302 Andrew J. London et al.
be resolved. Some isotopes have a very low isotopic abun- dance (below the quantification limit of the atom probe) and may reduce the reliability of the overlap separation. Thus, a minimum isotopic abundance is required for the isotope to
be considered present, e.g., a value of ~0.2% to exclude the 18O isotope.
Local Environment To define the local environment around each overlapped ion, either a sphere centered around the ion or a number of that ion’s nearest neighbors is used, as shown in Figure 2. Ions from this local neighborhood are used to create a mass spectrum and the counts of the ions from this spectrum are used to deconvolute the peak overlaps. A sphere of fixed radius gives a fixed spatial resolution around each ion, however, this sphere will contain a variable number of ions and hence have a varying degree of compositional uncertainty. Alternatively, specifying a number of nearest neighbors gives a fixed number of ions (fixed compositional certainty) but a variable spatial resolution.
Separating Overlaps
The counts for each mass peak are given by specific ranges of m/z values (Defining Overlaps section). This has the advantage over spectral fitting methods used by Cairney et al. (2015), which require a pure reference spectrum to perform the fitting. The approach used here assumes a linear com- bination of ion counts {n} divided into separate peaks as described by their isotopic abundance [A]. The counts {r} from each peak are contained within the specified range windows. For example, for two ions producing three mass peaks, this can be expressed in matrix algebra as
2 4
Ai;1 Aj;1 Ai;2 Aj;2 Ai;3 Aj;3
3 5
=
ni nj
8 <
: ;
r1 r2 r3
If the ions have a natural isotopic abundance, then [A]is
known. [A] contains one column for each ion and each row contains the contribution of that ion to one specific m/z position. {r} contains the counts for each peak observed from the atomprobe data. This leaves the ion counts {n}, which is unknown. Unless equation (1) is rank deficient (Defining Overlaps section), then there are at least as many ranges as unknown ions, meaning this is an over-constrained
ab r RESULTS
Case Study 1 ODS steels contain complex oxide phases which produce molecular oxygen-containing species during the atom probe experiment. Due to the differing chemistry of the oxide clusters, different ionic species can evaporate from different clusters. Figure 3a shows the mass spectrum from the 31–33Da range of two different clusters. These two clusters have very different ionic compositions, with one containing predominantly TiO2+ ions and the other containing onlyO2
+
ions in the 32Da peak. The ions from the 32Da peak of an ODS steel are shown in Figure 3b. Inspection of the global mass spectrum experiment shows small peaks from TiO2+ ions as well as a significant additional contribution to the 32Da peak from O2
+ ions overall. Figure 3c shows the per-
Figure 2. The local environment may be defined by (a) a sphere around the target ion, or (b) a number of nearest neighbors.
ion deconvoluted data using a 1nm spherical local environment. The deconvoluted data show that most of the oxide clusters contain TiO2+ ions, however, aO2-rich cluster
9 =
: (1)
mathematical problem. The simplest method to solve for {n} is to minimize the squared residual; this is called least squares (LS) (Lawson & Hanson, 1995). The application of LS to solve peak overlaps is described elsewhere (Gault et al., 2012; Larson et al., 2013). However, Miller et al. (1996) describe solving the same mathematical problem using maximum likelihood estimation (MLE). MLE is superior to LS when treating discrete data with limited counts (Maus et al., 2001), but is less numerically stable and somewhat more computationally intensive than LS. In our imple- mentation, MLE was roughly ten times slower than LS, but was considered an acceptable trade-off for the improved result. The formalism given by Miller et al. (1996) was an analytical solution for a specific species overlap and therefore of limited applicability. However, we have devised a method to solve any arbitrary set of overlapping peaks using the following likelihood function:
L r; n ðÞ= r ðÞ fg Tlog A fg½ n For any particular set of overlapping peaks (any [A]) and
an initial guess of {n}, this function can be maximized by a numerical method to return the most likely {n}, for a given {r}. From {n}, the most likely ionic composition of the overlapped mass peak is calculated.
Identity Assignment
Using the local ionic composition, computed by the method above, the identity of an overlapped ion can be reassigned. This reassignment is made probabilistically and weighted by the deconvoluted ionic composition. If an ion has a composition of 10% X and 90% Y, then ionic identity is assigned as X or Y at random but with a 10/90 weighted probability. Relabeling probabilistically gives the advantage that the resulting data can be used with conventional workflows such as composition line profiles and proximity histograms as demonstrated in the following section.
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