Reflections on the Projection of Ions in APT 239 GENERAL RECONSTRUCTION FRAMEWORK


System of Coordinates The reconstruction protocol in APT consists in a transfor- mation of the detector coordinates (XD, YD) and detection sequence N, as well as an instantaneous radius of curvature R into the coordinates of individual atoms (x, y, z) in the tomographic reconstruction built assuming that the speci- men is a spherical cap sitting on a truncated cone (Bas et al., 1995; Gault et al., 2011):


ðXD;YD;N;RÞ,ðx; y; zÞ: (1)


A point that was introduced by Gault et al. (2011) is the use of cylindrical coordinates to describe the system. We can write the relationship between cylindrical and Cartesian coordinates (Fig. 1) as


 on the detector, and


8 <


:


x=r cosψ =Rsin θ cosψ y=r sinψ =Rsin θ sinψ z=zcðNÞ +Rcos θ


(3)


at the specimen surface. R is the radius of the specimen, θ the launch angle, ψ the


azimuthal angle, and zc the z coordinate of the center of the apex of the spherical cap, which can conventionally be set to 0 at the beginning of the experiment, but varies with the sequence of evaporation. zc is a function of the detection sequence and can be seen as the analyzed depth. In this framework, an ion strikes the detector at the


coordinate (ρ, ψ) originating from a position at the surface of the specimen characterized by (r, ψ, z)or(R, θ, ψ, zc). For the remaining of this study, we will assume that the projections of the angle is azimuthal. This implies that there exist a particular point on the detector plane (but possibly outside the physical detector) around which the azimuthal angles (e.g., ψ) are preserved. This point will be called the center of the projection or projection center. The pseudo-stereographic


XD =ρ cosψ YD =ρ sinψ


(2)


projection model, for instance, is an azimuthal projection, the center of which is at the intersection of the axis of the specimen and the detector plane. Within this assumption, we can choose this point as the origin of the (ρ, ψ) detector coordinate so that ψ is preserved [which was implied in the use of the same symbol in equations (2) and (3)].


Tomographic Reconstruction


The transformation of the detector coordinates (ρ, ψ, N) into the (θ, ψ, zc) makes then use of the radius of curvature R. ψ is unchanged through an azimuthal projection so that we are left with


1. computing θ from ρ, i.e., adopting an angular projection model;


2. computing R, i.e., adopting an radius evolution model; 3. computing zc from N, θ, and R, i.e., adopting a depth of analysis model;


4. computing (x, y, z) through equation (3). This paper is concerned with the first item, namely the


angular projection model. We will thus assume that we use a satisfactory radius evolution model as well as a robust model for the depth increment. Classicalmethods to modelevolution of thespecimen


radius are based either on the usual relationship R=V/(kfFev) (usually referred to as voltage reconstruction mode), on the change in radius constrained by a constant shank angle, or on a micrograph of the specimen profile (respectively, shank angle or tip profile reconstruction mode in the most commonly used commercial data treatment software). In any case, one or more additional parameters must be introduced: kf for voltage mode, initial radius (R0) and the half-shank angle (α)for the shank angle mode or a complete profile of the specimen derived from a high-resolution electron micrograph of its outer shape. Finally, the depth of analysis zc of the current ion N can


be deduced from that of the previous ion N−1 through the following relationship:


zc N ðÞ=zc N - 1 ðÞ+ dzc dN : (4)


dzc dN is the so-called depth increment, which was defined by Bas


et al. (1995) as the (virtual) depth of a layer corresponding to the removal of a single atom. Its value is determined in such a way that the total analyzed volume is necessarily equal to the total number of atom multiplied by the atomic volume (corrected for the limited detection efficiency η). A waya to achieve this is to attribute to each atom a volume increment corresponding to the atomic volume Ω:


dV dN = dV


so that


dzc dN = Ω


Figure 1. Schematic view of the emitter and system of coordi- nates used in this study (not to scale).


aNon unique, but always used. η dV : dzc (6) dzc


dzc dN = Ω


η ; (5)


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