Ultramicroscopy 132 (2013) 92–99

Contents lists available at ScienceDirect

Ultramicroscopy

0304-39

http://d

n Corr

Enginee

Tel.: þ6

E-m

journal homepage: www.elsevier.com/locate/ultramic

Spatial decomposition of molecular ions within 3D atomprobe reconstructions

Andrew Breen a,b, Michael P. Moody c, Baptiste Gault d, Anna V. Ceguerra a,b, Kelvin Y. Xie e,Sichao Du b,f, Simon P. Ringer a,b,n

a School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australiab Australian Centre for Microscopy and Microanalysis, Madsen Building F09, The University of Sydney, NSW 2006, Australiac Department of Materials, University of Oxford, Parks Road, OX13PH, Oxford, UKd Department of Materials Science and Engineering, McMaster University, 1280 Main Street West, Hamilton, Ont. L8S4L8, Canadae Johns Hopkins University, Department of Mechanical Engineering, Baltimore, MD 21218, USAf School of Physics, The University of Sydney, NSW 2006, Australia

a r t i c l e i n f o

Available online 7 March 2013

Keywords:

Atom probe tomography

Complex molecular ions

Delaunay tessellation

Nearest neighbour analysis

91/$ - see front matter & 2013 Elsevier B.V. A

x.doi.org/10.1016/j.ultramic.2013.02.014

esponding author at: School of Aerospace, M

ring, The University of Sydney, NSW 2006, A

1 2 9351 2351; fax: þ61 2 9351 7682.

ail address: simon.ringer@sydney.edu.au (S.P.

a b s t r a c t

Two methods for separating the constituent atoms of molecular ions within atom probe tomography

reconstructions are presented. The Gaussian Separation Method efficiently deconvolutes molecular ions

containing two constituent atoms and is tested on simulated data before being applied to an

experimental HSLA steel dataset containing NbN. The Delaunay Separation Method extends separation

to larger complex ions and is also tested on simulated data before being applied to an experimental

GaAs dataset containing many large (43 atoms) complex ions. First nearest neighbour (1NN)

distributions and images of the reconstruction before and after the separations are used to show the

effect of the algorithms and their validity and practicality are also discussed.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

In the course of an atom probe tomography experiment, atoms areideally individually ionised and field evaporated from the specimen.However, very often this is not the case. Molecular ions (also knownas complex ions or cluster ions) are routinely observed in atom probemass spectra [1] and are usually represented as a single point in theresulting reconstruction rather than each constituent atom beingindividually located — a clearly unphysical model which has so farlargely been conveniently ignored.

This poses a problem, not only for the sake of visual clarity inthe resulting reconstruction, an obvious yet important issue, butalso for many statistical analysis tools at the heart of atom probeanalysis. For some of the current statistical analysis tools avail-able, atoms contained within complex ions are essentially invi-sible to the same species of atom that were evaporated as singularions and this will skew the results of the analysis — the severityof which is dependent on the size and number of complex ionsdetected. A partial solution to this problem is decomposing themolecular ions into their constituent atoms at the same point andindeed this is already being done. The commercially available

ll rights reserved.

echanical and Mechatronic

ustralia.

Ringer).

IVASs software allows ‘atom-based’ rather than ‘ion-based’ ana-lysis to be chosen for many of its analysis tools. However,positioning multiple atoms at a single point can lead to asignificant bias in statistical analysis of the results.

Of particular concern is the nearest neighbour (NN) analysiswhich histograms the distance between neighbouring atoms ofchosen atomic species [2–4]. Even if an atom-based approach isused, some atoms will have a 1NN separation of 0 nm. While anNN distribution may not be particularly useful in itself, other thanto indicate to segregation or anti-segregation of a particularatomic species, it is a fundamental component of many otherhigher order analysis techniques. Clustering algorithms [5,6], forinstance, commonly use NN distributions for parameter selection,when applied to isolate and characterise nanoscale segregationphenomena [3,7,8]. The results of such an analysis are thereforeinextricably linked to the quality of the NN analysis. The visua-lisation, cluster morphology and cluster size distribution are someaspects of this analysis that could be misleading if currentmolecular ion representations are used. NN distributions havealso been used for phase composition measurements [4,9]. Onesuch method derives a protocol based on the first nearestneighbour (1NN) distribution of a particular atomic species, todeconvolute the contributions from the different phases and thusdetermine the phase composition, but these measurements willbe inaccurate if the solute species is contained withinmolecular ions.

Fig. 1. Schematic demonstrating the splitting of complex molecular ions within an

atom probe reconstruction: (1) before separation when molecular ions are

represented at a single point and (2) after constituent atoms have been separated.

