Point process statistics to analysis atom probe data

1. Study on Atome
Probe
Tomography
W.J.Jannidi
TU-Kaiserslautern
Supervised by: Prof.Dr:Claudia Redenbach
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2. Introduction:
The atom probe was introduced in 1967 by Erwin Wilhelm Müller. It combined a field ion
miscroscope with a mass spectrometer having a single particle detection capability.
For the first time, an instrument could determine the nature of one single atom seen on a metal
surface and seleceted neighboring atoms at the descretion on the observer.
APT involves applying either ultra-fast voltage pulses or laser pluses to a needle-shaped specimen,
stripping away atoms located at the trip of the needle and converting them into charged ions in a
process know as field evaporation. These ions are then accelerated by an electric field towards a
position-sensitive detector that registers the time it takes each ion to travel from the sample to the
detection system, as well as its impact position.
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3. 3

4. Study Flow:
Statistical methods to solve the problems
Relations between AP data and point process
Identify the problems
Applications of APT
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5. Research Studies
[1] Ilke Arslan,Emmanuelle A.Marquis, Mark Homer, Michelle A. Hekmaty, Norman C. Bartelt Towards better 3-D
reconstructions by combining electron tomography and atom probe
tomography. University of Oxford UK 2008
[2] T.Philippe,O.Cojocaru-Mire‘din,S.Duguay & D.Blavette Clustering and nearest neighbour
distances in atom probe tomography: the infuence of the interfaces. Institut
Universitaire de France 2009
[3] F.De Geuser,W.Lefebvre,D.Blavette 3D atom probe study of solute atoms clustering during
natural ageing and pre ageing of an Al-Mg-Si alloy
[4] T.Philippe,Maria Gruber,Francois Vurpillot & D.Blavette Clustering and local magnification effects
in atom probe tomography: A statistical approach. Institut Universitaire de France 2010
[5] W.Lefebvre,T.Philippe &F.Vurpillot Application of Delaunay tessellation for the
characterization of solute rich clusters in atom probe tomography. Universite‘
de Rouen France 2010
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6. Research areas in APT:• Solute atoms clustering
• Nearest neighbour distances
• Local magnification effects
•Trajectory overlaps
•The influence of the interfaces
• 3-D reconstructions
•
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7. WHY ?... Point process statistics
A spartial point process is arandom set of points in d dimensional space. Spatial point process
(d=3) and their applications can be used for data treatment in atom probe tomography.
In APT dataset, the points represent the locations of atoms after reconstruction.
Use point process statistics to assess deviation from randomness, derive phase composition or
select and characterize solute enriched clusters.
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8. Statistical tools for APT
experiments
•NN method
•The mean distance to NN
•Chi-square test
•Dmax method
•Correlations and pair correlation function
•Delaunay tessellation
•Voronoi cells
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9. Nth nearest neighbour distance
distribution
The probability density function to have the first nn at the distance r within dr [6]:
The probability that no point occurs inside the sphere follows poisson distribution:
This leads to:
Thus
And
The probability density function fn(r) of the nth nn distances [6]
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10. Testing randomness
1. Mean distances [6]
Mean distances to nearest neighbour are measures of space and randomness is dependent upon
the boundaries of the space.
The test consists in comparing the observed mean distance to nth first nearest neighbour (On) to
the expected mean distance (dn = En (r)) if the population is randomly distributed.
The ratio (Rn) of the observed mean distance to the expected one can be used to demonstrate the
occurence of a non random distribution.
Rn =1 perfectly random
Rn <1 clustering
Rn >1 uniform or ordered
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11. 2. Test of significance[6]
The significance of departure of the observed mean distance(On) from the expected under
condition of randomness can be tested applying ordinary normal probability theory to the mean
dn and standard deviation .
The standard error of the normal curve is given by
In practice, non randomness is concluded when >1.96. This means that the observed mean
distance is not included in the confidence interval : dn ± 1.96
As the C.I is inversely proportional to , it is clear that large volumes are desirable.
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12. distribution
Define,
[7]
The probability density function[6]
)
This leads to distribution of x with 2n degree of freedom.
Note : xm the mean value of x over N
Nxm ~ with 2Nn degree of freedom
From fisher approximation , follows a normal disrtibution[6]
Using this approximation, the confidence interval for E(x) can be estimated.
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13. Pair correlation function
The pair correlation function derives from second moments properties. In APT, the pair
correlation function is often used to detect ordering or clustering [6].
Monte Carlo simulaions are used to generate point process of same density. Simulated pair
correlation function are then compared to the observed one.
Recently propose a best-fit method based on a theoritical function to derive phase composition
and spatial information related to clustering.
It can be shown that for a mono-dispersed system of spherical particles enriched in solute atoms
B and embedded in a matrix α, the pair correlation function can be expressed as [8]
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14. Cluster identification
In 3D, Delaunay tessellation uses empty circumscribed sphers, which define tetrahedral as the
cells. For an APT data set, atoms are hence the 4 vertices of the Delaunay cells.
We can use the propertis of the Delaunay tesselation to characterize solute enriched clusters
using a new metholodogy based on the density distribution function of the radius RD of the
circumscribed spheres. For a homogeneous poisson process, it can be shown that
It is possible to write f(RD) of a bi-phased system as
Fig 2. [5]
Delaunay tessellation [6].
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15. Example for Cluster selection
procedure
Compute density function of RD constructed on Delaunay tessellation
Chose a criteria R‘D to select specific objects
Fig 5. [6]
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16. Voronoi cells
The partitioning of a plane with points into convex polygons such that each polygon contains
exactly one generating point and every point in a given polygon is closer to its generating point
than to any other.
Voronoi volume becoms the number of atoms that are closer to a given atom than to any other
atom.
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17. Principle of the analysis [9]
Kolmogorov-Smirnov test[9]
Test for randomness of Voronoi volume distribution.
The supremum of the differences De,r between the cumulative curve of the two distributions Fe
and Fr is compaired.
a. Atomic disrtibution
b. Atomic density
c. Atomics are classified as clustered or unclustered
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18. ............cnt
The null hypothesis , no significant deviation from random is rejected if
Atoms classified as clustered are allocated to individual clusters by analysing their
neighbourhood relationships.
Fig 2. [9]
Fig 5. [9] 18

19. Properties of a cluster selection
algorithm in APT
A custer selection algorithm would be relevant if it presents the following properties.[5]
•High sensitivity (large signal over noise ratio)
•Low background noise (real clusters are detected)
•Independent of clusters or precipitates morphology
•Ability to reproduce the actual morphology of objects
•Statistical description of the results
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20. Other References
[6] T.Philippe,S.Duguay,G.Grancher,D.Blavette Point process statistics in atom probe
tomoprapgy
[7] H.R.Thompson Disribution of distance to Nth neighbour in apopulation od random
disrtibuted individuals
[8] T.Philippe,S.Duguay,D.Blavette Ultramicroscopy 2010
[9] P.Felfer,A.V.Ceguerra,S.P.Ringer, J.M.Cairney Detecting and extracting clusters in atom
probe data: A simple automated method using Voronoi cells
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21. Thanks For Your Attention!.
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