The second order Bragg's diffraction of $X$-rays with $\lambda = 1 \ \mathring{A}$ from a set of parallel planes in a metal occurs at an angle of $60^\circ$. The distance between the scattering planes in the crystal is ................ $\mathring{A}$.

Calculate the density of an element with molar mass $27 \ g \ mol^{-1}$ having $4$ atoms in a unit cell with edge length $405 \ pm$. (in $g \ cm^{-3}$)

Calculate the number of atoms present in $1 \text{ g}$ of a metal that forms a simple cubic unit cell structure, given that the product of density and volume of the unit cell is $66.5 \times 10^{-24} \text{ g}$.

Ionic radii of cation $A^{+}$ and anion $B^{-}$ are $102 \ pm$ and $181 \ pm$ respectively. These ions are allowed to crystallize into an ionic solid. This crystal has cubic close packing for $B^{-}$ and $A^{+}$ is present in all octahedral voids. The edge length of the unit cell of the crystal $AB$ is $pm$.

An element with a molar mass $27 \text{ g mol}^{-1}$ forms a cubic unit cell. Calculate the number of atoms present in a unit cell if the density of the metal is $2.7 \text{ g cm}^{-3}$. Given: $[a^3 \times N_A = 40 \text{ cm}^3 \text{ mol}^{-1}]$

$NH_4Cl$ crystallizes in a $bcc$ lattice. If the edge length of the unit cell is $387 \, pm$, the distance between oppositely charged ions is ........ $pm$.