Crystal microstructure under irradiation consists of two qualitatively different defect types: mobile (single vacancies, SIAs, and SIA and vacancy clus­ters) and immobile (voids, SIA loops, dislocations, etc.). The concentration of mobile defects is very small (^ 10 10—10 6 per atom), whereas immobile defects may accumulate an unlimited number of PDs, gas atoms, etc. The mathematical description of these defects is, therefore, different. Equations for mobile defects describe their reactions with immo­bile defects and are often called the rate (or balance) equations. The description of immobile defects is more complicated because it must account for nucle- ation, growth, and coarsening processes.

1.13.4.1 Concept of Sink Strength

The mobile defects produced by irradiation are absorbed by immobile defects, such as voids, disloca­tions, dislocation loops, and GBs. Using a MFA, a crystal

can be treated as an absorbing medium. The absorption rate of this medium depends on the type of mobile defect, its concentration and type, and the size and spatial distribution of immobile defects. A parameter called ‘sink strength’ is introduced to describe the reac­tion cross-section and commonly designated as k(, k(, and k? cl (x) for vacancies, SIAs, and SIA clusters of size x (the number of SIAs in a cluster), respectively. The role of the power ‘2’ in these values is to avoid the use of square root for the MFPs of diffusing defects between production until absorption, which are correspond­ingly k-1, k-1, and kcKx). There are a number of publications devoted to the derivation of sink strengths.40,59-61 Here we give a simple but sufficient introduction to this subject.

1.13.4.2 Equations for Mobile Defects

For simplicity, we use the following assumptions:

• The PDs, single vacancies, and SIAs, migrate 3D.

• SIA clusters are glissile and migrate 1D.

• All vacancy clusters, including divacancies, are immobile.

• The reactions between mobile PDs and clusters are negligible.

• Immobile defects are distributed randomly over the volume.

Then, the balance equations for concentrations of

mobile vacancies, Cv, SIAs, Ci, and SIA clusters,

Cg. l(x), are as follows

dCv = Gnrt (1 — er)(1 — ev) + G;h

— k? dycy — mRDiCiCv N

Gnrt(1 — er)(1 — ei)-k? DiCi

mRDi Ci Cv

G«l(x)-k.’,(x)DidC‘l.

x 2. 3. ••• xmax [ 12 ]

where G* is the rate of thermal emission of vacancies from all immobile defects (dislocations, GBs, voids, etc.); Dv, Di, and Dicl(x) are the diffusion coefficients of vacancies, single SIAs, and SIA clusters, respec­tively; and mR is the recombination coefficient of PDs. Since the dependence of the cluster diffusivity, Dicl(x), and sink strengths, k? d(x), on size x is
rather weak,45,46 the mean-size approximation for the SIA clusters may be used, where all clusters are assumed to be of the size <xf) . In this case, the set of eqn [12] is reduced to the following single equation

^ =<x?)-1Gnrt(1 — er)eg — k? lAdC? d [13] where eqn [9] is used for the cluster generation rate. To solve eqns [10]—[13], one needs the sink strengths k?, k2, and k2d, the rates of vacancy emission from various immobile defects to calculate G*, and the recombination constant, mR. The reaction kinetics of 3D diffusing PDs is presented in Section 1.13.5, while that of 1D diffusing SIA clusters in Section

1.13.5. In the following section, we consider equa­tions governing the evolution of immobile defects, which together with the equations above describe damage accumulation in solids both under irradiation and during aging.

1.13.4.3 Equations for Immobile Defects

The immobile defects are those that preexist such as dislocations and GBs and those formed during irradi­ation: voids, vacancy — and SIA-type dislocation loops, SFTs, and second phase precipitates. Usually, the defects formed under irradiation nucleate, grow, and coarsen, so that their size changes during irradiation. Hence, the description of their evolution with time, t, should include equations for the size distribution func­tion (SDF), f (X, t), where X is the cluster size.

