radius-ratio rules
Why does NaCl adopt the rock salt structure with six neighbours, while the larger CsCl prefers eight and the smaller ZnS settles for four? The radius-ratio rules give a beautifully simple first answer: it all comes down to how big the cation is compared with the anion. The bigger the cation relative to the anion, the more anions can crowd around it before they start bumping into each other.
The reasoning is pure geometry. Imagine the cation just touching the anions packed around it. There is a minimum size the cation must have to hold a given number of anions apart without forcing the anions to overlap. Working out the touching geometry gives clean threshold ratios (cation radius divided by anion radius): below about 0.225 only three anions fit (trigonal); from 0.225 to 0.414, four fit (tetrahedral, as in zinc blende); from 0.414 to 0.732, six fit (octahedral, as in rock salt); and above 0.732, eight fit (cubic, as in cesium chloride). So you measure or look up the two ionic radii, take the ratio, and read off the predicted coordination number and hence the likely structure.
These rules are a useful guide and a wonderful teaching tool, but be honest about their limits — they fail surprisingly often. They assume hard, perfectly spherical ions and purely ionic bonding, neither of which is strictly true, and ionic radii themselves depend on coordination number, making the prediction slightly circular. Many compounds adopt a coordination number the rule does not predict (lithium iodide should be tetrahedral on the numbers but is rock salt). Treat the radius ratio as a sensible expectation that is right perhaps two times out of three, not as a law — when it fails, covalency, polarization, or the energetics of the whole lattice have outvoted simple geometry.
For NaCl, the radius of Na+ (about 102 pm) divided by that of Cl- (about 181 pm) is roughly 0.56, which lands in the 0.414-to-0.732 octahedral band — correctly predicting the six-coordinate rock salt structure. The same arithmetic for CsCl gives about 0.93, above 0.732, predicting eight-coordination, which is right too.
The radius ratio correctly sorts NaCl into six-coordinate and CsCl into eight-coordinate — but it is right only roughly two times in three.
Radius-ratio rules are a guide, not a law: they assume hard spheres and pure ionicity and ignore covalency and polarization, so they fail for a sizeable minority of compounds. The thresholds derive from geometry, not from any deep physical principle.