Determine the radius of a cation having \(\mathrm{CsCl}\) structure, when all anions are in contact with the cation and touch each other. The critical radius of the anion is \(1.90 A\).

The cubic crystal system is one of the most straightforward and symmetrical forms of crystal structures. It features atoms arranged at the corners of a cube.
There are different types of cubic systems, but the CsCl structure falls under the simple cubic category, specifically known as a body-centered cubic (BCC) structure.
In this structure, there is one type of atom, the cation, located at the center of the cube, and another type, the anion, positioned at each corner.
This arrangement is essential for maintaining the structure's stability and balance. Here are some characteristics of a cubic crystal system:

• High symmetry and regular geometry
• Equal axes and 90-degree angles between them
• Repetition of the basic unit throughout the entire crystal

Understanding this arrangement helps in calculating various distances and dimensions crucial for solving problems related to the CsCl structure.

In the context of a cubic crystal system, the body diagonal is a crucial concept.
The body diagonal refers to the line connecting one corner of the cube to the opposite corner, passing through the center of the cube.
This diagonal is longer than the edges of the cube. The length of the body diagonal in a cubic crystal system can be calculated using the formula \[\text{diagonal} = \sqrt{3} \ cdot a\] where \(a\) is the side length of the cube.
This equation comes from applying the Pythagorean theorem in three dimensions.
Here’s a breakdown:

• First, calculate the diagonal of a face of the cube, which is \sqrt{2} \cdot a
• Then, apply the Pythagorean theorem again with this face diagonal and the side length \(a\), providing the body diagonal formula

Understanding this concept is critical because, in the CsCl structure, the cation at the center is in contact with the anions at the cube's corners via half of the body diagonal length.

In the CsCl structure, the arrangement and contact of anions and cations are crucial to understanding the crystal's stability.
The cation at the center of the cube is surrounded by eight anions located at the cube's corners.
These anions and cations are in direct contact, maintaining the structure’s integrity.
To determine the radius of the cation, consider that the distance from the center of the cation to the center of an anion is half the body diagonal.

• Given: Anion radius \(r_a = 1.90 \text{ Å}\)
• Key relationship: \(\frac{\sqrt{3} \cdot a}{2} = r_c + r_a\)
• Calculate side length: \(a = 2 \cdot r_a = 2 \cdot 1.90 \text{ Å}\)

Substituting \(a\) into the body diagonal equation, and solving:
\[\frac{\sqrt{3} \cdot 3.80}{2} = r_c + 1.90\]\
Simplify to find the cation radius \(r_c\): \[r_c = \sqrt{3} \cdot 1.90 - 1.90 ≈ 1.39 \text{ Å}\]\ Understanding how these ions are in contact helps in visualizing and calculating distances within the crystal lattice accurately.