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Question

Which of the following is correct for a 3-D hexagonal closed packing lattice ?
Here
Zeff=Total number of atoms in hcp latticeOv=Total number of octahedral voidsTv=Total number of tetrahedral voids

A
TvZeff=0.5
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B
OvZeff=0.25
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C
TvZeff=2
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D
OvZeff=1.5
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Solution

The correct option is C TvZeff=2
Total no. of spheres in a hcp unit cell Zeff=(12×16)+(2×12)+3=6
where,
(12×16) is for corners atoms of two hexagonal layers
(2×12) is for the centre atoms of two hexagonal layers
3 is for the atoms of middle layer

Zeff=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 particle is 13.
So, the total number of tetrahedral voids in hcp lattice are :
Tv=8+(12×13)=12

TvZeff=126=2

Also, There are six octahedral voids which are present completely inside the unit cell. Each fully contributes to the unit cell.
Ov=6
OvZeff=66=1

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