Problem $4.$ Below are listed diffraction angles for the first three peaks $($first order i$.$e$.\backslash ($ n$=1\backslash )$ in the Braggs Equation$)$ of the $\backslash ($ x $\backslash )-$ray diffraction pattern for some metal. Monochromatic $\backslash ($ x $\backslash )-$radiation having a wavelength of $0\mathrm{.}1254$ nm was used.

$($a$)$ Determine whether this metal's crystal structure is FCC$,$ BCC$,$ or SC$.$ Explain the reason for your choice.

$($b$)$ If the crystal structure is either BCC or FCC$,$ identify which of the metals in the Table below gives this diffraction pattern. Justify your decision.

Table $3\mathrm{.}1$ Atomic Radii and Crystal Structures for $16$ Metals

a $\backslash (\backslash$mathrm$\{$FCC$\}=\backslash )$ face$-$centered cubic; $\backslash (\backslash$mathrm$\{$HCP$\}=\backslash )$ hexagonal close$-$packed; $\backslash (\backslash$mathrm$\{$BCC$\}=\backslash )$ body$-$centered cubic.

$\backslash (\{\}\_\{$b$\}^\{$b$\}\backslash )$ A nanometer $($ nm $)$ equals $\backslash (10^\{-9\}\backslash$mathrm$\{$~m$\}\backslash )$; to convert from nanometers to angstrom units $(\backslash (\backslash$mathcal$\{$A$\}\backslash )),$ multiply the nanometer value by $10.$