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

To solve this problem, we need to understand the arrangement of atoms in face-centered cubic (FCC) and body-centered cubic (BCC) unit cells, and how the radius of atoms relates to the edge length of the unit cell, denoted by 'a'.

1. Face-Centered Cubic (FCC) Structure:
   • In an FCC structure, atoms are located at each of the corners and the centers of all the faces of the cube.
   • Each face diagonal passes through the centers of two face-centered atoms.
   • For FCC, the formula that relates the unit cell edge length 'a' to the radius 'r' of the sphere (atom) is: \(a = 2\sqrt{2}r\). This is because the face diagonal is equal to \(4r\) and the face diagonal in terms of edge length 'a' is \(\sqrt{2}a\).
2. Body-Centered Cubic (BCC) Structure:
   • In a BCC structure, there is an atom at each cube corner and one atom at the center of the cube.
   • In a BCC structure, the body diagonal passes through the center atom and each corner atom.
   • For BCC, the formula that relates 'a' to 'r' is: \(4r = \sqrt{3}a\). This is because the body diagonal equals \(4r\) and it can also be expressed in terms of the edge length as \(\sqrt{3}a\).

Given these relationships, the correct answer is:

• \(2\sqrt{2}r = a\) for FCC
• \(4r = \sqrt{3}a\) for BCC

Therefore, the option stating \(2\sqrt{2}r = a\) and \(4r = \sqrt{3}a\) is the correct relationship between unit cell edge length 'a' and radius of sphere 'r' for face-centered and body-centered cubic structures.