Problem points$).$ Diffusion.

The expression for the vacancy diffusion coefficient in a pure element A along one axis, here the $x$ axis, is

given by:

$D_x=\frac{1}{2}{\displaystyle \underset{s=1}{\overset{z}{}}}{}_s{x}_s^2$

where $s$ labels all the lattice sites that can be reached in one jump $($their total number is $z)$; ${}_s$ is the

jump frequency to a particular lattice site $s,$ and $x_s$ is the projection on the $x$ axis of the jump distance to

the site $s.$

a$/$ Calculate the diffusion coefficient along a direction, $D_{?}$ for a body centered cubic $($BCC$)$

structure of lattice parameter $a,$ assuming that the vacancy can only jump to first nearest neighbor sites

$($there are $8$ such sites$).$

$\frac{b}{?}$ Calculate the diffusion coefficient along a direction, for a body centered cubic $($BCC$)$

structure of lattice parameter $a,$ assuming again that the vacancy can only jump to first nearest neighbor

sites $($there are $8$ such sites$).$

c$/$ Compare the results obtained in $\frac{a}{?}$ and $\frac{b}{?}$ and discuss them in light of the symmetry properties of the

BCC lattice.

d$/$ Calculate $D$ in the basal plane and normal to the basal plane of an HCP crystal of lattice parameters $a$

$($in basal plane$)$ and $c($normal to the basal plane$).$ Assume that the vacancy jumps to first nearest neighbor

sites in the basal plane with a frequency ${}_b($there are $6$ such sites$)$ and to first nearest neighbor sites out

of the basal plane with a frequency ${}_n($there are $6$ such sites$).$