Abstract
When concentration of vacancies in a CZ silicon crystal is defined by molar fraction $X_{B}$, the degree for supersaturation $\sigma$ is given by $[X_{B}-X_{BS}]/X_{BS}=X_{B}/X_{BS}-1=ln(X_{B}/X_{BS})$ because $X_{B}/X_{BS}$ is nearly equal to unity. Here, $X_{BS}$ is the saturated concentration of vacancies in a silicon crystal and $X_{B}$ is a little larger than $X_{BS}$. According to Bragg-Williams approximation, the chemical potential of the vacancies in the crystal is given by ${\mu}_{B}={\mu}^{0}+RT$ ln $X_{B}+RT$ ln ${\gamma}$, where R is the gas constant, T is temperature, ${\mu}^{0}$ is an ideal chemical potential of the vacancies and ${\gamma}$ is and adjustable parameter similar to the activity of solute in a solute in a solution. Thus, ${\sigma}(T)$ is equal to $({\mu}_{B}-{\mu}_{BS})/RT$. Driving force of nucleation for the vacancy agglomeration will be proportional to the chemical potentialdifference $({\mu}_{B}-{\mu}_{BS})/RT$ or ${\sigma}(T)$, while growth of the vacancy agglomeration is proportaional to diffusion of the vacancies and grad ${\mu}_{B}$.