Which of the following is the correct relationship between edge length (a) and radius of sphere (r) located in a bcc unit cell?

All of these

The correct answer is option 4. All of these

Body centered cubic unit cell (BCC)

Atom at the centre will be in touch with the other two atoms diagonally arranged.

In ∆ EFD,

b² = a² + a² = 2a²

b = \(\sqrt{2}\)a

Now in ∆ AFD

c² = a² + b² = a² + 2a² = 3a²

c = \(\sqrt{3}\)a

The length of the body diagonal c is equal to 4r, where r is the radius of the sphere (atom), as all the three spheres along the diagonal touch each other.

Therefore, \(\sqrt{3}\)a = 4r

a = \(\frac{4r}{\sqrt{3}}\)

Also we can write, r = \(\frac{\sqrt{3}}{4}\)a