physical analog of fractal dimension

[Roger Bagula]

Hypothetical fractal process in the chemistry of ionic crystals:
Cubic crystals are well known in nature.
Sodium Chloride is a very good example.
The idea is to make up a statistical analog crystal to different types
on Menger sponges
using substitution of lithium or potassium.
Example:
mix:
7 moles of LiCl ( or KCl)
with
20 moles of NaCl
so that the metal ion
Moran dimension in sodium is
Log[20]/Log[3]
Random Menger cubes with 7 of one and 20 other would result
in the crystal that results overall from the solution.
( differential solubilities may flaw the result further: experiments
would have to be done)
There is little hope that a Menger structure would result,
but the result should be statistically Menger in properties.
The fracturing of the crystals of this type would depend on the relative
amounts of Lithium to Sodium.
Comparison with pure Sodium Chloride fractures to
different integer fractional ( rational) mole amounts of impurities of
LiCl or KCl
would give an experimental method of gaging fractal
statistical dimension properties so that a model
for crystal could be formulated.

A program in Mathematica to make up a random Menger cube is possible
with a slight alteration of the original program to put in 7 voids in a
27 cube
matrix of cubes.
The Random von Koch in Falconer's "Fractal Geometry"
and percolation fractals are very much like this process.
Picture Link:
http://profile.imeem.com/GUmj0c/photo/Qh05u0X9I5/
Mathematica:
Clear[a, menger]
(* Random Menger cubes : by Roger Bagula 26 Aug 2007© : Similarity
Dimension \
2.771243749161422*)
(* random voids in a Menger cube statistic of 27 cubes and 7 random voids*)
a := Complement[
Flatten[Table[{i, j, k}, {i, 0, 2}, {j, 0, 2}, {k, 0, 2}],
2], Table[{Random[Integer, {0, 2}], Random[Integer, {0, 2}], \
Random[Integer, {0, 2}]}, {n, 1, 7}]]
N[Log[Length[a]]/Log[3]]
2.771243749161422`
menger[cornerPt_, sideLen_, n_] :=
menger[cornerPt + #1*(sideLen/3), sideLen/3, n - 1] & /@ a;
menger[cornerPt_, sideLen_, 0] :=
{EdgeForm[], Cuboid[cornerPt, cornerPt + sideLen*{1, 1, 1}]};
Show[Graphics3D[Flatten[menger[{0, 0, 0}, 1, 3]]], Boxed -> False]

Respectfully, Roger L. Bagula
11759Waterhill Road, Lakeside,Ca 92040-2905,tel: [phone]
:http://www.geocities.com/rlbagulatftn/Index.html
alternative email: ***@sbcglobal.net

[Stewart Robert Hinsley]

Post by Roger Bagula
Cubic crystals are well known in nature.
Sodium Chloride is a very good example.
The idea is to make up a statistical analog crystal to different
types on Menger sponges
using substitution of lithium or potassium.
7 moles of LiCl ( or KCl)
with
20 moles of NaCl
so that the metal ion
Moran dimension in sodium is
Log[20]/Log[3]
Random Menger cubes with 7 of one and 20 other would result
in the crystal that results overall from the solution.
( differential solubilities may flaw the result further: experiments
would have to be done)
There is little hope that a Menger structure would result,
but the result should be statistically Menger in properties.
The fracturing of the crystals of this type would depend on the
relative
amounts of Lithium to Sodium.
Comparison with pure Sodium Chloride fractures to
different integer fractional ( rational) mole amounts of impurities of
LiCl or KCl
would give an experimental method of gaging fractal
statistical dimension properties so that a model
for crystal could be formulated.

This sort of thing has been investigated. Look for, for example,
articles on double chlorides and solid solutions. In the particular
cases of LiCl/NaCL and NaCl/KCl is appears that you don't get a crystal
with more than trace quantities of a second alkali metal. Instead you
would get separate crystals of the two chlorides. Or at least that's
what the abstract at

http://cat.inist.fr/?aModele=afficheN&cpsidt=16102068

seems to imply.

There are cases where two ions can be freely substituted in a crystal.
An example that comes to mind is olivine, which is a solid solution of
Mg2SiO4 and Fe2Si04. Although I seem to recall mention of ordered phases
in solid solutions I expect that here the Magnesium and Iron ions are
disordered. However olivine is orthorhombic, not cubic.

A solid solution is favoured if the substituting ions are of similar
charge and size. Li+, Na+ and K+ differ significantly in size.

A classic cases of double salts are the alums, which are double
sulphates of a singly charged ion (Na+, K+, Cs+, Rb+, NH4+) and a triply
charged ion (Al+++, Cr+++).

[Roger Bagula]

Post by Stewart Robert Hinsley
I
This sort of thing has been investigated. Look for, for example,
articles on double chlorides and solid solutions. In the particular
cases of LiCl/NaCL and NaCl/KCl is appears that you don't get a
crystal with more than trace quantities of a second alkali metal.
Instead you would get separate crystals of the two chlorides. Or at
least that's what the abstract at
http://cat.inist.fr/?aModele=afficheN&cpsidt=16102068
seems to imply.
There are cases where two ions can be freely substituted in a crystal.
An example that comes to mind is olivine, which is a solid solution of
Mg2SiO4 and Fe2Si04. Although I seem to recall mention of ordered
phases in solid solutions I expect that here the Magnesium and Iron
ions are disordered. However olivine is orthorhombic, not cubic.
A solid solution is favoured if the substituting ions are of similar
charge and size. Li+, Na+ and K+ differ significantly in size.
A classic cases of double salts are the alums, which are double
sulphates of a singly charged ion (Na+, K+, Cs+, Rb+, NH4+) and a
triply charged ion (Al+++, Cr+++).

I was looking for a system in a cubic or tetrahedral group (preferably
cubic as
it would be able simulate it with software fairly easy).
Maybe HCl ( not strictly ionic!) and LiCl would over come the
size difference problem. HCl abducts are well known in salt crystals.

As I remember Alums are pretty crystals and fairly common.
http://chemistry.about.com/cs/howtos/ht/alumcrystal.htm
http://chemistry.about.com/od/growingcrystals/ht/purplecystal.htm
http://scripts.iucr.org/cgi-bin/paper?S0021889875010588
"Sodium-chromium `anhydrous alum', NaCr(SO_4 )_2 is monoclinic, space
group /C/2//m/."
Monoclinc ( C2h is probably the point group of the unit cell?)
Low symmetry like this even in crystals is not a very promising
structure for fractal dimension experiments.

It looks like maybe a thought experiment and simulation would be ahead
of physical analogs.
At least until further research on models. Tetragonal systems with a
C4 axis and some other system
might work better ( works better with cubes!).
Trigonal crystal systems that are C3v might also be
worth while.
Transition metal substitutions like Cr, Mg and Fe do over come the size
problem, but also
are very close in attraction potential as well.

What is needed is a higher symmetry crystal with relative interchange of
ions.
One that can be adapted to some sort of Menger sponge simulation in a random
A system like LiBr and KCl has four variables instead of just two but
might work better
for sizes.

Another thing is "cheapness" of the materials, so it can be duplicated
by many people ( like the bath tub and cleanser experiment for
sedimentation).
Full a tub with medium hot water , take a bath.
Before you pull the plug, take a lot of cleanser and salt it in the water.
Let it drain relatively slowly. ( probaly important: as a rapid draining
doesn't allow the cleanser to
fall out as the water drains.)
Observe the patterns in the cleanser that sediments out.
This is essentually like sand in rivers and lakes.
Roger Bagula