Given a copper $($Cu$)$ single crystal with a Poissons ratio of $0\mathrm{.}34,$ Youngs modulus of $110$ GPa, shear modulus of $48$ GPa, and a critical resolved shear stress of $0\mathrm{.}59$ MPa, where the width of the dislocation varies as $1$b$,1\mathrm{.}5$b$,2$b$,$ and $2\mathrm{.}5$b $($where b is the Burgers vector$)$:

$1.$Calculate the minimum tensile stress required in the $[001]$ direction of the single crystal to induce slip in the $(111)[0-11]$ slip system. Provide the answer in MPa.

$2.$If the size of the Burgers vector of a dislocation in the crystal is $2\mathrm{.}6,$ and the Peierls$-$Nabarro stress is equal to the critical resolved shear stress of the single crystal, calculate the width of the dislocation in nm$.$