1.13.4.3.1 Size distribution function

The measured SDF is usually represented as a func­tion of defect size, for example, radius, X = R : f (R, t). In calculations, it is more convenient to use x-space, X = x, where x is the number of defects in a cluster: f (x, t). The radius of a defect, R, is connected with the number of PDs, x, it contains as:

4p 3 _

—R3 = xO 3

pR2 b = xO [14]

for voids and loops, respectively, where O is the atomic volume and b is the loop Burgers vector. Cor­respondingly, the SDFs in R — and x-spaces are related to each other via a simple relationship. Indeed, ifsmall dx and dR correspond to the same cluster group, the number density of this cluster group defined by two functions f (x)dx and f (R)dR must be equal, f(x)dx = f(R)dR, which is just a differential form

of the equality of corresponding integrals for the total number density:

1

f (x)dx

x=2

The relationship between the two functions is, thus,

f (R]=f (x)dR [16]

For voids and dislocation loops

Note the difference in dimensionality: the units of f (x) are atom — (or m- ), while f (R) is in m — atom-1 (or m-4), as can be seen from eqn [15]. Also note that these two functions have quite different shapes, see Figure 1, where the SDF of voids obtained by Stoller et a/.62 by numerical integration of the master equation (ME) (see Sections 1.13.4.3.2 and 1.13.4.4.3) is plotted in both R — and x-spaces.

1.13.4.3.2 Master equation

The kinetic equation for the SDF (or the ME) in the case considered, when the cluster evolution is driven by the absorption of PDs, has the following form

@f @x’ f) = Gs(x) + J(x — 1, t)-J(x, t), x > 2 [18]

where Gs(x) is the rate of generation of the clusters by an external source, for example, by displacement

Diameter (nm) Figure 1 Size distribution function of voids calculated in x-space, fvcl(x) (x is the number of vacancies), and in d-space, fvcl(d) (d is the void diameter). From Stoller et al.62

cascades, and J(x, t) is the flux of the clusters in the size-space (indexes ‘i’ and ‘v’ in eqn [18] are omitted). The flux J (x, t) is given by

J(x, t) = P(x, t)f (x, t) — Q(x + 1, t)f (x + 1, t) [19]

where P(x, t) and Q(x, t) are the rates of absorption and emission of PDs, respectively. The boundary conditions for eqn [18] are as follows

f (1) = C

f (x! 1) = 0 [20]

where C is the concentration of mobile PDs.

If any of the PD clusters are mobile, additional terms have to be added to the right-hand side of eqn [19] to account for their interaction with immobile defect which will involve an increment growth or shrinkage in the size-space by more that unity (see Section 1.13.6 and Singh eta/.22 for details).

The total rates of PD absorption (superscript!) and emission (t—) are given by

11 J! = P(x)f (x), Jto = Q (x )f (x) [21]

x= 2 x= 2

where the superscript arrows denote direction in the size-space. J! and are related to the sink strength

of the clusters, thus providing a link between equa­tions for mobile and immobile defects. For example, when voids with the SDF fc(x) and dislocations are only presented in the crystal and the primary damage is in the form of FPs, the balance equations are

dQ = gNRT — £r)

dt

— [uRACiCv + ZdrdDv(Cv — Cvo)]

— [Pc(1)fc(1, t) — Qvc(2)f (2, t)]

x=1

-E(pc(x)f (x’t)

x=1

— Qvc(x + 1)fc(x + 1, t)) [22]

GNRT (1 — er)

— [mRDiCiCv + ZfpdDiCi]

x=1

-^2 Aic(x + 1)fc(x +1’t) where рА and are the dislocation density and its efficiencies for absorbing PDs, mR, is the recom­bination constant (see Section 1.13.5); the last two terms in eqn [22] describe the absorption and

emission of vacancies by voids and the last term in eqn [23] describes the absorption of SIAs by voids. The balance equations for dislocation loops and sec­ondary phase precipitations can be written in a similar manner. Expressions for the rates P(x, t), Q(x, t), the dislocation capture efficiencies, Zdv, and mR are derived in Section 1.13.5.