In this problem, consider diffusion in a face$-$centered orthorhombic structure, as shown

below Lattice parameters along the $x,y,$ and $z$ directions are $a,b,$ and $c,$ respectively, where $a$

$bc.$

The jump frequency in this crystal depends on the site to which the atom jumps. For

instance, for the jump of an interstitial atom $($in black$)$ to position $1,$ the activation energy barrier

will be from the atoms $M$ and $N.$ Similarly, for jumps to $2$ and $7$ the barriers are N$-$P and P$-$R

respectively. We will denote the jump frequencies as ${}_1^{\text{'}},$ for positions $2,4,10,$ and $12$;${}_2^{\text{'}},$ for $1,$

$3,9,$ and $11$ ; and ${}^{\text{'}},$ for $5,6,7,$ and $8.$ Use this information to compute the interstitial

diffusivities along the $x,y,$ and $z$ directions, denoted $D_{\text{I}\text{,}x},D_{\text{I}\text{,}y},$ and $D_{\text{I}\text{,}z},$ in terms of $a,b,c,{}_1^{\text{'}},{}_2^{\text{'}},$

and ${}_3^{\text{'}}.$

Hint: the interstitial diffusivity can be calculated from the average jumps of many different

atoms. Since there are $12$ interstitial sites in the orthorhombic lattice,

$D_{\text{I}\text{,}x}=\frac{1}{12}{\displaystyle \underset{i=1}{\overset{12}{}}}{}_1^{\text{'}}{x}_i^2$

where $x_i^2$ is the component of the jump distance to site $i$ in the $x$ direction.