Detecting Clusters in Atom Probe Data 277
2.6%. The next simulation had denser, smaller, and uniform weight clusters, where the cluster widths were equal to 0.8 (~1nm in diameter) and the cluster weights were set to 0.4% (Fig. 9). For these more complex precipitate distributions GEMA outperformed the maximum separation method by 18 and 66%, respectively, at a fixed false positive rate of ~3%. It should be noted that in the third simulation, the ROC
plot for the maximumseparation method jumps sporadically as Dmax is varied. This illustrates that slight changes in Dmax can have drastic effects on the power when using the maximum separation method. This undesirable property is a product of having very sensitive parameters. Lastly, a fourth simulation was created in which cluster
size was varied. In this simulation half of the clusters (w = 2% and σ = 2.6) were only ~25% denser than the matrix (w = 80%). GEMA’s success in identifying these clusters shows promise in its ability to identify clusters within high solute concentration alloys (Fig. 10). However, more thorough testing of the limits and capabilities of GEMA is the subject of on-going research.
Atom Probe Data
affect GEMA’s performance, as these fluctuations are effectively modeled by the background cluster. However, large crystallographic features, such as poles, were avoided for this analysis. Due to detector efficiency, it is impossible to quanti-
Next GEMA was applied to a model RPV steel, provided by Rolls-Royce Plc. Needle-shaped specimens of the steel were created from matchsticks via a standard two-stage electropolishing process (Gault et al., 2012). The samples were then analyzed via APT on a LEAP 3000X HR (CAMECA Instruments Inc., Madison, WI, USA). The sample was run at 50Kwith a pulse fraction of 20%. The data reconstruction was performed using IVAS 3.6.8. Small atomic density fluctuations do not significantly
tatively determine the success of GEMA when applied to real APT data sets. However, by visual inspection, GEMA appears to successfully identify each of the clusters present (Fig. 11). For this sample, cluster identification was performed by
GEMA on the order of minutes with a standard laptop (3.1 GHz). GEMA’s computation scales linearly with the number of solute atoms and quadratically with k. Therefore, run time for a given data set depends on the percentage of solute atoms within the data set as well the optimal number of clusters as determined via the BIC.
DISCUSSION
Accurately identifying and extracting clusters within atom probe data sets is critical to the field of APT. In order for the field to continue to grow as a reliable technique within the
greater scientific community, cluster analysis results cannot vary from user to user; an algorithm that produces reliable
and reproducible results is essential. GEMA can provide this reliability and reproducibility. The three key virtues of GEMA are its principled
probabilistic framework, generality, and extensibility. The probabilistic nature of this model has scientific merit, in that it quantifies genuine physical uncertainty regarding atoms at precipitate/matrix interfaces. By contrast, the maximum separation method simply provides an “in” or “out” verdict. The generality of GEMA enables the algorithm to be
applied to a wide range of APT data sets without requiring user input. This is illustrated in the first simulationwhere the maximum separation method performs very well for simple, homogeneous clusters, whereas GEMA excels in all studied circumstances. In addition, the modular nature of this approach
facilitates modifications to the model. Potential extensions of GEMA are numerated in the following section.
Potential GEMA Extensions
GEMA currently models clusters as spherical Gaussians. Although this assumption does not noticeably hinder GEMA’s performance, the covariance matrix could be generalized to allow ellipsoidal clusters in subsequent versions of GEMA. In addition, GEMA, in its current state, only considers
solute atoms. If GEMA is to be used widely within the atom probe community, itwould be necessary to incorporate a function which addresses nonsolute atoms. The simplest extension would presumably automate the L and E step of the maximum separation method. Although seemingly straightforward, an intelligent automation is nontrivial. In its current form, GEMA is limited to analyzing
precipitates. However, this technique could presumably be adapted to analyze interfaces, such as grain boundaries, aswell as other microstructural features. The probabilistic backbone of GEMAgeneralizesmore flexibly than other state-of-the-art cluster detection approaches. Nonetheless, there exist several immediately apparent challenges. For instance, unlike precipitates, a grain boundary would likely be poorly modeled by a Gaussian distribution. This potentially could be circumvented by allowing the grain boundary to be modelled by multiple Gaussians and then joined post hoc,or by including uniform slabs as mixture elements. Doing this correctly is likely possible and fruitful, but far from straight- forward. In addition, precipitates found along the grain boundary would likely require adding a deconvolution step to the algorithm. Consistently and correctly extracting grain boundaries and other microstructural features from APT reconstructions is an important challenge within the field, which seems possible to overcome via a machine learning approach similar to GEMA. Another potential extension, and perhaps the most
interesting, is that one could extend GEMA so that it learns not only the location, size, and width of each precipitate, but also information regarding phase. Through unsupervised learning the algorithm could simultaneously characterize
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