there are no Coulombic charge issues asso­ciated with interstitial dislocation loops along 3 in alumina. These dislocation loops consist simply of a pair of Al layers interleaved with two O layers.

Figure 37 shows tensile data of various AuSS (316L, 304L, 316F, JPCA, and 316L-EBW) after SPN irradiations at 100-350 °c223’228’229’249-253 as well as neutron irradiations at around 300 °q254-257 The increase in yield and tensile stress for tests at the irradiation temperature appears to be somewhat lower following SPNI compared to mixed spectrum neutron irradiation, which also produces high levels of He. However, since the alloys used in the different studies are not the same, no clear conclusion can be drawn from this observation. The strength increases plateau above about 10 dpa. In all cases, the uniform elongation falls to less than 1% above about 5 dpa.252 Similar trends are also generally observed in fast reactor irradiations of AuSS.257^258

Figure 38 shows the effect of LANSCE irradia­tions below 160 °C on the fracture toughness of SS316L tested between 50 and 80 °C.223 The fracture toughness for tests at the indicated temperatures for EC316LN irradiated in STIP below 340 °C and SS316L and JPCA irradiated at HFIR at «300 °C is

 

1000

 

—— n (200 °C) —— n (300 °C) —— n (400 °C) ■ 100°C • 170 °C ♦ 250°C ▲ 400°C

 

800 —

 

1600 appm

 

500 appm

 

1200 appm

 

200

 

350appm

 

4 6 dpa1/2

 

10

 

Figure 36 Comparison of the predicted shift in the Master Curve reference temperature, AT0, with the data shown in Figure 35 showing the drastic embrittling effect of high concentrations of He.

 

1000

 

800

 

□ YS, SPN ■ UTS, SPN YS, neutron Ф UTS, neutron

 

100 £ Tt» T £ 350 °C

 

0

 

0

 

Figure 37 Irradiation dose dependence of yield stress (YS), ultimate tensile stress (UTS), strain to necking (STN), and total elongation (TE) of AuSS irradiated with SPN and fission neutrons and tested at temperatures between 100 and 350 °C.

also shown. The decreases Kjq with dpa have similar trends, although the STIP data show a less regular pattern; however, at higher dpa the toughness falls at 50 MPaVm or less. A similar trend has been previously reported for both fast reactor and mixed spectrum reactor irradiations and is considered to be due to reductions in the uniform strain and strain hardening rates that are not strongly influenced by He.2

1.01.4.1 Atomic Structure of Self-Interstitials

The accommodation of an additional atom within a perfect crystal lattice remained a topic of lively debates at international conferences on radiation effects for many decades. The leading question was the configuration of this interstitial atom and its surrounding atoms. This scientific question has now been resolved, and there is general agreement that this additional atom, a self-interstitial, forms a pair with one atom from the perfect lattice in the form of a dumbbell. The configuration of these dumbbells can be illustrated well with hard spheres, that is, atoms that repel each other like marbles.

Let us first consider the case of an fcc metal. In the perfect crystal, each atom is surrounded by 12 nearest neighbors that form a cage around it as shown on the left of Figure 13. When an extra atom is inserted in this cage, the two atoms in the center form a pair
whose axis is aligned in a [001] direction. This [001] dumbbell constitutes the self-interstitial in the fcc lattice. The centers of the 12 nearest neighbor atoms are the apexes of a cubo-octahedron that encloses the single central atom in the perfect lattice, and it can be shown19 that the cubo-octahedron encloses a volume of V0 = 10O/3. However, around a self-interstitial dumbbell, this cubo-octahedron expands and distorts, and now it encloses a larger volume of V001 = 4.435O. The volume expansion is the difference

DV = V001 — V0 = 1.10164O [28]

which happens to be larger than one atomic volume. We shall see shortly that the volume expansion of the entire crystal is even larger due to the elastic strain field created by the self-interstitial that extends through the entire solid.

We consider next the self-interstitial defect in a bcc metal. Here, each atom is surrounded in the perfect crystal by eight nearest neighbors as shown on the left of Figure 14. When an extra atom is inserted, it again forms a dumbbell configuration with another atom, and the dumbbell axis is now aligned in the [011] direction, as shown on the right of Figure 14. The cage formed by the eight nearest neighbor atoms becomes severely distorted. It is surprising, however, that the volume change of the cage is only

DV = 0.6418 O [29]

less than the volume ofthe inserted atom to create the self-interstitial in the bcc structure.

