Assertion: Only T-voids are present in CCP lattice structure.
Reason: Packing efficiency in CCP lattice structure is 74%.
Answer & explanation
Correct answer: option 4
The correct answer is option 4. Assertion is wrong statement, but reason is correct statement.
Two types of voids are present in a CCP lattice
Tetrahedral voids (T-voids) are located at the body diagonals, two in each body diagonal, at one-fourth distance from each end.
Total number of tetrahedral voids per unit cell = 8
Octahedral voids (O-voids) are located at body-center and at edge centers of cubit unit cell.
Total number of octahedral voids per unit cell = \(\frac{1}{4}\) x 12 + 1 = 4
Thus, in CCP, total number of voids per unit cell = 8 + 4 = 12
Face centered cubic unit cell (FCC) is present in CCP Lattice structure
let the unit cell edge length be ‘a’ and face diagonal AC = b.
In ∆ ABC
AC2 = b2 = BC2 + AB2 = a2 + a2 = 2a2 or b = \(\sqrt{2}\)a
If r is the radius of the sphere, we find b = 4r = \(\sqrt{2}\)a or a = \(\frac{4r}{\sqrt{2}}\)
a = 2\(\sqrt{2}\)r
we can also write, r = \(\frac{a}{2\sqrt{2}}\)
We know that each unit cell in ccp structure i.e., FCC unit cell has effectively 4 spheres.
Total volume of four spheres is equal to 4 x \(\frac{4}{3}\)πr3 and volume of the cube is a3 or (2\(\sqrt{2}\)r)3.
Therefore,
Packing efficiency = \(\frac{\text{Volume occupied by four spheres in the unit cell}}{\text{Total volume of the unit cell}}\) x 100%
Packing efficiency = \(\frac{4 × \frac{4}{3}πr^3}{(2\sqrt{2}r)^3}\) x 100% = 74%