Which of the following is correct for a 3-D hexagonal closed packing lattice ?
Here
\(Z_{eff}=\text{Total number of atoms in hcp lattice}\\O_v=\text{Total number of octahedral voids}\\T_v=\text{Total number of tetrahedral voids}\)

\(\text{Total no. of spheres in a hcp unit cell }Z_{eff}=\left(12\times\dfrac{1}{6}\right)+\left(2\times\dfrac{1}{2}\right)+3=6\)
where,
\(\left(12\times\dfrac{1}{6}\right)\text{ is for corners atoms of two hexagonal layers}\)
\(\left(2\times\dfrac{1}{2}\right)\text{ is for the centre atoms of two hexagonal layers}\)
\(3 \text{ is for the atoms of middle layer}\)

\(Z_{eff}=6\)


In hcp, there are total 12 tetrahedral voids out of which 8 tetrahedral voids are completely inside the unit cell and 12 voids are present at the edge centres.
Contribution of each edge centre void is \(\dfrac{1}{3}\).
So, the total number of tetrahedral voids in hcp lattice are :
\(T_v=8+\left(12 \times\dfrac{1}{3}\right)=12\)

\(\dfrac{T_v}{Z_{eff}}=\dfrac{12}{6}=2\)

Also, There are six octahedral voids which are present completely inside the unit cell. Each fully contributes to the unit cell.
\(O_v=6\)
\(\dfrac{O_v}{Z_{eff}}=\dfrac{6}{6}=1\)