$2.$X$-$ray diffraction pattern of gold as shown below has the FCC crystal structure. A monochromatic X$-$ray with a wavelength of $0\mathrm{.}1542nm$ was used $(6$ points$).$

$2\mathrm{.}1.$ Determine the inter$-$planar spacing for the peak of Au $(111)(2$ points$).$

$2\mathrm{.}2.$ Determine the atomic radius of Au based on the diffraction peak of Au $(111)(2$ points$).$

$3$

Determine the interplanar spacing of $(422)$ planes $(2$ points$).$