A. Breen et al. / Ultramicroscopy 132 (2013) 92–99 93

Radial distribution functions (RDFs) also suffer similar draw-backs to an NN analysis when the reconstruction being analysedcontains complex ions. RDFs histogram the radial distancebetween chosen atomic species and can be used for observingcrystallographic information as well as segregation phenomena[10,11]. However, it follows that if some atomic species areoccupied within molecular ions, any measured RDF distributionconcerning these atomic species will be skewed.

Grid based analysis techniques that divide the reconstructioninto voxels represent yet another group of data mining toolsparticularly susceptible to inaccuracies due to conventionalmolecular ion representation. One important example is thefrequency distribution analysis algorithms [12–14] that dividethe reconstruction up into a grid of blocks that all contain thesame number of ions. Histograms of the number of ions of aparticular species contained within each block are then created.Current available data analysis software generally do not offer anatom-based frequency distribution analysis and consequentlymolecular ions are counted as a single contribution to the sizeof the block. This not only causes the number of actual atoms ineach block to vary but also renders atomic species tied up inmolecular ions invisible to those that are not. Previously, Gaultet al. used their own atom-based technique whereby the mole-cules within a reconstruction of a thermoelectric material weresimply split into single atoms occupying the same spatial coordi-nates, facilitating species-specific frequency distribution analysesof Zn and Sb atoms [15]. This is acceptable except in cases wherethere are large molecular ions and fine voxel sizes such that thereis increased chance of multiple atoms contributing to one voxelwhen they should be spread over neighbouring voxels. Other gridbased analysis tools such as proxigrams [16], contingency tableanalysis [17] as well as density and concentration profiles willhave similar issues.

There are still many other analytical tools that stand to benefitfrom representing constituent atoms within molecular ions indi-vidually — lattice rectification techniques [10,18], that use thecrystallographic information contained within an atom probereconstruction to restore the original crystal structure of thesample is a notable example. Molecular ions inevitably compli-cate this process, even if they are decomposed into their con-stituent atoms at the same point, but by separating theconstituent atoms into their likely locations, the rectificationprocedure is facilitated particularly in cases where datasetscontain large molecular ions.

As atom probe tomography (APT) is applied to an increasingarray of sophisticated multicomponent materials, complex iondetection is becoming more common and hence, potentially moreof an issue. A prominent example is the analysis of steels, wherethe detection of complex ions such as C2, NbN and FeN in variouscharge states are commonly observed [19]. Semiconductors, inparticular III–V such as GaSb, are another material group that facesimilar challenges where Sb2, Sb3 and even larger molecular ionsin various charge states are routinely detected [20].

It is therefore apparent that the representation of complexmolecular ions within atom probe reconstructions should beavoided. Minimising the detection of complex ions in the experi-ment in the first place may seem like a reasonable approach,however, it is often inconvenient to change the experimentalparameters to minimise complex ion detection, as other aspectsrelated to the quality of data will be adversely affected [21]. Still,research has been undertaken studying the physics governing thefield evaporation of complex molecular ions [1,20,22,23], infor-mation that could be used to control the severity of theirdetection. The experimental conditions have been found to playa critical role in molecular ion evaporation — higher laser pulseenergies for instance have often been attributed to higher

proportions of molecular ion detection, as well as more massivemolecular ions themselves. It should also be noted that the fieldevaporation behaviour of metals is significantly different to thatof semiconductors and other material groups. Much remainsunknown, however, about the formation of these molecular ionson an atom probe tip.

A more practical approach may be to spatially separatecomplex ions within the reconstruction so that each atom hasits own unique position within the reconstruction. Fig. 1 illus-trates this concept. By considering the crystallography andevaporation behaviour of the specimen as well as observedatomic distributions within the reconstruction itself, the spatialseparation of complex ions within the reconstruction can besimulated accurately. At present it is impossible to preciselylocate each individual atom from the molecular ions but theapproach suggested could restore expected distributions onaverage and hence help to get better results from the analysistechniques aforementioned.

For small complex molecular ions of 2 atoms, a method toseparate the constituent atoms based on the measured 1NNdistribution of singular ions occupying the same phase is pro-posed. The separation distance and orientation of the molecularpair are selected so that they follow a Gaussian curve fitting the1NN distribution. For larger complex ions, this approach becomesanalytically cumbersome and a technique that considers theavailable local volume surrounding the complex ion is moreappropriate. In this case the use of Delaunay tessellations isproposed. Delaunay tessellations have many practical applica-tions including pattern recognition techniques and finite elementmethods, and their implementation to a range of analyticalstudies related to APT is starting to be realised [24–26]. ADelaunay tessellation is formed by joining all pairs of pointsbelonging to the set P¼{p1, p2, y, pn} where nZ4 such that nopoint in P is inside the circumsphere of any tetrahedra of thetessellation [27]. By measuring the volume of the tetrahedrasurrounding the detected co-ordinates of a molecular ion, posi-tions for the spatial redistribution of the constituent atoms can bedetermined that minimise any localised density fluctuations.

