  Molecular dynamics simulation is used to study vacancy cluster formation in
$\beta$- and $\alpha$-$Si_3N_4$ with varying vacancy contents (0 - 25.6 at%).
Vacancies are randomly created in supercells, which were subsequently
heat-treated for 114 nanoseconds. The results show that both $\beta$ and
$\alpha$ can tolerate vacancies up to 12.8 at% and form clusters, confirming
previous experimental data indicating 8 at% vacancy in $\alpha$-$Si_3N_4$.
However, 25.6 at% vacancy in $\beta$ results in complete amorphization, while
the same amount in $\alpha$ results in a transformation of a semi-amorphous
$\alpha$ phase to a defective $\beta$ phase, leading to the removal of the
clusters in newly formed $\beta$. This clearly explains why cluster vacancies
are not experimentally observed in $\beta$, considering that $\beta$-$Si_3N_4$
ceramics are produced from $\alpha$. Furthermore, the lattice parameters of
both modifications increase with increasing vacancy content, revealing the
cause of different lattice constants that were previously reported for
$\alpha$-$Si_3N_4$.
