How Can We Measure the Level of Order in Particle Arrangements on Surfaces?

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Jamesss
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Hello,
I'm trying to determine the level of order of a pattern of particles on a sample surface. One idea was to calculate the mean distance between one particle and those adjacent and compare them to the idealised (perfect grid arrangment) distance if I take the area of the surface divided by the number of particles. Would this be the right path to go down ? If so, what statistical criterion could I use to determine whether a particle is adjacent or not?

All the best,
James
 
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Why would there be a statistical criterion for adjacency? Either they are or are not adjacent, I don't see any stastics in that decision. What is the definition of adjacent that you are using?
 
For example: A nearby particle B may be in the viccinity of A but not necessarily adjacent. There may be another particle between A & B. Question is whether a radial distance (and how far it should extend) should be used to include particles in the viccinity of a given particle in the determining of the mean particle spacing, whether they are adjecent or not.

Thanks,
James
 
I'm trying to determine the level of order of a pattern of particles on a sample surface. One idea was to calculate the mean distance between one particle and those adjacent and compare them to the idealised (perfect grid arrangment) distance if I take the area of the surface divided by the number of particles. Would this be the right path to go down ? If so, what statistical criterion could I use to determine whether a particle is adjacent or not?
Really, it doesn't matter what you do to get a statistic for the "level of order"; as long as you can determine how a "random" distribution of particles will score, you can use your statistic to test for a deviation from "random".

If you're dead set on your approach, then I would suggest just coming up with something simple to calculate and easy to analyze, rather than spend a lot of time worrying about the "right" way to determine adjacency.

Incidentally, my first idea would have been to divide your surface up into regions of equal area and count the number of particles in each region. The score would be the sum of

(observed # of particles - expected # of particles)²

for each region. (You probably want to divide by something clever) A grid-like arrangement of particles would be "too perfect", and score much lower than random. Other arrangements might score higher. I don't know if this would detect the sort of "organization" you're looking for, though.
 
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