Here, these algorithms are first applied to simulated recon-structions for proof of concept before being demonstrated onexperimental data — first to a low-alloyed steel containing NbNcomplex ions and then to a GaAs nanowire containing very largemolecular ions. Focus is placed on the effect of this reconstructionartefact to the 1NN distribution because these analyses representa simple and directly comparable way of testing the performanceof the algorithms. Further, 1NN distributions are an importantcomponent of other analysis techniques and improvement in theaccuracy of these distributions will have flow-on benefits to theclustering algorithms, phase compositional measurements and

A. Breen et al. / Ultramicroscopy 132 (2013) 92–9994

radial distribution functions. Care should be taken in the inter-pretation of the results and spatial separation only performedwhen molecular ions are likely to bias the results from theanalysis tools aforementioned.

2. Theory

2.1. The Gaussian separation method (GSM)

The Gaussian separation method has been developed for theseparation of molecular ions that consist of two constituent atoms(dimers). This approach is based on the assumption that theconstituent atoms in the molecular ions were originally neigh-bouring atoms on the surface of the atom probe tip at the time ofevaporation. It follows that the Dz component of the separationsshould be small while the overall separation distances should besimilar to that of the atoms within the same phase on the surfaceof the atom probe tip.

The separation distance distribution should match the 1NNdistribution of the surface plane of atoms. For this to be accurate,the detector efficiency and a Dz limit should be taken intoconsideration. One approach for crystalline materials (and theone that has been used in this study) is to make a simulatedreconstruction of the lattice in the same crystallographic orienta-tion of the experimental reconstruction with 100% detectorefficiency and apply atom probe like noise — then create a 1NNplot where Dzo0.05 nm of all the atoms. Information on deter-mining the crystallographic orientation of APT reconstructions isdescribed elsewhere [10,28]. The Dz cut-off value was selectedbecause it is usually below the plane spacing commonly observedin atom probe data [29,30] and therefore creates a distributionmore closely resembling that for the surface plane of atoms.

The resulting curve can be closely fitted to a Gaussiandistribution:

f ðxÞ ¼ a exp �ðx�bÞ2

2c2

!ð1Þ

where a is the amplitude, b is the mean and c is the standarddeviation.

The atoms composing the complex ions are then separated — aschematic of the process is presented in Fig. 2. For each dimer

Fig. 2. The separation of a dimer complex ion using the GSM: the dark blue atoms

represent the constituent atoms of the complex ion where as the light blue atoms

are surrounding atoms in the reconstruction. The dsep is the separation distance

between the constituent atoms; the midpoint is kept as the original detected co-

ordinates of the complex ion. d1NN_1 and d1NN_2 are the distances to the nearest

neighbouring atoms after the separation and are used to determine whether the

orientation of the separation will be accepted. (For interpretation of the references

to color in this figure legend, the reader is referred to the web version of this

article.)

complex ion in the reconstruction, the separation distance (dsep)is chosen at random from a set of distances matching the fittedGaussian probability distribution. The original co-ordinates of thecomplex ion are kept as the midpoint/centre of mass between thetwo constituent atoms and the orientation is also randomisedsuch that dsep ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDx2þDy2

p. The move is accepted only if the

distances from the separated atoms to the nearest neighbouringatoms within the reconstruction are within 20% from a valueselected at random from the Gaussian distribution — otherwisethe orientation selection is repeated until this criteria is met. Thisresults in the constituent atoms having a separation distributionsimilar to the theoretical 1NN distribution of atoms on the surfaceplane within the phase of interest.

2.2. The Delaunay separation method (DSM)

The Delaunay separation method separates complex ions withtwo or more constituent atoms and uses the Delaunay tessellationof detected atom co-ordinates around a complex ion to determinethe local volume available to distribute the atoms containedwithin the complex ion. It is based on the assumption that theconstituent atoms were in close proximity to each other withinthe sample and that regions of larger empty volume surroundingthe molecular ion are more likely to be where the constituentatoms were originally located.

The algorithm identifies the simplexes (tetrahedra formed byDelaunay tessellation) that share a vertex with the detected co-ordinates of a complex ion, forming a convex hull within whichthe constituent atoms are distributed into the surroundingvolume. The first constituent atom is placed at the original co-ordinates of the complex ion. The volume of each simplex is thencalculated and the remaining constituent atoms are placed at thecentroids of the largest tetrahedra. Fig. 3 is a schematic showingthis process and clearly demonstrates how the constituent atomsare distributed into the available volume.

