The Model for Cluster Formation and Dissolution

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

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

$\displaystyle \frac{\partial C}{\partial t} = K_{\mathrm{fi}} \frac{I^{\mathrm{...
...mathrm{isfc}}_*} \left(C + \alpha I\right)^{\mathrm{cf}} - K_r C^{\mathrm{cr}},$ (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
$\displaystyle K_{\mathrm{fi}}$ $\displaystyle =$ $\displaystyle \mathrm{kfi0} \cdot e^{-\mathrm{kfi}E/kT},$ (5.19)
$\displaystyle K_{\mathrm{fc}}$ $\displaystyle =$ $\displaystyle \mathrm{kfc0} \cdot e^{-\mathrm{kfc}E/kT},$ (5.20)
$\displaystyle K_r$ $\displaystyle =$ $\displaystyle \mathrm{kr0} \cdot e^{-\mathrm{kr}E/kT},$ (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 clusters5.7. 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.