Q.In an ionic crystalline solid atoms of element Y form hcp lattice. The atoms of element X occupy one third of tetrahedral voids. What is the formula of the compound? (X₂Y₃)

Constituent particles of a crystal pack as tightly as possible, treated as hard touching spheres; the coordination number (number of touching neighbours) measures how compact that packing is. Building up from one dimension (a simple touching row) to two dimensions gives two packings: square close packing (spheres aligned in rows and columns, 2D coordination number 4, 'AAAA...' stacking) and hexagonal close packing (staggered rows, 2D coordination number 6, more efficient than square packing). Stacking these 2D layers in three dimensions gives three structures: stacking square layers directly above one another ('AAAA...') gives the simple cubic structure, coordination number 6 (polonium is the only metal that adopts it); stacking hexagonal layers so a third layer realigns with the first ('ABAB...') gives the hexagonal close-packed (hcp) structure; stacking hexagonal layers so a third, distinct layer covers the octahedral voids instead ('ABCABC...') gives the cubic close-packed (ccp, equivalent to fcc) structure -- both hcp and ccp reach the maximum coordination number of 12 (6 in-layer + 3 above + 3 below). Stacking two close-packed layers also creates interstitial voids: a tetrahedral void, enclosed by 4 touching spheres (3 from one layer + 1 from the next) whose centres form a tetrahedron; and an octahedral void, enclosed by 6 touching spheres (3 from each of two layers) whose centres form an octahedron. For N close-packed atoms there are always 2N tetrahedral voids and N octahedral voids -- a fixed 2:1 ratio used to derive the formula of a compound whose second element occupies a given fraction of these voids.