3. Simulated datasets

3.1. Method

Simulated datasets of a face-centred cubic (FCC) binary alloywere used to test the performance of the GSM and DSM algo-rithms. The original simulated dataset contained 3 million atomswith composition Solvent (Sv) �1.0 at% Solute (Su), lattice para-meter a0¼0.404 nm and Warren-Crowley parameter (which

Fig. 3. The separation of a large molecular complex ion using the DSM. Each

vertex of the tetraheda represents the co-ordinates of an ion within the

reconstruction — a molecular ion has been detected at the centre of the sub-

volume. One constituent atom is placed at the original reconstructed co-ordinates

of the molecular ion and the rest are positioned at the centroids of the largest

tetrahedra (highlighted in red). (For interpretation of the references to color in this

figure legend, the reader is referred to the web version of this article.)

Fig. 5. Showing the effect of the GSM on the measured Su–Su 1NN distribution:

(black) correct distribution — no complex ions present, (red) when Su2 ions are

simulated but only Su ions are used in the calculated distribution, (light blue)

When Su and Su2 ions are used in the calculated distribution and (dark blue) after

Su2 ions have been separated using the GSM. (For interpretation of the references

to color in this figure legend, the reader is referred to the web version of this

article.)

Fig. 6. Showing the effect of the DSM on the measured Su–Su 1NN distribution:

(black) correct distribution — no complex ions present, (red) when Su4 ions are

simulated but only Su ions are used in the calculated distribution, (light blue) Su4

and Su ions are used in the calculated distribution and (dark blue) after Su4 ions

have been separated using the DSM. (For interpretation of the references to color

in this figure legend, the reader is referred to the web version of this article.)

A. Breen et al. / Ultramicroscopy 132 (2013) 92–99 95

controls the clustering/segregation of Su) a¼0.01 [31]. 1% ofrandom neighbouring Sv atoms on the (001) planes (i.e. perpen-dicular to the z-axis within the simulation) were converted to Suto simulate the true position of Su atoms contained within Su2

dimer complex ions. The resulting Su distribution is somewhatsimilar to that in the early stages of precipitate formation. 43% ofall ions were then removed to simulate detector efficiency — ifone of the Su ions within the Su–Su pairs was selected then theother Su ion was also removed. A complementary dataset wasthen created where approximately half of the converted Su–Supairs remaining were replaced by Su2 complex ions at the centreof mass. A dithering function was then applied to the atom co-ordinates, with standard deviation sx¼sy¼sz¼0.05 nm, toresemble APT like spatial noise. The GSM was then used toseparate the simulated Su2 complex ions before comparing theSu distribution between the datasets.

To simulate the true atomic distribution of larger complex ionscontaining 4 atoms, 0.25% of Sv atoms were selected at randomfrom the original simulation along with 3 of their first nearestneighbours and were converted to Su. 43% of all ions were againremoved to simulate detector efficiency — if one of the Su ionswithin a cluster of 4 Su atoms was selected then the other Su ionswithin the cluster were also removed. Another dataset was alsocreated where the clusters of 4 Su atoms remaining wereremoved and replaced with Su4 complex ions. As before, adithering function was then applied to simulate atom probenoise. The DSM was then used to separate Su4 and the Sudistribution was compared between the datasets.

3.2. Results

The GSM was shown to separate dimer complex ions into theirconstituent atoms very close to the expected spatial distribution.Fig. 4 shows fitting a Gaussian curve to the 1NN of all atoms(Dzo0.05 nm) within the original simulation after APT like noisehas been added. The resulting probability distribution was thenused to model the separation distances and orientation of theconstituent atoms using the GSM. Although there is some differ-ence between the curves (R-square¼0.9742), the approximate fitis deemed appropriate for the algorithm to achieve sufficientlyaccurate results.

The 1NN distribution is markedly more accurate after separat-ing the simulated Su2 molecular ions with the GSM algorithm.Fig. 5 shows a range of 1NN distributions involving Su atoms

Fig. 4. A Gaussian curve (red) is fitted to the 1NN distribution of all atoms using a

depth threshold of Dzo0.05 nm (blue) within a simulated dataset. The mean

(b¼0.26 nm) and standard deviation (c¼0.072 nm) of the Gaussian curve are used

as inputs to the separation algorithm. (For interpretation of the references to color

in this figure legend, the reader is referred to the web version of this article.)

before and after the GSM is used. The Su–Su 1NN before Su2 isseparated (red) is significantly different to the correct distribution(black). Even if all Su containing ions are considered (light blue),essentially the same as an ‘‘atom-based’’ approach, the correct1NN distribution is still not restored. However, after using theGSM, the resulting 1NN Su–Su distribution (dark blue) is muchcloser to the correct distribution.

