$($a$)$ Calculate the positions of the first $5$ diffraction peaks $($in $q,$ unit $={}^{-1})$ formed by nanoparticles with

$10nm$ metal core and $5nm$ thickness of organic shell on the particle. Assume no contribution from the

resolution limit set by the instrument. The nanoparticles form FCC superlattice crystals.

$($b$)$ For the same structure as in $($a$),$ calculate the broadness of the first peak $(FWHM=q$ for nanoscale

face centered cubic lattice $(1$ micron, namely, ${10}^4$ Angstrom, domain size$)$

$[$Hints:

For $($a$)$: $(1)$ The permitted diffraction Miller indices for FCC lattice are either all odd or even.

$(2)$ The interplanar spacing equation for cubic system: $\frac{1}{d_{hkl}^2}=\frac{h^2+k^2+l^2}{a^2}.$

For $($b$)$: Refer to the revised slide #$66$ for the relationship between the domain size $D$ and $q$