Question

Which among the following options give the correct relationship between the edge lengths ‘a’ and the radius of a sphere ‘r’ in a body-centered cubic structure.

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

In $△EFD$,
b = a + a = 2a
Now in $△AFD$
$c^2=a^2+b^2=a^2+2a^2=3a^2$
$\to c=\sqrt{3a}$
The length of the body diagonal 'c' is equal to 4r, where r is the radius of the sphere (atom), as the three spheres along the diagonal touch each other.
Therefore, $\sqrt{3a}=4r$
$a=\frac{4r}{\sqrt{3}}$
$\to r=\frac{\sqrt{3}}{4}a$