The DSM is also shown to improve the measured 1NN Su–Sudistribution in the presence of larger molecular ions. Fig. 6provides a similar range of 1NN distributions to the previousfigure to show the effect of the DSM algorithm and the results areessentially the same — after using the DSM the resulting 1NN Su–Su distribution (dark blue) is much closer to the correct distribu-tion (black).

4. Experimental datasets

4.1. Method

The GSM was then applied to experimental data to separateNbN complex ions in a high strength low alloy (HSLA) steelproduced using the CASTRIPs process [32]. The dataset was

A. Breen et al. / Ultramicroscopy 132 (2013) 92–9996

collected on a Cameca LEAP 3000 under voltage pulsing at 25%pulse fraction and 90 mm flight path at 25 K. A simulated BCCferrite reconstruction with atom probe like noise orientated in asimilar crystallographic direction to that observed in the experi-mental reconstruction was then used to produce the expecteddistribution of neighbouring atoms on the surface of the tip andwas fitted with a Gaussian curve that was used when separatingthe Nb and N atoms. The mean and standard deviation of thisdistribution were found to be 0.204 nm and 0.127 nmrespectively.

Similarly, the DSM was applied to an experimental datasetcollected from a GaAs nanowire containing many molecular ionssuch as As2, As3, As4, As5 and GaAs4. This dataset was collected ona Cameca LEAP 3000 Si under 0.05 nJ laser pulsing, 90 mm flightpath at 25 K. Each dataset was reconstructed using the commer-cially available IVASs software. Significant trajectory aberrationswere observed on the edges of the nanowire and were thereforecropped from the analysis region.

Fig. 8. The effect of the GSM on the measured Nb–Nb and N–N 1NN distributions

in the HSLA steel sample: Nb–Nb 1NN distributions before and after using the

GSM (purple), NbN–NbN 1NN before using the GSM (blue) and N–N 1NN after

using the GSM (green). (For interpretation of the references to color in this figure

legend, the reader is referred to the web version of this article.)

4.2. Results

A significant difference in the atomic distributions of the Nband N atoms contained within the HSLA steel dataset wasobserved after applying the GSM. The visual aspects of thisseparation are shown in Fig. 7. Larger-scale spatial features ofthe reconstruction have been conserved such as the noticeablesegregation of the Nb and N atoms before and after using the GSMalgorithm (Fig. 7 a, b). A close up of individual NbN positions(Fig. 7 c, d), clearly reveals how the individual atoms have beenseparated with varying dsep and orientation along the x-y planerelative to the detector, facilitating statistical analysis of Nb and Natomic distributions.

Fig. 8 reveals how separating the complex ions affect the 1NNdistribution of N and Nb atoms. The Nb–Nb 1NN distributionbefore the complex ions are split is significantly affected by alarge fraction of Nb atoms within NbN complex ions — this wasmeasured to be 23%. After separating the complex ions, theresulting distribution is a much more realistic representation ofthe true Nb–Nb 1NN distribution. No N was ranged in the originaldata since the Nþ peaks overlapped with Si2þ peaks in the massspectrum but, after separating the complex ions, individual Natoms can be represented in the reconstruction. Their 1NNdistribution after the split is almost identical to the original1NN distribution of the NbN complex ions.

Fig. 7. Separating NbN complex ions within an experimental HSLA steel reconstruction:

and N atoms have been separated using the GSM — segregation and larger-scale featur

before separation and (d) after separation shows how the GSM separates individual N

After applying the DSM to the GaAs nanowire, As representa-tion within the reconstruction became much clearer. Fig. 9 showsthe reconstruction before (a) and after (b) the DSM algorithm wasapplied. In particular, an overwhelming majority of the As atoms(99%) are contained within complex molecular ions. After separa-tion, the complex ion species are replaced by a noticeably denserdistribution of As atoms.

This facilitates further statistical analysis of atomic distribu-tions and architecture — particularly anything concerning As.Fig. 10 shows the 1NN distribution for any ions containing Asbefore and after applying the DSM model. The frequency has beennormalised so that the volume under each curve is equal to 1, thisenables a better comparison between the distributions since sucha large fraction of the As atoms were contained within molecularions. This essentially transforms the curves into probabilitydensity functions. It is clear the true As–As 1NN distribution isseverely affected by the large complex ions — before the separa-tion, the mean 1NN As distance between As containing ions was0.398 nm with a standard deviation of 0.158 nm and after theseparation the mean was 0.136 nm with a standard deviation of0.048 nm.

The theoretical As–As 1NN distribution for the reconstructionis also given. This distribution was based on equations given byPhilippe et al. [4]. The probability density function of finding a1NN distance of r for any pair of atoms in a random solid solution

(a) original reconstruction showing the Nb and NbN complex ions and (b) after Nb

es are still conserved after the separation. (c) Small sub-volume of reconstruction

bN complex ions.

