Identify the structure from the cubic unit cell

The figure described is the standard rock-salt (NaCl type) arrangement: one ion forms an FCC lattice and the other occupies all octahedral voids.

Since the cation $M$ is smaller than the anion $X$, the larger anion $X$ forms the FCC framework and the smaller cation $M$ occupies octahedral holes.

Find the empirical formula

In an FCC lattice of anions $X$:

• Corner anions: $8 \times \frac{1}{8} = 1$
• Face-center anions: $6 \times \frac{1}{2} = 3$

So total anions per unit cell: $1+3=4$

Octahedral holes in FCC = 4 per unit cell, all occupied by $M$.

So total cations per unit cell = 4.

Hence, $M:X = 4:4 = 1:1$

Therefore, the empirical formula is: $\boxed{MX}$

So A is correct.

Check coordination geometries

In NaCl-type structure:

• Each cation $M$ is surrounded by 6 anions $X$.
• Each anion $X$ is surrounded by 6 cations $M$.

Thus, both have octahedral coordination.

Therefore, cation and anion do not have different coordination geometries.

So B is incorrect.

Find the ratio of $M-X$ bond length to cube edge length

In rock-salt structure, nearest $M-X$ contact occurs along the edge of the cube.

If cube edge length is $a$, then the distance between nearest cation and anion is: $\frac{a}{2}$

Therefore, $\frac{\text{M-X bond length}}{\text{edge length}} = \frac{a/2}{a} = \frac{1}{2} = 0.5$

So option C, which says $0.866$, is incorrect for NaCl structure.

Check radius ratio

For octahedral coordination (coordination number 6), the limiting radius ratio is: $\frac{r_+}{r_-} = 0.414$

But this is only the minimum radius ratio required for stable octahedral coordination, not an exact value that must hold for every compound.

The statement says the ratio is $0.414$, which is not necessarily true.

So D is incorrect.

Final derived answer

Correct option(s): $\boxed{A}$

Comparison with stored correct answer

Stored answer: $A, C$

My derived answer is only $A$.

I disagree with the stored answer because in the NaCl-type cubic unit cell, the nearest $M-X$ distance is $a/2$, not $0.866a$.

The value $0.866 = \sqrt{3}/2$ is associated with half the body diagonal, which does not represent nearest cation-anion contact in this structure.