The radius of the largest sphere which fits properly at the edge of a body centered cubic unit cell is (Edge length is represented by ‘a’)

Concept:

• The body-centered cubic unit cell has atoms at each of the eight corners of a cube plus one atom in the center of the cube.
• Each of the corner atoms is the corner of another cube so the corner atoms are shared among eight unit cells.

Calculation:

For body-centered cubic bcc structure,

Radius, (R) = (a×√3)/4    or a×√3 = 4×R  --- (1)

Where, a = edge length.

According to the question, the structure of a cubic unit cell can be shown as follows:

∴ a = 2(R + r)

On substituting the value of R from Eq. (1) and Eq (2), we get

\(\frac{a}{2}=\frac{√{3}}{4}a+r\)

\(r=\frac{a}{2}=\frac{√{3}}{4}a=2a-\frac{√{3a}}{4}\)

\(r=\frac{a\left( 2-√{3} \right)}{4}\)