Archive for December, 2006

An RDF example using the Apex RDF plugin

December 12, 2006

Apex a43 includes a new RDF generation plugin. Here’s an example of some data taken with the plugin. Thanks to Dieter Isheim of Northwestern University for a sample dataset. In this case, it is the Fe alloy with Ni, Cu and Al, and a number of other components. In the picture below, Cu atoms are in red, and visually they appear in clusters.

First, I use ‘Select Points’ mode to make a simple selection in the dataset. Thats the rectangular area shown where the atoms are drawn slightly darker. This selected range includes about 118,000 atoms, 2093 of which are Cu atoms. These 2093 atoms will be used as the reference points in the RDF calculation.

From Clipboard.gif

I choose the ‘RDF Export’ menu item and select the parameters I want — Cu atoms for reference, 1600 bins and cutoff radius of 40 Ångstroms. I specify the name for the output file, in this case ‘Cu44 3.rdf’. The calculation takes 40 seconds. In that time, Apex has gone through each of the 2093 Cu atoms, found all of the other atoms in the dataset in a 40 Ångstrom radius, and measured the distance to each reference atom and binned all the results, writing out a tab-delimited file. I go to my graphing application proFit , and choose ‘Import’. Here I can specify that the file contains its own column headers,and this is the window that gets created:

RDFChartData.gif

In short order, I’ve got a graph that looks like this:

RDFGraph.gif

A few things to note. The red line is the most interesting. What it means is that the density of Cu is a lot higher near your average Cu atom than further away: That’s the same thing as seeing the Cu atoms in clusters in the atom map, but in the RDF, you can see how big the clusters are. The plot drops down significantly between 5 and 10 Ångstroms, which is indicative of clusters of about 20 Ångstroms in size. This kind of data is fun to interpret, because it is an average over all the reference atoms, some of which are in the center of a cluster, some of which are near the edge of a cluster, and the clusters aren’t necessarily spherical or uniform in size.

Second thing to note is that the Ni and Al also have slightly higher concentrations near Cu than farther away, but that the Aluminum concentration drops off faster than the Ni — out at a radius of 20 Å, the Ni concentration is double that of the Al. So there is a much stronger correlation of the Al with the Cu than of Ni with Cu.

the Fe concentration stays pretty flat despite a higher density of the other components near Cu atoms. Does this mean that there is actually a higher atomic density in the sample near the Cu clusters? Well, no. There are any number of Atom Probe conditions which might cause this sort of data anomaly: one possibility is that the radius of the tip may change as a function of chemical composition of the tip — and thus the magnification changes, too, producing changes in the apparent density. This effect is quite common. The efficiency of detection may also change as a function of composition — the Atom Probe may only collect 50% of the atoms in any given sample — other are lost due to bad ion optics, or evaporation outside of the pulse window, or multiple atom evaporation, or other electrical noise. Another effect is that some of the reference atoms may be close to edges in the dataset itself, which means that those reference atoms are not surrounded by a 40 Å radius of atoms, which would result in a decrease of the total density with increasing radius.

A common ploy to correct all these effects is to make a plot of concentration, rather than density (Dieter says he prefers that, actually). It is easy to do that in this case, as the total number of atoms is already supplied in the dataset in column 2.

Some RDF datasets also show short range order: that is, down in the 2 -3 Ångstrom range, it is possible to see the effects of first-nearest neighbor positions vs, second or third nearest neighbor positions. This dataset doesn’t show any such interesting pattern.

Lastly, one feature of the RDF that we calculate is the gradually changing binsize. As noted here previously, data analysis in the Atom Probe is often a tradeoff of positional error vs. statistical sampling error. Because the number of atoms at a given radius increases with r squared, we expect to have less error in the measurement at higher radii. In our calculation, we show both lower sampling error and lower statistical sampling error as the radius increases, by decreasing the binsize with increasing radius. This is reflected in the data: At lower radii, the scatter in the data is higher, and there is slightly more distance between the datapoints.

Radial Distribution Function Plugin

December 10, 2006

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:

RDFParams.gif

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:

ProfitInput.gif

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.