5.6.3.1 The Model for Cluster Formation and Dissolution

In [109] the equation describing interstitial cluster kinetics is described as

$$\frac{\partial C}{\partial t} = 4\pi\alpha aD_IIC - \frac{CD_I}{a^2} e^{-E_b/kT},$$ (5.17)

where $a$ denotes the interatomic spacing, $\alpha$ the capture radius expressed in units of the interatomic spacing, $D_I = D_0e^{-E_m/kT}$ the interstitial diffusivity, $I(t,x)$ the concentration of unclustered interstitials, $C(t,x)$ the concentration of clustered interstitials, $t$ the time, $T$ the annealing temperature in Kelvin, and $k$ the Boltzmann constant. TSUPREM-IV uses a general clustering model that uses many of the models proposed in literature as subsets. The main formula for the change of the concentration of clustered interstitials is [13]

$$\frac{\partial C}{\partial t} = K_{\mathrm{fi}} \frac{I^{\mathrm{... ...mathrm{isfc}}_*} \left(C + \alpha I\right)^{\mathrm{cf}} - K_r C^{\mathrm{cr}},$$ (5.18)

where $I_{*}(t, x)$ denotes an equilibrium concentration of interstitials. All other symbols of (5.18) are parameters that need to be adjusted. $K_{\mathrm{fi}}$, $K_{\mathrm{fc}}$, and $K_r$ are the reactions constants which have the form

$$K_{\mathrm{fi}} = \mathrm{kfi0} \cdot e^{-\mathrm{kfi}E/kT},$$ (5.19)

$$K_{\mathrm{fc}} = \mathrm{kfc0} \cdot e^{-\mathrm{kfc}E/kT},$$ (5.20)

$$K_r = \mathrm{kr0} \cdot e^{-\mathrm{kr}E/kT},$$ (5.21)

with coefficients $\mathrm{kfi0}$, $\mathrm{kfc0}$, and $\mathrm{kr0} > 0$. Since the coefficients are all positive the first two terms of (5.18) describe the formation of clusters whereas the last term describes its dissolution. The term $K_{\mathrm{fi}}(I^{\mathrm{ifi}}/I^{\mathrm{isfi}}_*)$ describes the joining of two clusters^5.7. Therefore, the parameter values $\mathrm{ifi}$ and $\mathrm{isfi}$ are expected to be $2$. The second term, $K_{\mathrm{fc}}{I^{\mathrm{ifc}}}/{I^{\mathrm{isfc}}_*} \left(C + \alpha I\right)^{\mathrm{cf}}$, describes the case where an unclustered interstitial joins a cluster. A value of $1$ for $\mathrm{ifc}$ and $\mathrm{isfc}$ is assumed.

2003-03-27