Salt Crystals.

GuanYin

I'm just wondering if anyone knows anything about the mathematics involved in proving that salt forms a cubic shape. there seems to be some problem between the translation of reductionist theory and what actually occurs.

Salt (Sodium Chloride) forms a simple lattice structure. The formation of salt crystals has to be immune to the vibrations an atom undergoes for the lattices to form in the first place. According to quantum theory, these lattices are cubical (because the quantum mechanics forces them to be).

There seems to be a problem in proving the mathematically. Anyone know about this?

[edit] I've googled a good bit so please, don't just regurgitate stuff that you find goggle-ing, its not helping anyone[/edit]

ecksor

All I know is that chemists seem to be interested in group theory for talking about the symmetry of molecules, but I don't know if they use it to explain any shapes. Chapters had a book on the subject in their second hand section at one stage recently.

Victor

Not all salts are cubic, I think Potassium Chloride crystals are distinictly different (do a search for McDonalds salt on after Hours ).

Capt'n Midnight

No I can't prove the cubic crystal.
But I can disprove it - there exists at least one class of crystal which does NOT have a periodic structure.

A lot depends on valency and ion size , remember negative ions have bigger electron shells than in their ground state and divalent ones are bigger again.
IIRC Na+ and Cl- are similar sizes so cubic.
Remmber we get eclipes because the moon and the sun take up the same size in the sky.

Crystals usually have a two/three/four/six symmetry



Look up Penrose and his way of filling up the plane with fivefold pattern - it can extend forever but never repeats or symmetrical .
A crystal of aluminum-manganese alloy was found to have a pseudo-fivefold semmetry - it was a quasicrystal. - "somehow the orientation of incoming atomic groups has to take into account the overall pattern" - it's not regular but anything but random

http://www.quasi.iastate.edu/Scope/Background.html
In the physics and materials science communities,* "quasicrystal" refers to a class of binary and higher-order metallic alloys, most of which contain 60 to 70 atomic per cent aluminum (Al). These alloys are well-ordered structures which fall outside the realm of conventional crystallography. Their uniqueness stems from the fact that they exhibit rotational symmetries, most commonly five-fold symmetry, which are not consistent with periodic structures.

http://homepages.uni-tuebingen.de/peter.kramer/

Capt'n Midnight

http://geology.csupomona.edu/drjessey/class/gsc215/minnotes9.htm
At the bottom lists the likely crystal structure depending on the relative size of the ionic radii - then it's a matter of how closely you can pack the spheres.

Crystal Field Theory
http://www.unine.ch/chim/chw/Chapter%204.html
stuff on crystal shapes when you start going up the periodic table - D orbital

http://www.chm.davidson.edu/ChemistryApplets/Crystals/IonicSolids/NaCl.html
The ionic radius of the sodium ion is 1.16 angstroms and that of the chloride ion is 1.67 angstroms. The ratio of radii for the cation and anion is thus r+/r- = 1.16/1.67 = 0.695.

With a radius ratio of 0.695, the cubic holes are too large (rhole/r = 0.732) to be suitable. The sodium ions will prefer to occupy octahedral holes in a closest-packed structure. As it happens, the chloride ions in NaCl pack in a cubic closest-packed structure.

http://160.94.61.144/courses/2301/min07_xtalchem1.html
Ionic sizes - not a good link - but..

The maths is totally different - its all about packing spheres the relative numbers and sizes of which are determined by the valency and ionic radii of the ions involved.
http://www.math.pitt.edu/articles/harriotkepler.html
http://www.hypermaths.org/quadibloc/math/pakint.htm

===============================
Archimedean Polyhedra
http://www.uwgb.edu/dutchs/symmetry/archpol.htm
e^(i*pi)=-1

Capt'n Midnight

http://www.americanscientist.org/template/AssetDetail/assetid/15497
how to pack balls together in the densest possible way—was one of the oldest unsolved problems in mathematics ... In 1611, the German physicist Johannes Kepler stated what he felt to be the obvious solution: You make a triangular array, then fit another layer into the interstices between the balls in the first layer, and so on. In this arrangement, called the face-centered cubic lattice, just over 74 percent of the volume of the space is taken up by balls, and 26 percent by the spaces between the balls. Kepler never even tried to prove that this was the densest packing. ....Tom Hales of the University of Michigan claims to have a proof that no sphere packing can be denser than the face-centered cubic lattice. Like the proof of the Four-Color Conjecture, another notorious problem that was solved in the 1970s, his argument relies heavily on computer calculations: roughly 100,000 of them, virtually all of them too lengthy to do by hand.

.... Someday, he predicts, mathematicians will find a computer-free proof of the Kepler Conjecture—although it might take another 400 years.