Limiting radius ratio for tetrahedral

In summary, the conversation is about proving that the tetrahedral structure with a coordination number of 4 has a limiting radius ratio in the range of 0.225-0.414. The speaker is able to prove 0.225, but is unsure how to prove 0.414. They also mention needing to find the maximum and minimum packing fraction values as a function of (r1/r2). They ask for help and mention that larger values of the radius ratio are possible, with the maximum value being the minimum value of the next coordination number. The conversation ends with a question about the next possible coordination number and its minimum radius ratio value.
  • #1
tanaygupta2000
208
14
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
 
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  • #2
tanaygupta2000 said:
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?
 

FAQ: Limiting radius ratio for tetrahedral

What is the limiting radius ratio for tetrahedral?

The limiting radius ratio for tetrahedral refers to the maximum ratio between the radii of the largest and smallest spheres that can be packed in a tetrahedral arrangement without overlapping.

Why is the limiting radius ratio important in tetrahedral structures?

The limiting radius ratio is important because it determines the stability and packing efficiency of tetrahedral structures. If the ratio is too high, the structure may collapse or become less stable. If the ratio is too low, there will be wasted space between the spheres.

How is the limiting radius ratio calculated?

The limiting radius ratio is calculated by dividing the radius of the largest sphere by the radius of the smallest sphere in a packed tetrahedral structure. The resulting value is known as the coordination number, which is typically between 0.225 and 0.414.

What is the ideal limiting radius ratio for tetrahedral packing?

The ideal limiting radius ratio for tetrahedral packing is 0.414, which corresponds to the densest possible packing of spheres in a tetrahedral arrangement. This ratio is also known as the Kepler conjecture, which was proven by Thomas Hales in 1998.

Can the limiting radius ratio be exceeded in tetrahedral structures?

Yes, the limiting radius ratio can be exceeded in tetrahedral structures. However, this may result in a less stable and less efficient packing arrangement. In some cases, exceeding the limiting radius ratio may be necessary to accommodate specific design requirements or constraints.

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