face centered cubic ( cubic close pack) and simple cubic

[Roger Bagula]

There are two types of structures for Ionic crystals:
Na Cl->Face centered cubic->space group Fm3m
CsCl-> simple cubic-> space group Pm3m
Picture in cubes of face centered cubic structure:
http://profile.imeem.com/GUmj0c/photo/DUcZfUBQQu/
The book says the ionic size ratios:
r(CL)/r(metal)<=(Sqrt[3]+1)/2=1.366
are simple cubic and all ratios above that are face centered cubic.
In metal alloys the face centered cubic, the body centered cubic and the
Hexagonal close pack
are the major crystal structure forms. It appear that rational
substitution ( ratio) forms
in alloys are fairly common. These are all very symmetrical cell structures.

A while back while investigating Menger cube iso-types I did one based
on the
face centered cubic structure that is kind of pretty.
Mathematica:
(*checker board Menger cube by Roger Bagula 27 Aug 2007©*)
(* symmetric isomer of the Menger cube*)
(* patterned from Menger
cube code by Szabolcs Horvát < ***@gmail.com >, University of \
Bergen in Mathematica newsgroup : Mon, 28 May 2007 09 : 10 : 50*)
Clear[pieces, menger]
p = {{1, 1, 1}, {1, 0, 0}, {2, 1, 0}, {0, 0, 1}, {2, 2, 1}, {1, 2,
2}, {0, 1, 2}, {0, 1, 0}, {1, 2, 0}, {0, 2, 1}, {2, 0, 1}, {2, 1,
2}, {1, \
0, 2}};
pieces =
Complement[
Flatten[Table[{i, j, k}, {i, 0, 2}, {j, 0, 2}, {k, 0, 2}],
2], p];
N[Log[Length[pieces]]/Log[3]]
2.402173502732879`
menger[cornerPt_, sideLen_, n_] :=
menger[cornerPt + #1*(sideLen/3), sideLen/3, n - 1] & /@ pieces;
menger[cornerPt_, sideLen_, 0] :=
{EdgeForm[], Cuboid[cornerPt, cornerPt + sideLen*{1, 1, 1}]};
Show[Graphics3D[Flatten[menger[{0, 0, 0}, 1, 1]]], Boxed -> False]
Show[Graphics3D[Flatten[menger[{0, 0, 0}, 1, 2]]], Boxed -> False]
gr = Show[Graphics3D[Flatten[menger[{0, 0, 0}, 1, 3]]], Boxed -> False]
Show[gr, ViewPoint -> {0.248, 0.008, 3.375}]
Show[gr, ViewPoint -> {1.946, -1.956, 1.959}]

[Roger Bagula]

I've put into the Menger sponge program several space group unit cells
stuctures as vertices :

1) cubic close pack/ face centered cubic ( NaCl and about half the metals)
2) zinc blend ( ZnS and carbon, Si, Ge...)
3) Body centered cubic ( some metals )
4) Hexagonal close pack ( Zn, Zr, Ti... other half of the metals)

What I discovered late yesterday
( which should not have come as a surprise)
if you use ratio 2 instead of Menger 3,
you get a crystal build up instead of a fractal.
That is the space group unit cells behave just like a space fill
with their voids included. (So much for
tiling units with voids not being tiles.)
It is interesting that this result implies that crystal formation is
a fractal self-similar process.