242 Frédéric De Geuser and Baptiste Gault
EXPERIMENTAL ANDMODELING ASSESSMENT OF THE PROJECTION MODELS
An experimental assessment of the accuracy of the angular projection can be made by using either FIM images or desorption maps from an APT data set, if it bears enough crystallographic information such as poles or zone axis. This is easiest with pure metals. In the set of crystallographic poles shown in Figure 3, with their position on the detector, it is possible to directly compare the observed positions to the positions predicted by the projection models.
Distance Versus Angle
best fits for each pole considered as the projection center, as a function of the distance of that pole to the detector center. It can be seen that (1) the pseudo-stereographic model always predict a higher ICF, which arises in order to compensate for the nonlinearity; and (2) the pseudo-stereographic is very dependent on the chosen projection center, in contrast with the equidistant model. Figure 4c assesses the quality of the
Figure 4a shows the relationship between the distance on the detector and the angle between crystallographic features. To plot this figure, we have separately considered each identified pole as the center of projection, and plotted the distance between the center and all the other poles as a function of the crystallographic angle. Overlaid are two best fit curves corresponding, in blue, to an equidistant projection, and, in red, to a pseudo-stereographic projection. As each model has been optimized separately, the corresponding ICFs are different (as shown, for instance, by the different slope at origin). It is readily visible that the experimental data are linear up to very large angles and that the best fit pseudo- stereographic projection cannot accurately reproduce the experimental distribution, which is particularly striking at large angles. Figure 4b shows the ICFs (ξ=L/k) resulting from the
fits by plotting χ2 = 1
pseudo-stereographic model gives consistently less accurate results than the equidistant model. Such an analysis shows that (i) the azimuthal equi-
NPðρexp - ρmodelÞ
distant projection closely matches the experimental data, better than the pseudo-stereographic projection; and (ii) the equidistant model is essentially immune (within accessible angles) to errors on the position of the projection center, which is virtually always unknown. In electrostatic simulations of ion trajectories, with a
geometry mimicking a full-size commercial instrument introduced by Loi et al. (2013), this linear relationship was also found to reproduce well the trend observed in the variation of the distance ρ of an ion impact to the center of the projection, which corresponds in this case to the center of the detector, as a function of the emitting angle (Larson et al., 2013) for a variety of specimen geometries. In Figure 5a, ρ is plotted as a function of the launch angle for specimens with shank angles varying from 2 to 14° and radii in the range of 20–170nm. For each distribution, a linear regression corres- ponding to an equidistant projection was calculated and is displayed as a solid line. In addition, a dashed line is displayed which shows the distribution expected from a pseudo-stereographic projection using the estimation of the average ICF derived by Loi et al. (2013). The ratio between the results of the simulations and both the linear regression and the pseudo-stereographic projection is shown in Figure 5b. For the equidistant projection, this ratio only varies within a narrow range of ±2% around unity with a standard deviation for the ratio 2σeq.=0.011, whereas for the pseudo-stereographic projection, more significant variations are observed, with 2σpseudo−st.=0.017. In order to have a better appreciation of how such
variations could affect experimental data, we have calculated the average positioning errors between the simulated dis- tance ρ on the detector and the one predicted by the pseudo- stereographic and equidistant projection models for each
2. It shows that the
Figure 4. a: Distance between the center of a set of 41 crystallographic poles observed in the analysis of a pure Al data set obtained on a LEAP 5000 XS (each pole considered as the projection center). b: Best fit image compression factor (ξ=L/k) obtained for pseudo- stereographic (red) and equidistant (blue) models, respectively, as a function of the position of the pole considered as projection center. c: Quality of the fits obtained in (b) as assessed by χ2 = 1 nomial trend lines as guides for the eye.
NPðρexp - ρmodelÞ 2. The dashed lines are poly-
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