Limiting radius ratio for tetrahedral

[tanaygupta2000]

Homework Statement

Prove that the tetrahedral structure (like ZnS) having a coordination number of 4 have a limiting radius ratio in the range 0.225-0.414

Relevant Equations

Pythagoras Theorem, d^2 = a^2 + a^2

I am able to prove that it is 0.225 but how do I prove that it is also 0.414?
I need to find the max. and min. packing fraction values, which I got as a function of (r1/r2)
Please help

 

[Steve4Physics]

Homework Statement:: Prove that the tetrahedral structure (like ZnS) having a coordination number of 4 have a limiting radius ratio in the range 0.225-0.414
Relevant Equations:: Pythagoras Theorem, d^2 = a^2 + a^2

I am able to prove that it is 0.225 but how do I prove that it is also 0.414?
I need to find the max. and min. packing fraction values, which I got as a function of (r1/r2)
Please help

I’m definitely no expert but maybe this will help...

Each coordination number has a minimum value of the radius ratio (RR). Larger values of the RR are possible - you can get the general idea from this diagram. https://upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Criticalradiusratio.png/800px-Criticalradiusratio.png

For a given coordination number, the maximum value of the RR is taken as the minimum value of the RR of the next coordination number. (I’m no expert so please don’t ask me why!)

What is the next possible coordination number after 4?
What is the minimum value of its RR?