Question

What is the critical radius ratio for the CSCl structure?

Answer 1

Hint :The limiting radius ratio is the smallest value for the ionic radii ratio $ \rho = (\dfrac{{{r^ + }}}{{{r^ - }}}) $ that this structure can have while remaining stable. The radius of the cation is $ {r^ + } $ , and the radius of the surrounding anions is $ {r^ - } $ . It's worth noting that anions are usually larger than cations. The importance of the radius ratio is that it can be used to predict the structure of ionic solids. The greater the radius ratio, the more cations and anions are coordinated.

Complete Step By Step Answer:
 $ C{l^ - } $ ions are located at the cube's eight edges, while the $ C{s^ + } $ ion is found at the body's center in a CsCl configuration. Along the body diagonals, the eight corner anions contact the central cation in critical condition. Around the cube's sides, the eight anions touch each other.
The body diagonals move through the central cation and two opposing corner anions.
The side of the cube has a length, a, where:
 $ a = 2{r^ - } $
We know the length of each side of the triangle: cube edge $ \left( 1 \right) $ , face diagonal ( $ \sqrt 2 $ ), body diagonal $ \left( {\sqrt 3 } \right) $ .
 $ 2{r^ - } + 2{r^ + } = 2{r^ - }\sqrt 3 $
Therefore,
 $ \dfrac{{{r^ + }}}{{{r^ - }}} = \sqrt 3 - 1 $
 $ = 0.732 $

Additional Information:
The stability of an ionic crystal structure is determined by the radius ratio. Larger cations, for example, will fill larger voids like cubic sites, whereas smaller cations will fill smaller voids like tetrahedral sites. As a result, the radius ratio rule can be used to figure out the structure of ionic solids.

Note :
Remember the ionic radii ratio $ \rho = (\dfrac{{{r^ + }}}{{{r^ - }}}) $ . Also keep in mind that critical by limiting radius ratio gives the critical radius ratio for the structure.