244 Frédéric De Geuser and Baptiste Gault
corresponding to Fig. 4a). To adjust the orientation of the specimen, we have minimized the distance between the observed and predicted poles only. An alternative methods would have been to adjust the orientation by pattern matching on the figure showing the zone axes. For known structures, building a database that could be used for pattern matching is feasible only if the projection is known. Such an effort would allow for automated determination of the orientation of the specimen by techniques similar to those used in electron microscopy orientation mapping, e.g., electron backscatter diffraction (Britton et al., 2016), nano-beam diffraction (Rauch & Véron, 2014), or transmission Kikuchi diffraction (Keller & Geiss, 2012; Trimby, 2012).
DISCUSSION ON THE EFFECT OF THE PROJECTIONMODEL ON THE RECONSTRUCTED VOLUME
It is difficult to determine a good metric for the accuracy of atom probe reconstructions as it really depends on what is being investigated. However, there are two general aspects that are usually, albeit implicitly, used as quality assessment of the reconstruction:
1. Angular distortions: if the volume contains planar features such as an interface, thin layers or platelets precipitates, one expects them to appear flat in the reconstruction.
2. Distance accuracy: it is very common to check for known interplanar distances as a mean of reconstruction calibration.
Both of those aspects had been discussed by Bas et al.
(1995) already. A perfect illustration of this is the expression of z in equation (3). It consists in the sum of zc, the analyzed depth, and R cos θ, which links the curvature of the sample to the position of the atoms in the final reconstructed volume. Schematically, we can say that the error on R cos θ is indi- cative of angular distortions, whereas the error on zc corres- ponds to depth calibration error. Interestingly, these two aspects are not independent. Errors on R cos θ will occur if the projection model is
inaccurate. Systematic errors can, to a certain extent, be compensated by an adjusted R with respect to its physical value. However, this will also influence every other aspects of the reconstruction, in particular the depth of analysis (Gault et al., 2009). If we know the actual projection law, it is possible to
reverse project any given object to test the reconstruction protocol. In Appendix A, we derive the analytic equations of two planes normal to the analysis direction and located at a distance τ from each other, mimicking the reconstruction of a thin layer of thickness τ. It should be kept in mind that this is in the absence of any other reconstruction artifacts such as local magnification or trajectory aberrations due to variation in local evaporation fields.We can apply this
to the simulated geometries used above. Again, we are in a very ideal situation where the sample has a completely smooth surface. In this context, no near-field effect can deteriorate the reconstruction (Vurpillot et al., 2000; Oberdorfer et al., 2013), and we should thus obtain the best possible reconstructed layer. In addition, we have a full knowledge of the sample geometry, so that no calibration should be needed. Equation (A.13) shows that the expression of the
reconstructed z is dependent on θ (whereas it should be constant for a plane normal to the direction of analysis). Again it has two contributions: (i) the effect of the error on R cos θ (proportional to R*
R
cos θ* cos θ ) and (ii) the effect of the error
on the depth of analysis (proportional to R
2 R*
sin θmax sin θ*
max ). The calibration of an APT reconstruction is very often
performed by checking a known interplanar distance, i.e., by calibrating the depth of analysis. We see from equation
(A.13) that for this we need to adjust R
2 R*
sin θmax sin θ*
max to make it
close to 1. The influence of θmax is very important here as its contribution is squared. Of course it can be adjusted by artificially adjusting R and/or the ICF, which is what is necessarily done in practice in the absence of other infor- mation on their value, but it can only result in some degree of angular distortions. It also means that, in the case where the radius has been measured, for instance by ex situ electron microscopy, the actual value may not be the one that should be used to optimize the calibration of the recon- struction. The same holds true for caseswhere the ICF can be measured. To illustrate this, we have used equation (A.13) to
reconstruct a 5-nm thin layer normal to the analysis in one of the simulated sample geometries (R=90 nm, α=14°). We show only the trace of this layer in the (x, z) plane. The result is shown in Figure 8. With the actual physical parameters as input, the equi-
distant projection models quite accurately predicts the shape of the layer. Its thickness is accurate. Only faint distortions appear at large angles. The pseudo-stereographic projection predicts a layer thickness about 8% thicker than the expected 5 nm, mostly linked to a wrong value of θmax that causes an inaccurate estimation of the analyzed surface. The distor- tions are very important, even at moderate angles. If one assumes an a priori knowledge of the thickness of the layer, one could adjust the reconstruction parameters to achieve it. For this, one could change the value of the radius and/or that of the ICF. This is illustrated by the two lower images in Figure 8 where the layer has been forced to a central thick- ness of 5 nm. It is obvious that there remain some distortions and the layer does not appear flat. Again, this is in a very ideal situation where the model should work at its best. This explains why, because of the chosen standard angular projection model, there can always be, even in the most accurately calibrated APT reconstruction, some remaining distortions which sadly may lead potential readers to be less confident in the results.
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