An improved reconstruction procedure for the correction of local magnification effects in three-dimensional atom-probe

Abstract

A new 3DAP reconstruction procedure is proposed that accounts for the evaporation field of a secondary phase. It applies the existing cluster selection software to identify the atoms of the second phase and, subsequently, an iterative algorithm to homogenise the volume laterally. This Procedure, easily implementable on existing reconstruction software, has been applied successfully on simulated and real 3DAP analyses.

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arXiv:0907.5067v1 [cond-mat.mtrl-sci] 29 Jul 2009
An improved reconstruction procedure for the
correction of local magnification effects in 3DAP
F. De Geusera,bW. Lefebvrea,∗F. DanoixaF. Vurpillota
B. ForbordcD. Blavettea
aGroupe de Physique des Mat´ eriaux – UMR CNRS 6634, Institut des Mat´ eriaux de Rouen,
76801 Saint-Etienne-du-Rouvray Cedex – France
bPechiney CRV (Alcan Group) – BP 27, 38241 Voreppe Cedex – France
cNorwegian University of Science and Technology – 7491 Trondheim, and SINTEF
Materials Technology – 7465 Trondheim – Norway
Abstract
A new 3DAP reconstruction procedure is proposed, which accounts for the evaporation
field of a secondary phase. It applies the existing cluster selection softwares to identify
the atoms of the second phase and subsequently an iterative algorithm to homogenise the
volume laterally. This procedure, easily implementable on existing reconstruction software,
has been applied successfully on simulated and real 3DAP analyses.
Key words: Atom probe, local magnification, data analysis
∗Corresponding author. Address: GPM – UMR CNRS 6634, Institut des Mat´ eriaux de
Rouen – BP 12, 76801 Saint-Etienne-du-Rouvray Cedex – France. Tel.: +33 232 95 51 41;
Fax.: +33 232 95 50 32; E-mail: williams.lefebvre@univ-rouen.fr
Preprint submitted to IFES 200429 July 2009

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1 Introduction
One of the major problems that one has to face to with 3DAP originates from
the difference of evaporation field between phases. This commonly happens when
studying precipitation for instance. The difference of evaporation behaviour can
provoke severe atomic density variations in the 3D reconstructed volume and bi-
ased estimationsof the morphologiesof the observed objects. Although corrections
of the data in 1DAP have been proposed ([1]), 3DAP data correction are difficult.
Simulations of ion trajectories ([2]) have shown that two different effects occur:
trajectories overlaps at the interface and the so called local magnification effect.
A new reconstruction procedure for correcting this latter effect is proposed. It can
be proceeded by adjusting the reconstruction parameters (i.e. evaporation field)
whether a matrix atom or a precipitate atom is considered. Combining this correc-
tion to an algorithm that corrects the coordinates in the plane perpendicular to the
analysis is the proposed procedure to account for the difference in magnifications
between the matrix and the second phase in the reconstruction protocol.
2 Local magnification effect in 3D atom probe analysis reconstruction
The basic principles of 3DAP are provided elsewhere [3]. The volume reconstruc-
tion methods are based on the fact that the atoms are evaporated atomic layer by
atomic layer. Simple geometric considerations are often enough to resolve atomic
planes, proving the efficiency of this simple method [4]. In a first approximation, a
3D atom probe can be considered as a point projection microscope. The sample is
prepared as a tip that can be characterised by its radius of curvature R. This radius
of curvature determines the magnification η of the projection. R is related to the
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total voltageV applied to the specimen by the relation:
R =V
Fβ
(1)
where F is the evaporation field and β a geometrical factor. According to this rela-
tion, the magnification of the projection can be computed for each evaporated atom
and its initial position (x,y) at the specimen surface can be deduced from the po-
sition of the impact on the detector. The third coordinate z is incremented for each
atom by the ratio between its atomic volume Ω and the analysis surface Sa. Sais
deduced from the detector surface Sdand from the magnification η.
In order to study the effect of the reconstruction procedure on basic systems, we
have simulated the field evaporation of an A−B alloy containing pure B spherical
precipitates embedded in a pure A matrix. The precipitates are coherent with the
matrix, and with an evaporation field 1.25 times larger than that of the matrix. The
atoms are detected on a virtual detector with a given surface. For each detected
atom, the position on the detector and the initial position in the volume were eval-
uated.
The volume reconstructed by the standard method is shown on Fig. 1a while Fig.
1b represents the positions of the detected atoms in the initial structure. The recon-
structed precipitate in Fig. 1a exhibits a lens shape and it is difficult to resolve any
atomic planes (they can be seen but are very close to each other). Strong atomic
density variations appear in the precipitate and at its interface with the matrix. This
effect originates from the difference in evaporation field between the two phases.
In order to evaporate the precipitate, the tip develops locally a different radius of
curvature according to Eq. 1, and the magnification changes accordingly.
It has been proposed in the past to adjust the position of the atoms to achieve an
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equal density in the volume ([5]). The method was based on a relaxation method.
