With Apex a43, there is now a plugin module that calculates a radial distribution function, or RDF. The RDF is one way of analyzing the local environment of atoms in the dataset. The calculation looks at all the neighboring particles of a particular atom (the reference atom), and collects the distance between that reference atom and each neighbor, and the atomtype of that neighbor. This information is compressed onto a single graph, the y axis representing the density of neighboring atoms of each type, and the x axis representing the distance of separation.

Usually, the data collected relative to a single atom is neither representative of the whole sample, nor is the statistical error low enough to have confidence in the results. Typically, the RDFs of many reference atoms are combined into a single plot. The RDF calculation in Apex operates on the current selection, and only on the atoms of a given atomtype.

When RDF calculation is enabled, there is a menu item called ‘RDF Export’ under the Plugins menu. Selecting this menu item offers the following choices for the calculation parameters:

Only atoms of the specified atomtype will be used for the RDF, and only those which are in the current selection. The distribution will be generated as a histogram with the given number of bins, and the last bin will represent radial separation of the specified cutoff radius.

The RDF will be output as a text file with tab-delimited columns and return delimited radius values. This file can be imported into a graphing application such as proFit with little effort.

If you are using proFit, select ‘Open…’ from the profit ‘File’ menu. Select the display ‘All Files’ option to be able to select the generated .rdf file. When importing, you will be presented with this dialog:

Choose the ‘with title’ option from the format popup menu. This will direct profit to use the first line of the text file as the column headers in the generated dataset.

One of the quirks of RDF generation is that as the radius increases, the number of atoms N in a shell of thickness Δr increases with r2. As is common with the interpretation of atom probe data, there is always a trade off between statistical error and positional error. If the binsize were to remain constant in the RDF histogram, this would mean that the statistical error would decrease as a function of 1/r , because the statistical error is proportional to sqrt(N). But because of the constant binsize, the positional error would remain constant through the data. This is non-ideal because at small radii, the statistical sampling error is much higher than the positional error, and at high radii, the positional error is higher than the statistical error. Instead, Apex generates an RDF with a continuously decreasing binsize, so that both the statistical error and positional error decrease with increasing radii proportional to sqrt(1/r). That is, as the radius increases, the binsize gets smaller by a factor of 1/sqrt(r) and the number of atoms per bin gets larger by a factor of r. The choice of number of bins and radial cutoff combine to determine the tradeoff between positional error and statistical sampling error.

As of Apex version a43, the value of the calculated RDF is in units of atoms per cubic Angstrom times the number of reference atoms. This will be changed in the future to be simply atoms per cubic Angstrom, so that the data is invariant to the number of atoms selected. In Apex a44, the units will be Atoms per cubic nanometer.

In addition, the radius value is given as the arithmetic average of the inner radius and the outer radius of the given shell. This is less than ideal for small radii, because for small radii, the number of atoms near the outer radius is higher. In other words, for the shell from radius = 0 to radius = 1, the current RDF labels the bin as r = 0.5, although the mean radius should be close to 0.8. In Apex a44, will have a radius value appropriately weighted toward the higher radius.

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