$($a$)$ Assuming homogeneous nucleation, calculate the critical radius $r^{**}$ and the activation free energy $G^{**}$ for the

homogenous nucleation of the solidification of nickel $(Ni).$ Values for the latent heat of fusion and surface free

energy of $Ni$ are : $-2\mathrm{.}53{10}^9\frac{J}{m^3}$ and $0\mathrm{.}255\frac{J}{m^2},$ respectively. The supercooling value for homogeneous

nucleation of $Ni$ is $319C.$ The melting temperature of $Ni$ is$,T_m=1455C.$

$($b$)$ Now calculate the number of atoms found in a nucleus of critical size. Assume a lattice parameter of $0\mathrm{.}360nm$

for solid nickel at its melting temperature.

Hint : Nickel has a f$.$c$.$c$.$ Bravais lattice and the lattice parameter is$,a=0\mathrm{.}360nm(0\mathrm{.}36{10}^{-9}(m)).(20$ points$)$