Each atom applies on its neighbours a force derived from a potential (e.g. harmonic
or Lennard-Jones type). The system is then solved numerically until equilibrium is
reached. One of the hypothesis is that the volume limits remain fixed. However, as
the atoms are moved isotropically until they are all equally spaced, the resolution
of atomic planes is necessarily lost. This leads to a kind of amorphization of the
volume. Furthermore, the fixed volume limits conditions is too strict to be applied
to large precipitates where the variation in the analysis surface is the determinant
factor. This is clearly demonstrated in Fig. 1b.
A possible way to overcome the local magnification effect is to apply the appro-
priate reconstruction parameters for the matrix and for the precipitates. A basic
requirement is to be able to select the atom of the second phase. For that purpose,
it is possible to run cluster selection type softwares ([6,7]) despite the local mag-
nification effect. An alternative option is to use the density variation for the cluster
identification using Fourier analysis [8]. Once the selection is achieved, it becomes
possible to run the reconstruction algorithm, taking the second evaporation field
into account.
Although this simple correction can be applied successfully to the coordinate along
the analysis direction (see Fig. 1c), the lateral coordinates correction is not straight-
forward because of the interdependence between the atoms coordinates ([4]). To
solve this problem, we use an iterative algorithm which homogenizes the volume
by superimposing a grid to the 3D atom map, and deforming this grid according to
the gradient of atomic density.
The nodes of the grid are only slided in the (x,y) planes, i.e. the dimension in
the z scale is not modified by this algorithm. Slice by slice, the atomic density is
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homogenised. The steps of the grid are larger than the lattice parameter, typically
between 1 and 2 nm, in order to keep the structural information at the lattice scale.
When a grid becomes too distorted (the limit is fixed in terms of angles), the new
atom coordinates are computed and another grid is superimposed to the resulting
volume. For each iteration, the mean quadratic difference between the density of
each box and the expected density is checked. The program proceeds until conver-
gence is achieved. An important advantage of this method is that no condition on
the reconstructed volume is needed.
The final reconstructed volume is shown on Fig. 1d. The atomic planes are shown
to be well resolved and properly spaced. The density is homogeneous. The shape
is very similar to that of the initial positions of the atoms in the sample (Fig. 1b),
which proves the efficiency of the proposed protocol. In particular, the variation
of the analysed surface is properly accounted for and corrected. This procedure
will now be applied to various systems where the difference in evaporation field
between matrix and precipitates is important.
3 Application to low field precipitates
Thealgorithmdescribed insection 2has been appliedon twotypical systemswhere
the evaporation field of the precipitates is lower than that of the matrix, namely the
FeCu and the AlZnMg system.
The FeCu specimen studied contains spheroidal pure copper precipitates that are
coherent with the iron matrix ([9]). When standard reconstruction procedures are
used, the lower evaporation field of the Cu precipitates leads to ellipsoidal shapes
of the precipitates, elongated along the analysis direction (Fig. 2a). Applying the
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improved procedure, the shape of the precipitates is more spheroidal, as expected
(Fig. 2b). Atomicplanes in the copper precipitates can be observed before and after
correction.
Another illustration is given from an AlZnMg alloy, which contains η′platelets
parallel to the {111} planes of the matrix and η precipitates that are more or less
spherical ([10]). Both η and η′have a lower field of evaporation with respect to the
matrix. The number density of atoms in the precipitates is 3 times larger than in the
matrix.After applicationoftheimprovedprotocol,theirmorphologiesaremodified
(Fig. 3b). The atomic density becomes homogeneous. Fig. 3c is a zoom on the η′
platelet that can be seen at the top left corner of the volume on Fig. 3b. This platelet
is shown to be still properly aligned on the {111} planes, which indicates that the
reconstruction is valid even at the atomic scale.
4 Application to high field precipitates
We have applied the method to an AlZrSc alloy. The microstructure of the alloy
consists in L12Al3(Zr,Sc) spherical particle in a FCC aluminium matrix. These
dispersoids have a much higher evaporation field than the matrix [11]. In the anal-
ysis conditions used, their size is in the same order of magnitude as the analysis
surface. Because of this, when they are intercepted, the standard reconstruction
procedure compensates the low density in the precipitate by squeezing the volume
in depth. As the area ratio between matrix and precipitate changes during the in-
terception of the precipitate, this effect is continuous. This can be seen on figure
4a where the matrix is subjected to a strong density variation in the vicinity of the
precipitate. The {220} planes of the matrix are resolved and the interplanar spac-
ing is coherent with the lattice parameter which indicates that the reconstruction
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parameter combination is correct for this phase. Next to the precipitate, however,
these atomic planes are closer and quite blurred. A former treatment ([11]) has
shown that an adjustment of the reconstruction parameters is necessary to reveal
the {110} superstructure planes in the precipitate. It was difficult, though, to re-
solve the superstructure sufficiently in the entire precipitate. This was due to the
continuous variation of the surface ratio on the detector between the two phases.
Furthermore, with this adjustment, it was not possible to image simultaneously the
atomic planes in the matrix.
