# Calculate an order parameter for a given set of atomic coordinates

Physics Asked by masher on January 1, 2021

I have a box which contains a few thousand atoms, and I would like to be able to calculate a single number which gives some indication of how ordered their arrangement is in 3D.

For example, if they were totally crystalline, then the order would be 1. If the atoms were randomly arranged, the order parameter would be 0.

At this stage, I don’t need to be able to differentiate between different types of order (eg bcc vs fcc), just the overall amount of order.

The system is an infinite network with identical atoms. The infinite network is built with periodic boundary conditions applied to the box.

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What do I need to be reading? I’m currently reading on spatial autocorrelation and Moran’s I.

Eqn 4.1 gives a crystal-independent measure of order $$T^* = frac{ int_{rho^{1/3}sigma}^{xi_C} left | h(xi) right | dxi }{xi_C - rho^{1/3}sigma}$$

where $$h(xi) = g(xi) - 1$$ where $$g(xi)$$ is the radial distribution function, $$xi=rrho^{1/3}$$ where $$r$$ is the radial coordinate, $$rho = frac N V$$ is the number density, and $$xi_C$$ is a cutoff limited by the size of the box.

Answered by masher on January 1, 2021

I would recommend calculating the structure factor $$S(mathbf{q})$$ $$S(mathbf{q}) = frac{1}{N} left| sum_{j=1}^N exp(-i mathbf{q}cdotmathbf{r}_j) right|^2$$ for a large number of wavevectors $$mathbf{q}$$ and finding the maximum value of $$S$$, call it $$S_{text{max}}$$, amongst those values, excluding $$mathbf{q}=0$$. Then, as explained on that Wikipedia page, $$S_{text{max}}/N$$ will be of order $$1$$ for a regular crystal, and of order $$1/N$$ for a disordered system, if you have $$N$$ atoms. The components of the wavevectors will be integer multiples of $$2pi/L$$ assuming a cubic box of side $$L$$. You only need to consider components up to $$2pi/a$$ where $$a$$ is the atomic diameter. There is no need to include atomic form factors in the calculation, since your atoms are identical.

This is similar to the experimental approach of looking for sharp peaks in the X-ray scattering pattern. If you were interested in identifying the crystal structure, the pattern of sharp peaks would help. But for your purposes, just the fact that there are sharp peaks would be enough.

Answered by user197851 on January 1, 2021