Comparison of APT Cluster Analysis Methods 369 Cluster
Cluster- matrix interface
Matrix
Table 2. Density of Atoms in Clusters in (i) Atom Probe Tomo- graphy (APT) Data and (ii) APT Data Following Simulation to Reverse the Effect of Local Magnification.
Atoms/nm3 in Solute Clusters As Radius (nm) 0 r 2r Distance from cluster centre of mass
Figure 3. Schematic representation of displacement function in X–Y plane to simulate the local magnification effect.
effective cluster radius R(z). The function satisfies the boundary conditions, 0 at the center of each cluster and far away from the cluster. A positive gradient corresponds to a compression of the distances between atom positions, a negative gradient to an expansion. The “transformation” also ensures that the atoms preserve their relative positions to each other. The exact form of the displacement function is not
known and so a simple trial function was considered Δd / k2Rz
ðÞd; ðÞ- d
have a maximum shift of k1. Δd=k1
4 k2 2Rz ðÞ jj
where k2 is a constant. In this case the maximum displace- ment will occur when d= k2Rz
or d = k2R(z). The maximum displacement is Δdmax / 1
2 and will equal 0 when d = 0 ðÞ
2. Hence, the shift can be normalized to
ðÞd 1 4 k2
k2Rz
ðÞ- d 2Rz
ðÞ
jj 2
: However, the maximum shift should increase with
increasing R(z). When R(z) = 0, the shift should equal 0, when R(z) = r (the cluster radius) the shift should be a maximum. The maximum should depend on the size of the cluster. This can be achieved by multiplying by Rz
can have any value between 0 and r). Thus, a better model for the displacements is
Δd= k2Rz = k2Rz
ðÞd 1 4 k2
ðÞ- d 2Rz
ðÞ 2r
ðÞd 1
ðÞ- d 4 k2
jj
jj 2
:Rz
ðÞ r : k - 1
: k - 1 ðÞ; simplifies to
with k defined as a compression coefficient (k1+1). Thus with k = 1.1 (a compression coefficient of 10%), Δdmax = 0.1R(z), corresponding to a displacement toward the cluster center making the cluster 10% smaller in the X–Y plane. With k2 = 2, [dmax is set to be 2R(z)] the equation
Δd= 2Rz
ðÞd r
ðÞ- d jj : k - 1 ðÞ:
Tests were performed to determine how robust the approach adopted is. APT data from an irradiated RPV steel
ðÞRz ðÞ
ðÞ r as (R(z)
Entire reconstructed volume
0.75–1.25 1.25–1.75 1.75–2.25 2.25–2.75
Reconstructed APT Data
36
49.0±10.4 58.1±13.1 65.8±9.7 66.4±12.3
Post Local Magnification Transformation with k1 = 1.25
36
39.2±11.5 33.9±9.3 39.3±7.2 40.6±5.7
containing a range of solute cluster sizes (0–5 nm diameter) was selected. The MSM method was used to identify atoms belonging to each cluster. Each cluster was then analyzed to determine the associated center of mass and principle axes of the best fit ellipsoid. The observed density of atoms in the solute enriched clusters was then determined by dividing the number of atoms within the best fit ellipsoid by the ellipsoid volume. The atoms included in these calculations can include both those assigned as cluster atoms and also matrix atoms. The resultant density data were analyzed as a function of cluster size. The transformation was applied, selecting a single value of k1 to spread out atoms within the clusters to a density consistent with the rest of the reconstructed data. The transformed data were re-analyzed, and the observed densities of atoms in each cluster size measured. The experimental data set selected was from Ringhals NPP and has been published (Styman et al., 2015). The results show that the apparent density of atoms in
clusters in this APT data are 60–70%greater than what would be expected from a typical steel with 37%detection efficiency. The same value of k was applied to all clusters in the data set irrespective of diameter. Table 2 and Figure 4 show that a single value of k reduced the density to close to the average for the entire reconstructed volume (36 atoms/nm3). A single value of k can be used for a range of cluster sizes.
Analysis of Solute Clusters
The MSM is one of the most commonly used cluster identification algorithms. Several authors have provided detailed explanations of its strengths and weaknesses and made recommendations regarding how best to optimize the parameters (Cerezo & Davin, 2007; Kolli & Seidman, 2007; Styman et al., 2013; Williams et al., 2013; Jägle et al., 2014). A second method, named “Isoposition method” (IPM)
was also used to identify clusters. The principle of this method which is based on solute concentration criteria is described elsewhere (Lefebvre et al., 2016). Roughly, a 3D
Magnitude of the atom displacement towards cluster centre
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