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