The application of the proposed method shows simultaneously the {220} planes
in the matrix and the {110} planes in the particle, with a distance of 0.145 nm and
0.29 nm respectively(see Fig 4c), in excellent agreement with thelattice parameter.
The spherical morphology is also shown to be better reconstructed. No remaining
atomic density variation can be observed.
5 Discussions
The proposed reconstruction procedure obviously improves the 3D reconstruction
of precipitates embedded in a matrix. With this method, the particles morphology
can be resolved without any preliminary assumption. The local structural informa-
tion contained in the analysis, such as atomic planes resolution for instance, is not
affected by the improvedprocedure. On thecontrary, it can resolvedetails that were
invisible on the standard reconstruction volumes.
Furthermore, it takes into account the anisotropy of the local magnification effect
considered along or perpendicularly to the analysis direction. In the standard re-
construction method, this anisotropy will lead to biases in the results from data
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analysis, such as concentration profiles or proxigrams ([12]), that consider all the
precipitates in the same concentration profile. Indeed, two precipitates of the same
dimensions but intercepted differently by the surface of analysis are subject to dis-
tinct effects because of their different interception area.
But even on independently treated precipitates, the bias on the geometry is not
negligible. We have plotted an erosion profile of the first particle. This profile (Fig.
5) corresponds to a binning of the atoms of the precipitates as a function of their
distancetothematrix.Theconcentrationisplottedasafunctionofthedistance.The
bins step is 0.15 nm. The error bars are the statistical errors on the concentration
due to sampling (related to the number of atoms in the bin). Fig. 5a is the profile
with the ”standard” volume and Fig. 5b with the ”corrected” volume. The expected
Zr–poor core [11,13] of the particle can be clearly indicated on Fig. 5b, whereas on
Fig. 5a the bins overlap and hence average it out. In addition, the Zr concentration
is continuously decreasing on Fig. 5a, instead of forming a plateau, as in Fig. 5b.
6 Conclusion
A new reconstruction protocol has been proposed. It uses existing cluster selec-
tion softwares on a primary reconstructed volume to attribute another evaporation
field to the second phase. After correction of the analysed depth, the atomic den-
sity in the (x,y) section is homogenised by an iterative algorithm. The resulting
volumes are shown to be closer to reality than the standard reconstruction. This is
confirmed by the performed simulation. Primary unresolved details can be seen on
reconstructed volume, such as atomic planes both in precipitates and matrix. Con-
centration gradients in particles are shown to be more accurate. This method can be
easily implemented on an existing reconstruction algorithm in order to be routinely
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used.
References
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[5] F. Vurpillot, PhD thesis, Universit´ e de Rouen, 2001.
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(2004).
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[10] A. Deschamps et al., Phil. Mag. 81, 2391 (2001).
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List of Figures
1 Simulation of a 3DAP analysis on an A-B alloy with a spherical
precipitate. The relative evaporation field of the second phase is
1.25. a. Standard reconstruction method. b. Initial positions of the
atoms c. Correction in depth d. Correction in depth and in the (x,y)
plane.11
2 3D reconstruction of a copper particle in a FeCu alloy: a. Standard
protocol b. Correction in depth and in the (x,y) plane. The
elongation of the particle disappear but the coherent planes remain.12
3 3D reconstruction of a AlZnMg alloy containing Zn- and Mg-rich
particles : a. Standard protocol b. Correction in depth and in the
(x,y) plane c. η′platelet parallel to the Al {111} planes.13
4 3D reconstruction of a Al3(Zr,Sc) particle following : a.
Standard protocol b. Correction in depth and in the (x,y) plane
c. {110} planes of the particle and {220} planes of the matrix
simultaneously resolved.14
5Erosion concentration profile for the zirconium on the Al3(Zr,Sc)
particle from Fig. 4. a. Uncorrected volume b. Corrected volume,
revealing a Zr-poor core. 15
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Fig. 1. Simulation of a 3DAP analysis on an A-B alloy with a spherical precipitate. The
relative evaporation field of the second phase is 1.25. a. Standard reconstruction method. b.
Initial positions of the atoms c. Correction in depth d. Correction in depth and in the (x,y)
plane.
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Fig. 2. 3D reconstruction of a copper particle in a FeCu alloy: a. Standard protocol b.
Correction in depth and in the (x,y) plane. The elongation of the particle disappear but the
coherent planes remain.
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Fig. 3. 3D reconstruction of a AlZnMg alloy containing Zn- and Mg-rich particles : a.
Standard protocol b. Correction in depth and in the (x,y) plane c. η′platelet parallel to the
Al {111} planes.
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Fig. 4. 3D reconstruction of a Al3(Zr,Sc) particle following : a. Standard protocol b. Cor-
rection in depth and in the (x,y) plane c. {110} planes of the particle and {220} planes of
the matrix simultaneously resolved.
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Fig. 5. Erosion concentration profile for the zirconium on the Al3(Zr,Sc) particle from Fig.
4. a. Uncorrected volume b. Corrected volume, revealing a Zr-poor core.
15