The cubic unit cell structure of a compound containing cation M and anion X is shown below. When compared to the anion, the cation has smaller ionic radius. Choose the correct statement(s).

The anion X occupies all the eight corners of the cube while the smaller cation M is present at the body-centre.

1. Ions per unit cell
  • Corner ion contribution = $$8 \times \frac{1}{8}=1$$ X
  • Body-centre ion contribution = 1 M
Therefore a unit cell contains 1 M and 1 X, giving the empirical formula $$MX$$. Hence Option A is correct.

2. Coordination geometry
A corner X is joined to the body-centre M and to the equivalent body-centres of the adjoining cells. Each ion thus has eight nearest neighbours of the opposite kind arranged at the vertices of a cube, i.e. both ions possess 8-coordination (cubic geometry). Because the geometries are identical, Option B is incorrect.

3. M-X bond length versus edge length
The nearest M-X distance is from the body-centre to any corner. This is one-half of the body diagonal:
$$d_{M\!-\!X}=\frac{\sqrt{3}\,a}{2}=0.866\,a$$
Thus the stated ratio 0.866 is right, so Option C is correct.

4. Radius ratio
If the two ions just touch along the body diagonal,
$$r_M+r_X=\frac{\sqrt{3}\,a}{2}$$
and the limiting (minimum) radius ratio for 8-coordination is $$r_M/r_X=\,0.732$$. The value 0.414 corresponds to 6-coordination (octahedral holes), not to this structure. Therefore Option D is incorrect.

Hence the correct statements are Option A and Option C.

Final answer: Option A (The empirical formula is MX), Option C (The bond-length to edge-length ratio is 0.866).