$(25\%)$

Q$3.$ Pure Ti undergoes following phase transformations.

$-T{i}_{(hcp)}{T}_{\text{a}\text{l}\text{p}\text{h}\text{a}\text{,}\text{b}\text{e}\text{t}\text{a}}=1158K-T{i}^{(bcc)}{T}_m=1940K$ Liquid

a$)$ Suppose that nuclei formed during solidification are cubic in shape rather than spherical. If $a$ is the dimension of the cube, derive the expression for the critical cubic edge length, $a^{***},$ in terms of $($surface free energy$)$ and $H_f($latent heat of fusion$)$ for the case of a homogeneous nucleation from a liquid phase.

b$)$ If Ti homogeneously nucleates at $1900K,$ calculate the critical cubic edge length, a$*,$ given values of $-2{10}^3\frac{J}{c}{m}^3$ and $2{10}^{-5}\frac{J}{c}{m}^2,$ respectively, for the latent heat of fusion and the surface free energy.

c$)$ Calculate the number of atoms in the critical$-$sized nucleus of the Ti at $1900K,$ using the data given as Atomic Radius of $Ti=1\mathrm{.}45$ where $1={10}^{-8}cm.$