On Monday 01 December 2008 15:07:56 Edward A. Berry wrote:
> Thanks, Ethan,
> For your third point- I realized (after sending) that the distribution
> would be stretched along the long axis- but actually I'm having
> a hard time coming to grips with that conceptually- if there
> are n atoms in the cell, they will necessarily be distributed
> more sparsely in projection along the long cell axis than the
> short axes, and you can't add more atoms along the long axis
> to increase it's density without increasing density along the other two.
Heh. A new "Monty Hall" problem to demonstrate how probability
distributions mess with our minds.
You are of course correct that you cannot increase the number of
atoms in a fixed cell without increasing the density in all directions
through the crystal. And I phrased my original description badly,
if I made it sound like somehow this was possible.
I should not have used the word "density"; perhaps
"mean distance between particles along the path" is a better wording.
The main point I was trying to make is that if you ask for a
"random distribution in 3-space" without mentioning crystals,
you are probably expecting certain properties. In particular,
unless otherwise stated, the criteria for "random" would
normally include isotropy. Or at least that's what I would assume.
A distribution that had significantly different properties in
different directions would not be considered "random" in this
context.
But a crystal lattice is the antithesis of "random" in this
sense, because it imposes by definition a requirement for an
exact direction-dependent repeat spacing determined by the lattice.
You cannot simultaneously satisfy a requirement for uniform
isotropic distribution in 3-space and a requirement for
crystalline symmetry, except in the degenerate case of density = 0.
> As for the rest, I think it is semantics or a question how precisely
> we want to say something. Yes, what I was describing was a randomly
> chosen sample from a uniform probability distribution, but it is this
> sample that the OP is requesting- so I would rephrase your question:
> does he want _a random sampling from_ a uniform probability distribution
> throughout the lattice, or ...
> Ed
>
> Ethan Merritt wrote:
> > On Monday 01 December 2008 10:28:34 Edward A. Berry wrote:
> >> Ethan A Merritt wrote:
> >>> On Friday 28 November 2008, Mueller, Juergen-Joachim wrote:
> >>>> Dear all,
> >>>> does anybody know a program to
> >>>> fill an unit cell a,b,c randomly by an arbitrary number
> >>>> of spheres (atoms)?
> >>> First you would need to define "random".
> >>> Uniform density throughout the lattice?
> >>> Uniform distribution of neighbor-neighbor distances?
> >>> Uniform fractional coodinates?
> >>> Must the placement conform to space group symmetry?
> >>>
> >> Although I am sure it was not intended, this might suggest
> >> to some that uniform is equivalent to random-
> >> actually they are the opposite: a random distribution would
> >> have large areas with nothing and other places where two or
> >> three spheres are almost on top of each other.
> >> A uniform distribution is, well, uniform.
> >
> > I fear you are muddying the waters rather than clarifying.
> > What you refer to as "random distribution" is better described
> > as random sampling from a uniform distribution.
> >
> >> Most programming languages have a function to generate a random
> >> number evenly distributed between 0 and 1.
> >
> > My point was that simple random sampling is not correct in the
> > context of crystallographic symmetry. If you use this procedure to
> > "fill the unit cell", as originally requested, you will violate
> > the crystal symmetry. If you use it to fill the asymmetric unit,
> > then the distribution that describes placement within the full
> > unit cell is no longer the same distribution as you sampled from,
> > since it is now perturbed by the additional placements generated
> > by crystallographic symmetric rather than by random sampling.
> > That may be acceptable, or it may not, depending on the
> > intended application.
> >
> >> Decide how many atoms
> >> you want, get three random numbers for each atom, and those are
> >> your fractional coordinates of your random spheres. Coordconv will
> >> convert to orthogonal angstroms given your cell parameters.
> >
> > That was the "uniform fractional coordinates" case that I listed.
> > It is unlikely to be the correct choice (although as always it depends
> > on the question). This problem is that since it is based on fractional
> > coordinates rather than the true cartesian coordinates, the resulting
> > density of atomic centers will be strongly anisotropic. The density
> > along each axis will be inversely proportional to the cell edge.
> > You would do better to define a cartesian coordinate grid that fills
> > the region of interest, and then assign an atom to each grid point with
> > probability 1/N. This produces artifacts of its own, of course, since
> > the distribution of interatomic distances is now discrete rather than
> > continuous.
> >
> > The question "what is random?" is very deep, and the answer
> > depends strongly on the intended application.
> >
>
>
--
Ethan A Merritt
Biomolecular Structure Center
University of Washington, Seattle 98195-7742
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