Fig. 9. Separating complex ions from a GaAs nanowire reconstruction: (a) sub-volume of reconstruction before complex ions were split showing Ga and As containing ions.

Approximately 292,000 As atoms are contained within only 80,000 reconstructed ions and (b) reconstruction after As containing complex ions were split.

Fig. 10. The As–As 1NN distribution before (blue) and after (red) the complex ions

in the GaAs reconstruction were separated. The theoretically predicted distribu-

tion is also shown (dotted black). (For interpretation of the references to color in

this figure legend, the reader is referred to the web version of this article.)

A. Breen et al. / Ultramicroscopy 132 (2013) 92–99 97

is given by

P rð Þ ¼ 4pr2QC0exp �4

3pQC0r3

� �ð2Þ

where Q is the detector efficiency and C0 is the real concentrationof the atoms. The maximum of P(r) can be found by makingdP=dr¼ 0:

r0 ¼1

2pQC0

� �1=3

ð3Þ

Therefore the measured concentration of As within the recon-struction was substituted for QC0 in Eq. (2). The dotted line inFig. 10 shows PðrAsÞ vs. rAs and is very similar to the resultingAs–As 1NN distribution suggesting that the DSM provides a validrepresentation of As within the reconstruction.

5. Discussion

5.1. The validity of the seperation algorithms

It is important to note that the separation algorithms devel-oped simulate the likely distribution of constituent atoms withinthe complex molecular atoms routinely seen in many materialsbut they do not restore the actual relative coordinates of theconstituent atoms. Such a representation however, improvesvisibility as well as facilitates more accurate statistical analysis

concerning the spatial distribution of atoms contained withinmolecular ions including atom-based frequency distribution,nearest neighbour analysis and more advanced analysis toolssuch as clustering algorithms. However, it is up to the researcherto gauge whether or not the separation is necessary for theanalysis they are performing.

The GSM makes the assumption that the species containedwithin a dimer complex ion are neighbouring atoms on thesurface of the atom probe tip at evaporation so that DzijE0,where i and j are the pair of atoms contained within the complexion, and that the separation distance follows the 1NN distributionof atoms on the surface of the tip. While much remains unknownabout the formation of complex molecular ions before fieldevaporation, one would assume that evaporation would usuallyoccur from the terrace edges, similar to that of singular ions andthat it would be more energetically favourable for molecular ionsto form with neighbouring atoms on the same terrace rather thanbetween terraces.

The DSM similarly assumes that constituent atoms wereoriginally in close proximity to each other but relaxes therestriction that they were all neighbouring atoms on the samelattice terrace at evaporation. This was primarily used because itbecomes difficult to distribute a large number of constituentatoms within a small area of the same x-y plane without causinglocalised density fluctuations. It seems more reasonable to dis-tribute the constituent atoms within the available volume of aconvex hull around the detected co–ordinates of the complex ion.

It seems logical to assume that a large molecular ion wouldhave a larger empty volume surrounding it than singular ions andwould also make the approach of the DSM more valid. This wasinvestigated for the GaAs reconstruction by comparing the aver-age volume contained within the convex hull surrounding asingular ion with no molecular ion neighbours to the convex hullsurrounding a molecular ion. Interestingly, it was found that theaverage volume surrounding the molecular ions was 0.0425 nm3

where as the average volume surrounding singular ions was0.0398 nm3, which confirms the initial prediction. This findingalso suggests that the molecular ions detected within the regionof interest did not succumb to significantly worse trajectoryaberrations than singular ions.

It should be noted, however, that there are many other factorsthat influence the relative spatial distributions of atomic specieswithin atom probe data and that these affects should be takeninto consideration with any subsequent statistical analysis. Thelocal magnification affect [33] and surface migration [34] are justtwo examples. Local magnification of the NbN molecular ions is

A. Breen et al. / Ultramicroscopy 132 (2013) 92–9998

known to have occurred in the HSLA steel sample used in thisstudy. Some surface migration of N may also have occurred butthis is considered negligible due to the N being strongly heldwithin Nb, C and N rich clusters [35]. The GaAs data was alsoaffected by these evaporation phenomena. The original recon-struction showed significant segregation of As containing ions tothe side walls of the hexagonal GaAs nanowire and is a strongindication that surface migration occurred. The shape of the tipwas also skewed around the edges suggesting either trajectoryaberrations or the limitations of the reconstruction algorithm thatassumes a tip in the shape of a hemispherical cap on a truncatedcone — not a hexagonal prism. It was for these reasons that onlythe central region of the reconstruction was cropped for analysis— the observation of lattice planes within this region suggeststhat at least a moderate level of spatial integrity was maintainedhere. The algorithms developed are only appropriate for complexion detection and if the accuracy of various statistical analyseswas of concern due to other effects, further data treatment maybe necessary.

