
Volume of HCP unit cell is:
A)
B)
C)
D)
Answer
503.1k+ views
Hint: The volume of the unit cell can be determined by taking the product of the area and height of the cell. For the hexagonal close packing structure , the volume is equal to:
.
Since the base of the hexagonal structure is a hexagon which is itself made of the six equilateral triangles, and the area of the equilateral triangle is .
Complete step by step answer:
In the hexagonal close packing structure , there are 6 atoms at the corner and in the top layer and bottom layer of the structure. There are a total of 12 atoms, 2 atoms at the base centred, and 3 atoms within the unit cell.
Each atom at the corner contribute since there are 12 atoms at the corner, therefore the total number of atoms per unit cell is :
Each base entered atoms have a total contribution of , the number of base-centred per unit cell is
The three atoms within the unit cell give the complete contribution.
Therefore, the contribution of atom in the hexagonal packed structure is calculated as;
We know that the volume of the unit cell can be represented as the product of the area of base and height of the unit cell.
The is a hexagonal prism structure. The base of is a hexagon. We know that the hexagon is made of the 6 equilateral triangles.
Therefore, the area of the base is given as,
Let’s the ‘h’ be the height of the hexagonal unit.
Therefore we have,
Now, substitute these values in the volume of we have,
We know that the edge length ‘a’ is equal to twice the radius of the atom. That is,
Substitute these values in the volume we get,
Therefore, the volume of the hexagonal close packing structure is
Hence, (A) is the correct option.
Note: We can further calculate the Atom packing efficiency of the HCP unit cell by taking the ratio of the volume of atoms in the unit cell to the volume of the unit cell. Here a total of 6 atoms per unit cell. Since atoms are sphere the APF can be calculated as follows,
Therefore, the atomic packing fraction for HCP is equal to the 0.74.
Since the base of the hexagonal structure is a hexagon which is itself made of the six equilateral triangles, and the area of the equilateral triangle is
Complete step by step answer:
In the hexagonal close packing structure
Each atom at the corner contribute
Each base entered atoms have a total contribution of
The three atoms within the unit cell give the complete contribution.
Therefore, the contribution of atom in the hexagonal packed structure is calculated as;

We know that the volume of the unit cell can be represented as the product of the area of base and height of the unit cell.
The
Therefore, the area of the base is given as,
Let’s the ‘h’ be the height of the hexagonal unit.
Therefore we have,
Now, substitute these values in the volume of
We know that the edge length ‘a’ is equal to twice the radius of the atom. That is,
Substitute these values in the volume we get,
Therefore, the volume of the hexagonal close packing structure is
Hence, (A) is the correct option.
Note: We can further calculate the Atom packing efficiency of the HCP unit cell by taking the ratio of the volume of atoms in the unit cell to the volume of the unit cell. Here a total of 6 atoms per unit cell. Since atoms are sphere the APF can be calculated as follows,
Therefore, the atomic packing fraction for HCP is equal to the 0.74.
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