The continuous production of SIA clusters in dis­placement cascades is a key process, which makes microstructure evolution under cascade conditions qualitatively different from that during FP producing 1 MeVelectron irradiation. In this case, eqns [10]—[12] should be used for the concentration of mobile defects. The equations for isolated PDs have been considered in detail in the previous section. In order to analyze damage accumulation under cascade irradiation, one needs to define the sink strengths of various defects for the SIA mobile clusters in eqn [12]. We give examples of such calculations for the case when cluster migrates 1D rather than 3D in the following section.

1.13.6.1 Reaction Kinetics of One-Dimensionally Migrating Defects

The 1D migration of the SIA clusters along their Burgers vector direction results in features that dis­tinguish their reaction kinetics from 3D diffusing defects. These were first noticed in and theoreti­cally analyzed for annealing experiments (Lomer and Cottrell,126 Frank et a/.,127 Gosele and Frank,128 Gosele and Seeger,129 and Gosele40) and, then, under irradiation (Trinkaus et a/.19,20 and Borodin130). In this section, we consider the reaction kinetics of 1D migrating clusters with immobile sinks and follow the procedure employed in Barashev et a/.25

Detailed information about the diffusion process of a 1D migrating particle is given by the function u(t, X, x), which is known as Furth’s formula for first passages and has the following probabilistic signifi — cance.1 1 In a diffusion process starting at the point X > 0, the probability that the particle reaches the origin before reaching the point x > X in the time
interval t1 < t < t2 is given by the integral over this interval. For particles undergoing random walk, this function is found to be equal to

1

u(t, X, x) = 2пУ ] i exp

i=1

where D1d is the diffusion coefficient. Using this function, one can write the probability for a particle to survive until time t, that is, not to be absorbed by the barriers placed at the origin and at the point x, as [1

•q(t, X, x) = dt0 [u(t0, X, x) + u(t0, x — X, x)]

_ 4y’exp[—(2i — 1)2р2Р1Р?|x2]sin fnX(2i -1) [111,

n 2i — 1 sin x [ J

i=1

The expected duration of the particle motion until its absorption is given by: 1

v(t, X, x)dt

0

Equation [112] is the classical result of the ‘gambler’s ruin’ problem considered by Feller.131

1.13.6.1.1 Lifetime of a cluster

In order to obtain the lifetime of 1D migrating clusters, one should average truin(X, x) over all possi­ble distances between sinks and initial positions of the clusters, that is, over x and X. For this purpose, the corresponding probability density distribution, ‘(x, X), is required.

Let us assume that all sinks are distributed ran­domly throughout the volume and introduce the 1D density of traps (sinks), L, that is, the number of traps per unit length. In this case, ‘(x, X) can be repre­sented as a product of the probability density for a cluster to find itself between two sinks separated by a distance x, L2x exp(—Lx), and the probability density to find a cluster at a distance X from one of these sinks, 1 =x:

‘(x, X) = L2exp(—Lx), 0 < x < 1,0 < X < x [113]

With this distribution, the cluster lifetime, t1D, and the mean-free path to sinks, l, are:

t1D = <Trum(X, x)>X, x = 1|2D1dL2 [114]

l =<X>X, x = 1|L [115]

where the brackets denote averaging: <>X, x =

o x

dx dX'(x, X)

00

1.13.6.1.2 Reaction rate

It follows from eqn [114] that the reaction rate between 1D migrating clusters and immobile sinks (e. g., Borodin13 ) is given by:

R1D = 2L2D1d C = — г Ad C [116]

l

This equation defines the total reaction rate as a function of L, determined by the concentration and geometry of sinks. If there are different sinks in the system, L is a sum of corresponding contributions Lj from traps of type j In a crystal containing disloca­tions and voids only,

L = Ld + Lc [117]

where subscripts ‘d’ and ‘c’ stand for dislocations and voids, respectively. These partial trap densities are found below.

Consider voids of a particular radius r, randomly distributed over the volume. Without loss of general­ity, the capture radius of a void for a cluster is assumed here to be equal to its geometrical radius, that is, rc, = r,. A void of radius r, is available to react with mobile clusters that lie in a cylinder of this radius around the cluster path. Hence, the partial 1D density of voids of any particular radius, Lci, and the total 1D void density, Lc, are given by

Lc, = prljf (r,) [118]

Lc = Yl, Lc, = Prc2Nc [119]

,

where f (r,) is the SDF of voids (^ f {ri) = Nc is the

,

total void number density) and rc2 is the mean square of the void capture radius. For dislocations

Ld = PrA p* [120]

where pd is the dislocation density defined as the mean number of dislocation lines intersecting a unit area (surface density) and rd is the corresponding capture radius. This can be shown in the following way. The mean number of dislocation lines intersect­ing the cylinder of unit length and radius rd around the cluster path equals the area of the cylinder sur­face, 2шА, times the dislocation density divided by 2. (The factor 2 arises because each dislocation intersects the cylinder twice.) It should be noted that the dislocation sink strength for 3D diffusing defects is usually expressed through the dislocation density, pd, defined as the total length of dislocation lines per unit volume of crystal (volume density). The relationship between p* and Pd depends on the
distribution of the dislocation line directions. For a completely random arrangement, the volume density is twice the surface density, Pd ~ 2p[j (see, e. g., Nabarro132). In this case, eqn [120] is the same as found by Trinkaus et at}9,20

Substituting eqns [117] — [120] into eqn [116], the total reaction rate of the clusters in a crystal contain­ing random distribution of voids and dislocations is found to be130:

R1D = 2(prdfd + Pr2NcJ AdC [121]

For the case, in which immobile vacancy and SIA clusters are also taken into account, the sink strength for 1D diffusing SIA clusters, kg, is equal to

kg = 2 ( 2"^ ^ prc Nc + svclNvcl + ffidAic^ [122]

where svcl and ffid are the interaction cross-sections and Nvcl and ЛА the number densities of the sessile vacancy and SIA clusters, respectively. svcl and ffid are proportional to the product of the loop circum­ference and the corresponding capture radius similar to rd for dislocations.

1.13.6.1.3 Partial reaction rates

A detailed description of the microstructure evolu­tion requires the partial reaction rates, Rj, of the clusters with each particular sink, for example, dis­locations or voids of various sizes.22 According to the definition of the parameters Lj and L, the ratio Lj/L is the probability for a trap to be of type j. Hence, the partial reaction rates are

R = L R [123]

A similar relation between total and partial reaction rates was used in Gosele and Frank.12 Using eqn [116], one can write the partial reaction rate of clus­ters with sinks of type j 2Lj LD1D C where lj = 1/Lj is the mean distance between a cluster and a sink of type j in 1D, cf. eqn [116]. Thus, the partial reaction rate of a specific type of sink depends on the density of that sink and also on the density of all other sinks. This correlation between sinks is characteristic of pure 1D diffusion — reaction kinetics in contrast to 3D diffusion where the leading term of the sink strength of any defect is not correlated with others (see eqn [54]).