Still, the algorithms developed have been shown to improvethe spatial distribution of atomic species contained within com-plex ions. This was clearly shown in Figs. 5 and 6 when thealgorithms separated complex ions within simulated data andimproved the resulting 1NN distribution of solute species. Itshould be noted however that both algorithms seem to slightlyunder-estimate the true separation distances i.e. the measuredSu–Su 1NN distribution after separation is slightly shifted to theleft for low 1NN values. This suggests that there is slightly morerandomness or noise added to the positioning of constituentatoms relative to their actual positions after separation — themean 1NN distribution tends to decrease as more randomness isadded to the system which is suggested from Eq. 3 where r0 isalways less than the theoretical 1NN distribution for a perfectcrystal. However, some extra randomness is expected since theexact positioning of the constituent atoms cannot be known. The1NN distribution does show a marked improvement however ascompared to that when the molecular ions are not spatiallyseparated which suggests a valid distribution of the atomscontained within the molecular ions has been achieved. This hassignificant implications — results from statistical tools employingthe NN analyses and similar type algorithms can provide valuableinformation on the structure–property relationships of materialsand can potentially be significantly improved for any materialwhere complex ion detection poses a limitation such as the HSLAsteel or GaAs nanowire used to demonstrate the algorithms.

5.2. Computational performance

The DSM and GSM can be performed within a practical amountof time for millions of atoms on most personal computers. Bothalgorithms voxelise the dataset into 1 nm3 cubes so that anymeasurements only occur between neighbouring voxels, not overthe entire dataset, significantly reducing computation time. TheGSM was coded in C and was able to separate 1426 Su2 complexions in the simulated dataset containing 2, 872,034 detected ionsin �14 s. The DSM was coded in Matlabs and took �23 mins 60 sto separate 1426 Su4 atoms from 2867, 756 total detected ions.The processing time increases almost linearly with number ofcomplex ions detected. The increased computer time for the DSMis largely due to the Delaunay tessellation of each voxel of atoms.All processing was done using a single 2.3 GHz Intel Core i7processor with 8 GB memory. Further improvements to perfor-mance could be achieved through parallel computing, GPUprocessing and compiling the DSM in C/Cþþ code.

While the DSM is necessary for complex ions containing morethan 2 atoms, the GSM is much faster. A programme that uses

both algorithms, depending on the size of the complex ion couldbe implemented in the future.

6. Conclusion

In summary,

�

Molecular ions are typically represented as single pointswithin atom probe reconstructions rather than each constitu-ent atom being represented individually. This has negativeimplications to the visualisation of the reconstruction as wellas many statistical analysis tools concerning the spatial dis-tribution of atoms contained within molecular ions. � Two techniques were developed in the study to separate

complex molecular ions existing within atom probe tomogra-phy data:

1. The GSM was developed to split complex ions containingtwo constituent atoms.

2. The more computationally intensive DSM was developed tosplit larger complex ions.

�

A series of simulated datasets were used to test the perfor-mance of the algorithms. 1NN plots were used to show thatcurrent molecular ion representation skews the observeddistribution of atoms contained within the molecular ionsand that the separation algorithms can be used to significantlyimprove this. � The GSM was then used on a HSLA steel dataset containing

NbN complex ions and facilitated Nb–Nb and N–N 1NNdistribution measurements significantly.

� The DSM was applied to a GaAs dataset containing many large

complex ions, significantly improving the visualisation of thereconstruction and facilitating more realistic measurementson atomic distributions — particularly the As–As 1NNdistribution.

� The algorithms can be implemented on most personal com-

puters within a practical amount of time.

Acknowledgements

The authors would like to thank Peter Felfer and LeighStephenson for the Australian Centre for Microscopy and Micro-analysis (ACMM) for fruitful discussions. The authors acknowl-edge the funding support from the Australian Research Council,scientific and technical support from the Australian Microscopyand Microanalysis Research Facility (AMMRF) at The University ofSydney.

References

[1] T.T. Tsong, Physical Review B 30 (1984) 4946–4961.[2] A. Shariq, T. Al-Kassab, R. Kirchheim, R.B. Schwarz, Ultramicroscopy 107

(2007) 773–780.[3] L.T. Stephenson, M.P. Moody, P.V. Liddicoat, S.P. Ringer, Microscopy and

Microanalysis 13 (2007) 448–463.[4] T. Philippe, F. De Geuser, S. Duguay, W. Lefebvre, O. Cojocaru-Miredin, G. Da

Costa, D. Blavette, Ultramicroscopy 109 (2009) 1304–1309.[5] E.A. Marquis, J.M. Hyde, Materials Science and Engineering: R: Reports 69

(2010) 37–62.[6] D. Vaumousse, A. Cerezo, P.J. Warren, Ultramicroscopy 95 (2003) 215–221.[7] T. Philippe, O. Cojocaru-Miredin, S. Duguay, D. Blavette, Journal of Microscopy

239 (2010) 72–77.[8] R.K.W. Marceau, L.T. Stephenson, C.R. Hutchinson, S.P. Ringer, Ultramicro-

scopy 111 (2011) 738–742.[9] F. De Geuser, W. Lefebvre, Microscopy Research and Technique 74 (2011)

257–263.

