Problem about lattice structure proof

[fisher garry]

I have looked at the cation anion ratio of cubic, octahedral and tetrahedral arrangments on an internet site. By a mathematical derivation they find the minimum value for the cation anion ratios for cubic, octahedral and tetrahedral arrangments. My problem is that even though I get the mathematics I don't get why the derivations show that there is an interval of values that the cation anion ratios for the cubic, octahedral and tetrahedral arrangments can have. I only get that they can have the values that they derive which is

cubic: 0.732
octahedral: 0.414
tetrahedral: 0.225

I need a mathematical justification on why it is an interval of values for the arrangments that in fact are

cubic: 0.732 to 0.999
octahedral: 0.414 to 0.732
tetrahedral: 0.225 to 0.414

I have uploaded a derivation online that says mathematically how the minimum values are found. In addition the question about how the intervals are found are posed there as well The answer is not mathematical. Can anyone help me in finding a mathematical answer to the intervals? The intervals are also in the attachment from an inorganic chemistry book.

 

[Christian0412]

I have done "small bit" of studying on this concept, I have not yet come across a mathematical answer that shows this "interval". It can only be understood physically, but expressed mathematically. I'm not saying that it's not possible (I can have an "interesting" math imagination), but I have yet come across mathematical "proof", so to speak. Assumptions (i.e. cation barely touches anion) allow us to mathematically determine the minimum ratios. (It's only considered minimum-ratio based on the chemical assumptions, other than that we're just doing geometry).

These ratios have upper limits because theoretically, a tetrahedral formation cannot have a stable radius ratio of 0.42. Because if this is the case, the theoretical most stable arrangement is now octahedral. Same goes for the octahedral arrangement. After 0.732, the theoretically most stable arrangement is the cubic - which goes up to 0.999. If there is a ratio greater than 0.999, it means that the cation is much bigger than the anion, which is usually not common considering that the electron cloud is much bigger than anions.

Furthermore (in case of your interest or maybe to further explain the concept)
Ratios can have values between 0.225 and 0.41. These are possible when the "type" of atoms are considered. For example, you can use actual radii values to determine its "actual" cation-anion radius ratio. If it falls within this interval, the simple model predicts the structure to have that particular arrangement. When the structure doesn't follow the model (i.e. real life), it gives researchers something to figure out about exceptions! lol

I hope this helps... Let me know if you have questions. Good luck!

 

[fisher garry]

Thanks for the great reply. I get the issue more now:)