The reason for this is that the bcc structure does not produce the most densely packed arrangement of atoms, and some ofthe empty space can accommodate the self-interstitial. In contrast, the fcc structure has in fact the densest arrangement of atoms, and disturbing it by inserting an extra atom only creates disorder and lower packing density.

As already mentioned, the large inclusion volume DV of self-interstitials leads to a strain field

throughout the surrounding crystal that causes changes in lattice parameter and that is the major source of the formation energy for self-interstitials. In order to determine this strain field, we treat in Appendix A the case of spherical defects in the center of a spherical solid with isotropic elastic properties. Although this represents a rather simplified model for self-interstitials, for vacancies, and for complex clusters of such defects, it is a very instructive model that captures many essential features.

1.02.3.1 Intrinsic Defect Concentrations

Introducing a doping agent to a crystal lattice can have a significant effect on the defect concentration within the material. As such, doping represents a powerful tool in the engineering of the properties of ceramic materials.

The concept of a solid solution, in which solute atoms are dispersed within a diluent matrix, is used in many branches ofmaterials science. In many respects, the doped lattice can be viewed as a solid solution in which the point defects are dissolved in the host

Charge

Site

t

Subscript denotes species in the nondefective lattice at which defect currently sits. Interstitial defects are represented by letter ‘i’

Examples

Vo

o"

AIMg

{peMg:NaMg}X

MgM g + oX — VMg + Vo + Mg°

Vacancy on an oxygen site with an effective 2+ charge oxygen interstitial with an effective 2- charge

Substitutional defect in which an aluminum atom is situated on a magnesium site and has an effective 1 + charge

Neutral defect cluster containing: Fe on Mg site (1+ charge) and Na on Mg site (1- charge). Braces indicate defect association

Defect equation showing Schottky defect formation in MgO

Figure 5 Overview of Kroger-Vink notation.

lattice. The critical issue with such a view is in defining the chemical potential of an element. This is straight­forward for the dopant species but is less clear when the species is a vacancy. This is circumvented by defining a virtual chemical potential, which allows us to write equations similar to those that describe chemical reac­tions. Within these defect equations, it is critical that mass, charge, and site ratio are all conserved. Using Kroger-Vink notation, we can describe the formation of Schottky defects in MgO thus: Null! VMMg + VO* or MgMg + °0 ! VMM g + vo* + MgO Note that in this case, the equation balances in terms of charge, chemistry, and site. As this is a reaction, it may be described by a reaction constant ‘K,’ which is related to defect concentrations by

Al2°3 ! 2AlMg + 300 + VMg Then,

it

Ahsol 3kT

3 2 exp

In general, the law of mass action6 states that for a reaction aA + bB $ cC + dD

m£-K. — exp(-AG) b exp kT

[vO*]

Ks

In the case of the pure MgO Schottky reaction, charge neutrality dictates that

[vO*]

7.7′ exp — JT

[v°*]

Now, consider the effect that solution of 10 ppm Al2O3 has on MgO. The solution reaction implies that a concentration of 5 ppm of VMg has been intro­duced into the lattice, that is, jvMg] = 5 x 10-6. Therefore, [vO*] =2 x 105 exp(-77) Thus, at 1000 °C, the [V°*] = 6.4 x 10-26 compared to an oxygen vacancy concentration in pure MgO of 5.66 x 10-16. The introduction of the extrinsic defects has therefore lowered the oxygen vacancy concentration by 10 orders of magnitude!

1.02.3.2 Effect of Doping on Defect Concentrations Similar reactions can be written for extrinsic defects via the solution energy. For example, the solution of CoO in MgO, where the Co ion has a charge of 2+ and is therefore isovalent to the host lattice ion, Co0 ! CoMg + °o + MgO

exp

Ksolution

where Ahsol is the solution enthalpy. As the concen­tration of CoO in CoO = 1,

1.02.3.4 Defect Associations So far, we have assumed that when we form a set of defects through some interaction, although the defects reside in the same lattice, somehow they do not interact with one another to any significant extent. They are termed noninteracting. This is valid in the dilute limit approximation; however, as defect concentrations increase, defects tend to form