A. Breen et al. / Ultramicroscopy 132 (2013) 92–99 99

[10] M.P. Moody, B. Gault, L.T. Stephenson, R.K.W. Marceau, R.C. Powles,A.V. Ceguerra, A.J. Breen, S.P. Ringer, Microscopy and Microanalysis 17(2011) 226–239.

[11] D. Haley, T. Petersen, G. Barton, S.P. Ringer, Philosophical Magazine 89 (2009)925–943.

[12] M.G. Hetherington, M.K. Miller, Journal De Physique 48 (1987) 559–564.[13] M.P. Moody, L.T. Stephenson, A.V. Ceguerra, S.P. Ringer, Microscopy Research

and Technique 71 (2008) 542–550.[14] M.K. Miller, Atom Probe Tomography: Analysis at the Atomic Level, Kluwer

Academic/Plenum Publishers, New York; London, 2000.[15] B. Gault, E.A. Marquis, D.W. Saxey, G.M. Hughes, D. Mangelinck, E.S. Toberer,

G.J. Snyder, Scripta Materialia 63 (2010) 784–787.[16] O.C. Hellman, J.A. Vandenbroucke, J. Rusing, D. Isheim, D.N. Seidman, Micro-

scopy and Microanalysis 6 (2000) 437–444.[17] M.P. Moody, L.T. Stephenson, P.V. Liddicoat, S.P. Ringer, Microscopy Research

and Technique 70 (2007) 258–268.[18] F. Vurpillot, L. Renaud, D. Blavette, Ultramicroscopy 95 (2003) 223–229.[19] W. Sha, L. Chang, G.D.W. Smith, C. Liu, E.J. Mittemeijer, Surface Science 266

(1992) 416–423.[20] M. Muller, D.W. Saxey, G.D.W. Smith, B. Gault, Ultramicroscopy 111 (2011)

487–492.[21] F. Tang, B. Gault, S.P. Ringer, J.M. Cairney, Ultramicroscopy 110 (2010)

836–843.[22] B.P. Gorman, A.G. Norman, Y. Yan, Microscopy and Microanalysis 13 (2007)

493–502.

[23] A. Cerezo, C.R.M. Grovenor, G.D.W. Smith, Journal De Physique 47 (1986)309–314.

[24] B. Gault, M.P. Moody, F. de Geuser, G. Tsafnat, A. La Fontaine, L.T. Stephenson,

D. Haley, S.P. Ringer, Journal of Applied Physics 105 (2009).[25] R.A. Karnesky, D. Isheim, D.N. Seidman, Applied Physics Letters 91 (2007).[26] W. Lefebvre, T. Philippe, F. Vurpillot, Ultramicroscopy 111 (2011) 200–206.[27] L. Guibas, D. Knuth, M. Sharir, Algorithmica 7 (1992) 381–413.[28] V.J. Araullo-Peters, B. Gault, S.L. Shrestha, L. Yao, M.P. Moody, S.P. Ringer,

J.M. Cairney, Scripta Materialia 66 (2012) 907–910.[29] B. Gault, M.P. Moody, F. de Geuser, D. Haley, L.T. Stephenson, S.P. Ringer,

Applied Physics Letters 95 (2009).[30] B. Gault, M.P. Moody, F. De Geuser, A. La Fontaine, L.T. Stephenson, D. Haley,

S.P. Ringer, Microscopy and Microanalysis 16 (2010) 99–110.[31] A.V. Ceguerra, R.C. Powles, M.P. Moody, S.P. Ringer, Physical Review B 82

(2010) 132201.[32] K.Y. Xie, T.X. Zheng, J.M. Cairney, H. Kaul, J.G. Williams, F.J. Barbaro,

C.R. Killmore, S.P. Ringer, Scripta Materialia 66 (2012) 710–713.[33] T. Philippe, M. Gruber, F. Vurpillot, D. Blavette, Microscopy and Microanalysis

16 (2010) 643–648.[34] B. Gault, M. Muller, A. La Fontaine, M.P. Moody, A. Shariq, A. Cerezo,

S.P. Ringer, G.D.W. Smith, Journal of Applied Physics 108 (2010).[35] K.Y. Xie, A.J. Breen, L. Yao, M.P. Moody, B. Gault, J.M. Cairney, S.P. Ringer,

Ultramicroscopy 112 (2012) 32–38.