Question

The correct relationships between unit cell edge length ‘a’ and radius of sphere ‘r’ for face-centred and body-centred cubic structures respectively are:

• A. \(2\sqrt 2r = a\) and \(4r = \sqrt 3 a\)
• B. \(r = 2\sqrt 2a\) and \(\sqrt 3 r = 4a\)
• C. \(r = 2\sqrt 2a\) and \( 4r = \sqrt 3a\)
• D. \(r = 2\sqrt 2a\) and \( \sqrt 3r =4a\)

Answer 1

The Correct Option is A

For both FCC (Face-Centered Cubic) and BCC (Body-Centered Cubic) structures, we can derive the relationships between the unit cell edge length \( a \) and the radius of the spheres \( r \).
For FCC (Face-Centered Cubic):
In the FCC structure, the relationship between the edge length \( a \) and the radius \( r \) is based on the geometry of the cube. In a face-centered cubic unit cell, the diagonal of the face equals four radii: \[ \sqrt{2}a = 4r \] Solving for \( a \): \[ a = \frac{4r}{\sqrt{2}} = 2\sqrt{2}r \] For BCC (Body-Centered Cubic):
In the BCC structure, the relation between the edge length \( a \) and the radius \( r \) is derived from the body diagonal. The body diagonal of the cube is equal to \( 4r \), so: \[ \sqrt{3}a = 4r \] Solving for \( a \): \[ a = \frac{4r}{\sqrt{3}} \] Thus, the correct relationships are \( a = 2\sqrt{2}r \) for FCC and \( a = \frac{4r}{\sqrt{3}} \) for BCC.