Symbols r^> and r^< represent the radius of the bigger and...

Symbols $r^{>}$ and $r^{<}$ represent the radius of the bigger and smaller atom, respectively, in a crystal of diatomic basis. Show that the critical ratio,
$\frac{r^{>}}{r^{<}}=\frac{\sqrt{3}+1}{2}$ for the $\mathrm{CsCl}$ structure
$=2+\sqrt{6}$ for the zinc blende structure

Key Concepts

Coordination Number and Lattice Contact

The coordination number, which is the number of nearest neighbors each ion has in a given structure, plays a pivotal role in determining the stability of a crystal. The contact between ions in a lattice is governed by the geometry corresponding to these coordination numbers. For instance, the CsCl structure has a simpler coordination environment (often eightfold) compared to the more complex zinc blende structure, and this difference is reflected in the different critical radius ratios derived through geometric analysis.

Crystal Structure Geometry

Crystal structure geometry involves the spatial arrangement of ions or atoms in a crystal lattice. Specific arrangements, like those of the CsCl and zinc blende structures mentioned, impose different geometric constraints due to their distinct coordination numbers and lattice symmetries. These geometries directly affect how the ions touch one another and thus dictate the calculated critical radius ratio for the structure.

Critical Radius Ratio

The critical radius ratio refers to the minimum size ratio of the larger ion to the smaller ion that is necessary for a given coordination geometry to be maintained in a crystal structure. It is derived by considering the geometric constraints – that is, the requirement that the ions must be close enough to touch each other. If the radius ratio falls below this critical value, the smaller ion can move without remaining in contact with several neighbors, leading to a less stable configuration.

Radius Ratio

The radius ratio is the ratio between the size of the larger ion and the smaller ion in an ionic or diatomic crystal. It is crucial because it determines how the ions pack together and the type of coordination that is possible. In many cases, only certain ratios will allow the ions to be in contact and achieve a stable structure without leaving too much free space or causing repulsion between like-charged species.

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