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Chemistry Forums for Students => Undergraduate General Chemistry Forum => Topic started by: cistran on March 30, 2006, 08:57:28 PM

Title: how to prove the PE of hexagonal closed packing structure to be 0.74?
Post by: cistran on March 30, 2006, 08:57:28 PM

Every general chemistry textbook discusses the method of calculating the packing efficiency (PE) of simple cubic, bcc and fcc structure. But I haven't found any book mentioning about how to prove the PE of hexagonal closed packing structure to be 0.74. Do you have any suggestion how to find it?

Title: Re:how to prove the PE of hexagonal closed packing structure to be 0.74?
Post by: AWK on March 31, 2006, 12:26:36 AM

Start from:
http://en.wikipedia.org/wiki/Kepler_conjecture
and
http://www.math.pitt.edu/~thales/kepler98/

Title: Re:how to prove the PE of hexagonal closed packing structure to be 0.74?
Post by: cistran on March 31, 2006, 02:29:57 AM

It's a very informative site and I really enjoy reading its. But so far, it seems to me that I could not use these to help my students understand this problem. Do you know any simpler and more understandable method?

Title: Re:how to prove the PE of hexagonal closed packing structure to be 0.74?
Post by: AWK on March 31, 2006, 03:33:47 AM

A proof of Kepler conjecture was done by Tom Hales about 10 years ago.

http://www.math.princeton.edu/~annals/issues/2005/Nov2005/Hales.pdf

Understaning of this proof  even transcends abilities of most matematicians.
(The refereeing process involved more than 12 referees over a five year period).
But stiil you can calculate density of packing for HCP, CCP or BCC by comparing volume of balls to volume of hexagonal prism or cube and treat this results as formal proof.

Title: Re:how to prove the PE of hexagonal closed packing structure to be 0.74?
Post by: Borek on March 31, 2006, 04:00:47 AM

Quote from: cistran on March 31, 2006, 02:29:57 AM

It's a very informative site and I really enjoy reading its. But so far, it seems to me that I could not use these to help my students understand this problem. Do you know any simpler and more understandable method?

Do I understand you correctly - are you looking for a way to just calculate the 0.74 value?

Take a look at the lattice - it is made of repeated identical regular tetrahedrons. You can easily calculate its volume, you should also have no problems with calculating what part of this tehrahedron volume is taken by the spheres located in vertices. That should be a starting point.

Take a look at the picture - it is only 2-D case, but hopefully you should get what I mean. Blue surface is tehrahedron minus red. Now do it in 3d.