$($a$)$ Derive the planar density expression for the FCC $(130)$ plane in terms of the atomic radius R $.$

$($b$)$ Briefly explain what the two atomic mechanisms for diffusion are and how Fick's first law $($for steady$-$state diffusion$)$ is derived.

$($b$)$ The yield stress for a copper alloy that has an average grain diameter of $55m$ is $180$ MPa $.$ At a grain diameter of $2m,$ the yield stress increases to $240$ MPa $.$ At what grain diameter will the yield stress be $210$ MPa $?$

$[3+3+3$ marks $]$