The correct relationships between unit cell edge length 'a' and radius of sphere 'r' for face-centred and body-centred cubic structures respectively are:

We need to find the correct relationships between the unit cell edge length $$a$$ and the radius of the sphere $$r$$ for face-centred cubic (FCC) and body-centred cubic (BCC) structures.

Face-Centred Cubic (FCC) Structure:

In an FCC unit cell, atoms touch along the face diagonal. Consider one face of the cube:

- The face diagonal passes through the center of the face atom and connects two corner atoms.

- Along this diagonal, we have: half of corner atom + full face atom + half of corner atom = $$r + 2r + r = 4r$$.

- The face diagonal of a cube with edge length $$a$$ has length $$a\sqrt{2}$$ (by the Pythagorean theorem: $$\sqrt{a^2 + a^2} = a\sqrt{2}$$).

Therefore:

This gives us: $$2\sqrt{2}r = a$$.

Body-Centred Cubic (BCC) Structure:

In a BCC unit cell, atoms touch along the body diagonal. The body diagonal connects two opposite corners and passes through the center atom.

- Along the body diagonal: half of corner atom + full center atom + half of corner atom = $$r + 2r + r = 4r$$.

- The body diagonal of a cube with edge length $$a$$ has length $$a\sqrt{3}$$ (by the 3D Pythagorean theorem: $$\sqrt{a^2 + a^2 + a^2} = a\sqrt{3}$$).

Therefore:

FCC: $$2\sqrt{2}r = a$$

BCC: $$4r = \sqrt{3}a$$

The correct answer is Option 4: $$2\sqrt{2}r = a$$ and $$4r = \sqrt{3}a$$.