380 Kristina Lindgren et al.


Figure 7. Nearest neighbor distributions (Mn and Ni, 29Da peak excluded) for the material irradiated to 2.0×1023 and 6.4×1023n/m2, laser pulsed analyses.


Figure 6. a: Proxigram of isoconcentration surfaces (Ni+Mn = 5.5%). Composition of matrix is 1.08 at%. b: Matrix Ni content as


a function of dmax. Laser pulsed analysis, Nmin = 18, material irra- diated to 2.0×1023n/m2.


Nmin (Williams et al., 2013). The randomization for Nrandom is obtained retaining the positions of the atoms and randomizing their identity. In the limit of no random clusters, this ratio approaches unity. The plots give an opportunity to put a threshold value or tolerance level,which can be set the same for different analyses giving different parameter choices. This gives the same statistical risk of defining randommatrix fluctuations as clusters. However, it does not give any indications to whether clusters are missed. For a specific Nmin, a suitable value of dmax can also be


found by comparing matrix solute levels with that from isoconcentration surfaces. In Figure 6a, a proxigram obtained from isoconcentration surfaces with a threshold of Ni+Mn = 5.5% is shown, indicating a matrix Ni content of around 1.08%. In Figure 6b, the concentration of Ni in the matrix is shown as a function of the choice of dmax, when applying the MSM with Nmin = 18. A comparison of the


matrix concentration suggests a dmax of 0.50 nm for the low fluence material (Fig. 6) and 0.55nm for the high fluence material. This method is dependent on manual determina- tion of a reasonable threshold for the isoconcentration sur- faces, and will depend slightly also on the parameters used, such as voxel size and delocalization. NNDs would be useful for finding the clustering para-


meters as they are not dependent on dmax and Nmin, but on the reconstructed data itself. Stephenson et al. (2007) used the NNDs for finding dmax. The NND is a superposition of two complete spatial randomness (CSR) distributions; one


for the clusters and one with a larger nearest neighbor (NN) distance for the matrix. By fitting a sum of two CSR dis- tributions and using the cluster part of this fit, a dmax can be found. Jägle et al. (2014) chose a different approach to pick parameters, also using NND. For each dmax, the clusters are removed and the matrix is compared with the matrix part of the full analysis CSR distribution. The random distribution and the matrix distribution are compared and the difference between them is minimized. For the materials analyzed here this method is hard to apply for the first-orderNNdue to the NND not being bimodal, see Figure 7. Thus, the fits of CSR distributions were arbitrary. An attempt to use the method developed by Jägle et al. was made to instead compare the distribution with a random distribution of the total analysis. This is not strictly correct as the full analysis contains slightly more Ni and Mn than the matrix, but the resulting values


should be reasonably close. This gives dmax of 0.55nm (low fluence sample) and 0.50nm (high fluence sample).


Higher Orders Higher order NNs can be used in the cluster analysis. This makes the analysis less local for high orders, but the con- tribution of clusters is more clearly resolved in the NND. In Figure 8a, the N = 8 NND is bimodal (contribution from clusters for a lower dpair and from matrix for a higher dpair), whereas the N = 1 NND has no bimodal shape since the contribution from clusters and matrix are overlapping. Chen et al. (2014) picked the order based on Gaussian dis- tributions fitted to the NNDs and by comparing the ampli- tude of the cluster contribution and the matrix contribution. For the materials discussed here, this method gives the eighth order as optimal. When changing the order, the other parameters need to be re-optimized; a dmax of 0.75nm is a reasonable choice following the approach presented above. However, the relative number of nonrandom clusters also depends on the order, see Figure 8b. The major contribution to the lower values of the relative number of nonrandom clusters for the same Nmin, for a higher order, stems from the higher probability of finding a random cluster. Possibly, this is an effect of increasing dmax.Hence, Nmin needs to be larger for higher orders for this material in order to avoid classi- fying random matrix composition fluctuations as clusters,


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