True Atomic-Scale Imaging in Three Dimensions 213


FIM sequences. In a first approach, the lateral scale can be approximately estimated. Using Dreschler’s method (Drechsler &Wolf, 1958) or employing the relationship V = kFFR, the curvature radius of curvature, R, of a tip can be deduced, with V the DC voltage applied, F, the effective evaporation electric field of the material studied and kF the electrostatic compression factor (Gault et al., 2012; Larson et al., 2013; Lefebvre-Ulrikson et al., 2016). The depth scale is readily determined by counting the number of evaporated atomic planes. 3D images are composed of an assembly of lentil-shaped 3D “spots,” with a 0.3nm lateral dimension and 0.05nm in depth. Close to high Miller index pole directions, where the lateral distance between atoms is sufficient, each atom can be spatially resolved. Atomic resolution is thus achieved in 3D in these {hkl} planes. This simple method is not very accurate as the tip’s


surface is highly curved, and the magnification is not linear over the surface. Nevertheless, image formation in FIM is described by simple projection laws (Vurpillot et al., 2007; Gault et al., 2012; Larson et al., 2013; Miller & Forbes, 2014; Lefebvre-Ulrikson et al., 2016). Using these projection rela- tionships, a more refined 3D FIM reconstruction protocol has been developed. The assumptions employed are similar to those of the standard APT reconstruction procedure (Bas et al., 1995). The 3D FIM protocol consists of a two-step methodology. The first step consists of calculating the lateral transformation appropriate for the recorded sequence. This transformation is obtained from the standard relationship,


L = kθθ, where L is the distance to the center of the image and θ the angle from the tip’s long axis assuming a hemi-


spherical apex. The quantity kθ is taken to be a constant and can be separately calibrated (Gault et al., 2012). Considering a given point P at the tip’s surface and assuming the center of the image is also the tip’s center, the corresponding angle θ is deduced from its position P(x,y,z) using the geometric formula


θ = a cos


px2 + y2 R


ffiffiffiffiffiffiffiffiffiffiffiffiffi ! : (1) From this angle, the corresponding position on the


detector is directly found. Note that different types of pro- jection laws may be used to improve the accuracy of the lateral reconstruction as this formula may present some limitations for wide angles (Brandon, 1964). As a result, an (x,y) regular map of intensities on the tip’s surface can be calculated for each successive FIM image. The second step consists of calculating the z-depth coordinate of the (x,y) positions previously defined. The corresponding z- coordinate is the sum of two contributions,ΔzR andΔze. The hemispherical shape of the tip’s apex introduces a depth shift, ΔzR, as a function of the distance to an image’s center:


ΔzR =R 1 - cos θ ðÞ: ðÞ (2) Moreover, as the tip is field-evaporated at a constant


evaporation rate when recording the FIM movie, Δze is deduced from the number of FIM images and from the total


Figure 3. Corrected volume reconstructed from the set of field- ion microscopy images presented in Figure 2, with the reconstruc- tion algorithm described in the text. The image is a slice crossing the precipitate observed in Figure 2g. Note the clear presence of the (011) atomic planes parallel to the horizontal axis.


depth (ZT) of the evaporated volume. This depth is calcu- lated from the number of evaporated planes:


Δze = ZT NT


´ i; (3)


where NT is the total number of images and i the index of an FIM image. This algorithm, applied to the images of Figure 2, produces a corrected volume with flat (011) atomic planes, and the correct morphology of the carbide is displayed in Figure 3. Compared with individual FIM images, more informa-


tion is contained in the 3D reconstructions. Indeed, the tomogram is the result of the dynamic field-evaporation process of a specimen, whereas classical FIM images are generally acquired in an essentially static state (i.e., using a low field-evaporation rate). The contrast effects obtained from the 3D exploration of the volume are generally easier to interpret, as sections or slices may be taken using any direction into the material. Improvements of the algorithm, using a variable radius of curvature in parallel with the number of images are being concurrently developed (Rademacher et al., 2009; Semboshi et al., 2009). The ability to provide a 3D image of precipitates, when a high contrast difference is found between precipitates and matrix, was demonstrated for small carbide precipitates in steels (Akré et al., 2009), in core–shell precipitates in Al–Sc–Zr alloys (Van Tendeloo et al., 2012), and in Cu–Ti or Cu–Fe alloys (Rademacher et al., 2009; Semboshi et al., 2009). It was also demonstrated that even small precipitates with small differ- ences in contrast compared with the matrix may be readily reconstructed in 3D. The precipitate size distributions and 3D morphologies of precipitates are determined in a straightforward manner without the necessity for complex interpretations (Akré et al., 2009; Rademacher et al., 2009; Semboshi et al., 2009; Cazottes et al., 2012). Image distor- tions are generally less severe than in APT reconstructions. The extent of local magnification effects and local trajectory overlaps are much smaller because the image gas is ionized from the static positions of atoms at the surface of a nanotip at a distance of a few tenths of a nanometer (Gomer, 1961). The ability to generate volumes with reduced deforma-


tion is essential for measuring accurate angles and morphologies of nanostructures embedded in 3D FIM tomograms. Another set of images was recorded at a different depth from the same model alloy sample (Fig. 4). The bright spots correspond to solute atoms in a precipitate


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