How to derive minimum radius ratio for a given coordination number

[Lamboman2008]

Homework Statement

Calculate the minimum radius ratio for CN = 3.

Givens: Minimum radius ratio for CN = 4 is 0.225, so that's also the max for CN = 3.

The Attempt at a Solution

I know that the radius ratio determines the CN, which is related to the geometry, and that for a CN of 3, the geometry is an equilateral triangle. If I understand it correctly, to find the minimum radius ratio. I've figured that to find the minimum radius ratio, r_c/r_a, the anion must be as large as possible, so the side of the triangle must be twice the length of the radius of the anion, or 2*r_a. As such, the total area of the triangle of the compound must be 0.5*pi*r_a² and the area of the empty space in the triangle is $$\sqrt{}3$$*r_a²-0.5*pi*r_a² or ~0.16125r_a².

Now, I can't figure out how to fit a geometry to that empty space that'll only take up the space a small circle would and I can't find another way to figure out the answer.

I know the answer 0.155, but I'd like to be able to find out how.

 

[KataKoniK]

I would approach this problem by first understanding the concept of radius ratio and coordination number (CN). The radius ratio is the ratio of the radius of the cation (rc) to the radius of the anion (ra) in a crystal lattice. The coordination number is the number of nearest neighbors surrounding a particular atom in a crystal lattice.

For a CN of 3, the geometry is indeed an equilateral triangle with the anion at each vertex. In order to find the minimum radius ratio, we need to consider the maximum possible size of the anion and the minimum possible size of the cation within this geometry.

As mentioned in the forum post, for a CN of 4, the maximum radius ratio is 0.225. This means that the cation can have a maximum radius of 0.225 times that of the anion. For a CN of 3, the cation must fit within the empty space in the equilateral triangle, which is approximately 0.16125ra^2 (as calculated in the forum post).

To find the minimum radius ratio, we can assume that the cation is a small circle that just fits within this empty space. The area of a circle is given by A = πr^2. Therefore, the radius of this small cation can be calculated as √(0.16125ra^2/π) or approximately 0.255ra.

The minimum radius ratio is then calculated as (0.255ra)/ra = 0.255.

However, we need to keep in mind that this is just an approximation and the actual minimum radius ratio may vary depending on the specific geometry and arrangement of atoms in the crystal lattice.