240 Frédéric De Geuser and Baptiste Gault
dV dzc
has the units of a surface and can be seen as the projected
analyzed surface. If the effective detected surface on the detector is a disk and the maximum observable angle is θmax, then dV
not exactly a disk, which is often the case when the end user selects a specific area of the detector for data reconstruction, a more general formula would be needed that includes a geometrical factor σ depending only on the shape of the selected area:
dzc =πR2sin2θmax. If the effective detection surface is
dzc dN =
ησπ sin 2θmaxR2 : Ω (7)
Interestingly, θmax depends on the angular projection model, which is itself strongly dependent on the specimen and microscope geometries. Therefore, θmax is not a constant, and actually is expected to evolve throughout a single analysis. Finally, please note that the formula introduced in Gault et al. (2011) to extend the concept introduced by Bas et al. (1995) equating dV
surface of the spherical cap is not generally applicable. dzc to the reverse projection of the detector onto the Azimuthal Equidistant Projections (Linear Model)
ANGULAR PROJECTIONMODELS If the specimen is assumed to possess a spherical endshape, the position of an atom at the surface of the sample can be given by its longitude and latitude angles, widely used in geographical mapping. In the notation of this work, the longitude is ψ. As we consider only azimuthal projections, its value is conserved through the projection. While the latitude represents the elevation from the equatorial plane, in APT, it is more conventional to access it through its complemen- tary angle θ which gives the angle between the axis of the specimen and the position of the atom. Defining an angular projection is simply finding the
relationship between ρ, the distance between the projection center within the detector plane and the detected hit position, and θ, the launch angle, i.e., ρ=f(θ).
Pseudo-Stereographic Projections (Standard Model)
The pseudo-stereographic projection is the de facto standard angular projection model for APT. It is a point-projection model. It can be envisioned as an intermediate situation between a gnomonic projection (i.e., where the origin is the center of the spherical cap) and a stereographic projection (i.e., where the origin is at the south pole, i.e., at twice the radius from the surface). In the pseudo-stereographic model, all trajectories originate from the same point situated on the specimen axis but behind the center of the spherical cap. The distance from the center to the apparent origin of the trajectories is written mR. Trigonometric considerations leads to the following expression for the projection:
ρ= L m+ cos θ sin θ:
(8)
A possible misconception originating from the ubi- quitous use of the expression image compression factor (ICF)
As shown independently by Newman et al. (1967), Wilkes et al. (1974), and Cerezo et al. (1999) using FIM, the relationship between distance on the detector and crystal- lographic angle is in fact better reproduced by a linear relationship, i.e.
ρ=k θ:
k= L ξ = L
θ= ρ k :
m+ 1 ; (12)
The convergence of the projections at small angles imposes that
(13)
but the projection models are only equivalent at small angles. The inverse projection is
(14) In geographical mapping, this model is called azimuthal
equidistant. Similarly to the pseudo-stereographic projec- tion, it is azimuthal. It is equidistant in the sense that the distance on the detector ρ (i.e., on the map) between the center and any other point is proportional to the actual dis- tance at the surface of the specimen’s spherical cap (i.e., Rθ). The azimuthal equidistant projection is not a point projec- tion and the apparent trajectories of the ions do not cross at a single projection point, as pointed out in the description of the ion projection in FIM by, i.e., Newman et al. (1967).
Comparison of the Models
Distance Versus Angle Figure 2a compares the standard pseudo-stereographic projection model with the azimuthal equidistant model. For convergence at low angle, we chose k=L/ξ. This choice imposes that the models share the same ICF a low angles. The angular compression model is also shown, and it is confirmed that, while a pseudo-stereographic projection
is that the standard pseudo-stereographic model for atom probe reconstruction is equivalent to a linear compression of the launch angles so that the ions are detected with an apparent angle θ′ such that
θ=ξθ0; (9)
with ξ being the ICF. In the pseudo-stereographic projection model, however, ξ=θ/θ′ is not a constant over the detector. It can be expressed as
ξ= atan sin θ : θ m+ cos θ
While this leads to the expected ξ=m+1 for small θ and is still a reasonable assumption up to any practical angles, it is not strictly equivalent to a simple angular compression. The inverse projection gives
θ=atan ρ L
+ asin msin atan ρ L
: (11) (10)
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