O

Consider the solution of Al203 in MgO. In this case, the Al ions have a higher charge and are termed aliovalent. These ions must be charge-compensated by other defects, for example,

into pairs, triplets, or possibly even larger clusters. Take, for example, the solution of Al2O3 into MgO. Al2O3 ! 2AlMg + 3O0 + VMg + 3MgO When the concentration is great enough, AlMg + VMg! {AlMg : VMg}	the scope of this chapter.7 Therefore, to illustrate the types of relationships that can occur, we use the example of the binary {TiJJg: VMg cluster resulting from TiO2 solution in MgO via TiO2 ! TiMg + 2O0 + VMg + 2MgO and
Ti** V" Ti** : V" 0 TiMg T VMg! TiMg : V Mg Then, using
As seen for solution energies, using the enthalpy associated with this pair cluster formation (the bind­ing energy Ahbind), the reaction can be analyzed using mass action:
exp
AlMg
But since,
yields
exp
a
If the concentration of titanium ions on magnesium sites is x so that Mg(1-2x)TixO is the formula of the material, then,
we have the relationship
Ahbind T Ahsol kT
{TiMg : VMg}"
which describes the solution process Al2O3 ! AlMg + 3O0 + {AlMg: VMg}
Substituting p into a yields the quadratic equation,
x = 0
Further, it is possible to form a neutral triplet defect cluster, {AlMg: VjMg: AlMj, which has a binding enthalpy of AEbind with respect to isolated defects so that
Solving this in the usual manner allows us to deter­mine [TiM’J as a function of Kbinding energy If we now assume that Ah>ind = 1 eV (a typical binding energy between charged pairs of defects in oxide cera­mics), relationships between the concentration of the clusters and the isolated substitutional ions can be determined as a function ofeither total dopant concen­tration, x, or temperature. These are shown in Figure 6, which assumes a fixed temperature of 1000 K and Figure 7, which assumes a fixed value of x = 1 X 10-6 and a range of temperature from 500 to 2000 K.
|AlMg : VMg : AlMg}
Kbinding triplet
VII VMg
Al*Mg
Ahbind exp[ kF
which leads to
П * II * fX] Ahbind T Ahsol AlMg : VMg : AlMg = exp — —kT —
We now investigate the relative significance of defect clusters over isolated defects as a function of temper­ature for a fixed dopant concentration. For most systems, there are a great number of possible isolated and cluster defects, and the equilibria between them quickly become very complex. Solving such equili­bria requires iterative procedures that are beyond

the host material, Fe3+ in the last case. Usually, this is dependent on how easily the cation can be oxidized or reduced. Associated with these reactions is the removal or introduction of oxygen from the atmo­sphere. For example, the reduction reaction follows

°o! 2°2(g)+ VO +2e

where e represents a spare electron, which will reside somewhere in the lattice. For example, in CeO2, the electron is localized on a cation site forming a Ce3+
ion. This is usually written as CeCe. Similarly, the oxidation reaction is

2°2(g) ! Oc> + 2h

where h represents a hole, that is, where an electron has been removed because the new oxygen species requires a charge of 2—. Thus in CoO, for example,

2O2(g) + 2Co0o! O0 + 2CoCo + v£ o

If the enthalpy for the oxidation reaction is Ah0x,

[C°C.]’ [VCo]
(Po, )’“

(Po, )is the partial pressure of oxygen, that is, the concentration of oxygen in the atmosphere. Since electroneutrality gives us

[coCo] = 2 [VC o]

[VCo] « (PO, )1/6

Now, if the majority of cobalt vacancies are associated with a single charge-compensating Co3+ species, that is, we have some defect clustering, then the oxidation reaction will be

,O,(g) + 2Co^o! Oq + CoCo + (CoCo : VCo}’

and

[CoCo] [(CoCo : VCo)’

(Po, )

which, given that (CoCo : VCo)’ implies that [(CoCo: VC„)’] is proportional to (Po2)1/4. Similar relations can be formulated for even larger clusters.

The defect concentration can be determined by measuring the self-diffusion coefficient. When this is related to the oxygen partial pressure on a lnPo2 versus ln [{CoCo:VCo}] graph, the slope shows how the material behaves. For CoO, the experimentally determined slope is 1/4, showing that the cation vacancy is predominantly associated with a single charge-compensating defect.8