  Impact ionization coefficients of anisotropic monoclinic $\beta
$-Ga$_{2}$O$_{3}$ are estimated along four crystallographic directions and the
plot for the $\left[ 010 \right]$ direction is shown. The approximation models
were fitted to Baraff's universal plot for ionization rate in semiconductors
and the values were obtained for $\beta $-Ga$_{2}$O$_{3}$. The phonon mean free
path of $\beta $-Ga$_{2}$O$_{3}$ was estimated to be 5.2604 \si{\angstrom}
using Gray medium approximation. The phonon group velocity takes the value of
longitudinal acoustic phonons. The ionization rate has a maximum value of $
3.98\times \,10^{6}\,cm^{-1}$ along the $\left[ 010 \right]$ direction over the
applied electric field range $(1.43-4)\times10^{7}\,V\,\cdot\,cm^{-1}$.
Contrary to expectations, the phonon mean free path along direction $\left[
\bar{2}01 \right]$ is the lowest since it has a lower thermal conductivity to
phonon group velocity ratio. The plots were compared with GaN and 4H-SiC, which
shows that as bandgap increases the field required for ionization increases.
The critical electric field was estimated to be $ 0.921
\times10^{8}\,V\cdot\,cm^{-1}$ along $\left[010 \right]$ direction.
