Question

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

• A. $\frac{O_v}{Z_{eff}}=1.5$
• B. $\frac{O_v}{Z_{eff}}=0.25$
• C. $\frac{T_v}{Z_{eff}}=0.5$
• D. $\frac{T_v}{Z_{eff}}=2$

Answer 1

The correct option is D $\frac{T_v}{Z_{eff}}=2$

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

$Z_{eff}=6$


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 $\frac{1}{3}$.
So, the total number of tetrahedral voids in hcp lattice are :
$T_v=8+(12\times \frac{1}{3})=12$

$\frac{T_v}{Z_{eff}}=\frac{12}{6}=2$

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