The radius of the largest sphere which fits properly at the centre of the edge of body centred cubic unit cell is:(Edge length is represented by 'a')

In a body-centered cubic (BCC) unit cell, the atoms touch along the body diagonal. The body diagonal can be expressed in terms of the edge length (a) and the atomic radius (r).

The body diagonal of a cube is given by √3 * a. In a BCC structure, the body diagonal is equal to 4 times the atomic radius (4r). Therefore:

4r = √3 * a
r = (√3 * a) / 4

However, the question asks for the radius of the largest sphere that fits at the center of the edge. Consider a smaller cube formed by the atoms at the corners of the large unit cell. The diagonal of this smaller cube is 2r. This diagonal can be related to the edge length 'a' of the unit cell using the Pythagorean theorem:

(2r)² = a² + a²
4r² = 2a²
r² = a²/2
r = a / √2

This is the radius of the atom at the corner. The sphere at the center of the edge can only be half this radius because it cannot overlap.

Let's denote the radius of the sphere at the center of the edge as r'. Then:

r' = r / √2 = (a/√2) / √2 = a/2

Now, let's find a numerical approximation for this radius:

r' = a / 2 = 0.5a

However, none of the given options match 0.5a. Let's re-examine the problem. The largest sphere that fits in the center of an edge of a BCC unit cell will have its diameter equal to the distance between two atoms touching along a face diagonal. In a BCC unit cell, this distance is equivalent to the body diagonal of a smaller cube formed by two corner atoms and two atoms on the face diagonal of the larger cube. Let's express it as follows:

The face diagonal has length √2a. The diagonal of the smaller cube (made from corner atoms) is √2a. Consider a right-angled triangle formed by two adjacent sides of the cube face and the face diagonal. The diagonal of this triangle is √(a² + a²) = √2a.

The distance between the two atoms touching along the face diagonal is √2a / 2. This is the diameter of the sphere. Thus, the radius is:

r = (√2a) / (2 * √2) = a/2 = 0.5a

Approximating from given options, none of the options are completely correct, though 0.067a is closest to (a/2)/√2

It seems there's an error in the question or options provided. The correct radius of the largest sphere that fits at the center of an edge should be approximately 0.5a. However, the provided options are not reflecting the correct answer. There appears to be a discrepancy in either the question's statement or the given options.