STATISTICAL ANALYSIS OF ATOM PROBE DATA

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STATISTICAL ANALYSIS OF ATOM PROBE DATA

L. Alvensleben, R. Grüne, A. Hütten, M. Oehring

To cite this version:

L. Alvensleben, R. Grüne, A. Hütten, M. Oehring. STATISTICAL ANALYSIS OF ATOM PROBE DATA. Journal de Physique Colloques, 1986, 47 (C7), pp.C7-489-C7-494. �10.1051/jphyscol:1986782�.

�jpa-00225977�

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STATISTICAL ANALYSIS OF ATOM PROBE DATA

L. v. ALVENSLEBEN, R. G R U N E , A. HUTTEN and M. OEHRING Institut fiir Metallphysik der Universitat Gottingen, 0-3400 Gottingen, F.R.G.

and Sonderforschungsbereich 126, 0-3392 Clausthal-Zellerfeld.

(GBttingen), F.R.G.

Abstract

-

If the phases of a decomposed alloy show no contrast in field ion images, only random area atom probe (AP) analyses are possible. The AP data have to be evaluated by statistical methods to obtain the decomposition parameters. Especially, the autocorrelation analysis of AP concentration profiles has often been used. We have studied the applicability of this sta- tistical method using computer simulated concentration profiles. The limits of the reliability of autocorrelation analysis are tested by varying the input values for the mean concentration of the precipitate and matrix, the aperture diameter and mean diameter of an assumed particle distribution and its stan- dard deviation. Furthermore, the conditions for a successful use of this statistical method are discussed.

I

-

INTRODUCTION

Field ion microscopy (FIM) in combination with atom probing (AP) is a suitable method for the observation of the very early stages of the decomposition of alloys I - . In comparison to transmission electron microscopy combined with analytical facilities smaller particles can be resolved and analysed with respect to their chemical composition.

Nevertheless, the volume analysed by AP is relatively small due to the high resolution (in normal observations 800 nmf 1. With FIM-AP one has to take care that the analysed volume is large enough to give representative informations about the most important decomposition parameters: particle diameter, concentration, number density, volume fraction and mean distance. These parameters (except the concentration) can be determined directly from the FI micrographs, if the dif- ferent chemical compositions of precipitate and matrix leads to a visible contrast in the FI images. The compositions of both phases can be measured using selected area atom probing. If the specimen does not generate a contrast in the FI micro- graphs, the analysis becomes more difficult. The decomposition parameters can only be determined by random area atom probing. It is a great advantage of analyt- ical FIM to have an accurate depth scale given by the collapsing of low indexed crystallographic planes observable in the FIM.

The data set which is then obtained by AP is a concentration profile into the depth of the specimen. The length of the profile is given by the number of field evaporated atomic layers. If the number density of particles is higher than

lo2'

m-'

,

the particle diameter is larger than 4 nm and the difference in the con- centration between particles and matrix is significantly greater than the statisti- cal noise due to the limited number of atoms per atomic layer one can obtain the decomposition parameters directly from this concentration profile (e.g. in 151).

The purpose of this paper is to go to the opposite end where all these parameters are less obvious. In this case, statistical methods have to be applied to the concentration profiles. Two problems impede the straight-forward evaluation of concentration profiles: the cutting of precipitates and the noise onto the concen- tration profiles due to the small sample size (typically 5 0 to 100 detected atoms per field evaporated layer).

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986782

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C7-490 JOURNAL DE PHYSIQUE

In our case, it is not advantageous to apply statistical analysis of field evaporation event series (sample size: one atom) because it is more difficult to get direct informations about the decomposition parameters which are already present in the concentration profiles (sample size: one atomic layer). For graphical evaluation of AP event series ladder diagrams can be used which are well suited for applications of the high resolution of the AP as e.g. investigations of ordered phases in alloys /6/ or the sharpness of phase boundaries. Deviations from the statistical distributions of atoms in an alloy can be determined by Markov chain analysis / 7 / . However, there is no straight-forward method to obtain direct informations about cluster sizes and concentrations. The main objective of this article is the autocorrelation analysis of concentration profiles. Piller and Wendt /8/ have shown the feasibility of this method.

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-

COMPUTER MODEL

In order to show the possibilities and the limits of statistical methods it is necessary to have a computer model which allows a realistic simulation of concentration profiles under assumed input parameters. Therefore, we take a <200>

--

oriented fcc model-crystal. An assumed mean diameter of spherical particles is distributed normally. A randomly distributed set of ( x , y, 2)-coordinates is at- tached to each particle under the condition that particles do not overlap. A cylindrical tube equivalent to the analysed volume of an atom probe experiment is cut out of this crystal. A sharp interface between particle and matrix and no concentration variations in between are assumed.

The following parameters can be varied: a: lattice parameter (e. g. 0.35 nm)

,

d: mean diameter in a/2, o: standard deviation of d in a12 (i.e., 68% of particle radii lie between r-o and r+o), co: mean concentration, c : particle concentration, c

-

matrix concentration, n: length of profilz in al2, dap:

aperture diameter in a/2 m' , ^g: efficiency of the detector (e.g. 0.6).

The concentration in each (200)-layer is calculated from the covering fraction of the projected aperture area by the precipitate. Realistic noise is added to a concentration profile by simulating an AP event series for each atomic layer. This is simply carried out by deciding randomly with the respective probabilities for each atom whether it is detected and whether it is the alloy component A or B.

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-

AUTOCORRELATION FORMULA

The autocorrelation coefficient is defined by n-k

with

1 n- k

=

^Ci

:=

n-k

^i=l

and

1 n

C2 :=

-

_n

_-

_{k i=k}

=

^Ci

(ci: concentration of the i-th layer, n: length of profile).

Assuming c, ^. c and c, rr c with 1 n

c :=

-

^L ^Ci

n i=1

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n-k

i=I

The autocorrelation coefficient R(k) gives us a normalized value for the correlation between the concentration of the i-th and the (i+k)-th layer. The value 26 within whose limits 95,5% of the noise contributions are laying is 191:

2 2 a = 2 { var [ ~ ( k ) 1

)'I2

=

( n - k ) 112

In order to demonstrate the possibilities of the autocorrelation analysis we have simulated a concentration profile with large concentration variations (fig. 1).

The input parameters are included in the figure. The related autocorrelogram is shown in fig. 2. Using graphical extrapolation we obtain the value k correspond- ing to the mean diameter of the particles d and to the first followigg maximum k, which indicates the mean distance between two particles. A more extensive descrip-

tion of this method is shown in /8/. It has to be pointed out that the autocorrelation analysis does not only average over the apparent particle radii which would lead to a diameter of 17.5 a/2, but also takes into account the apparent layer concentration due to the cutting of particles by the analysed cylinder; ad- ditionally, it weighs in a way that input parameters can be redetermined correctly by this method. If all R(k)-values with k>O lie within the 20 limits, the analysed set of data is randomly distributed and no decomposition is detected (e.g.

in /lO/).

A discrete value for k is commonly evaluated by a graphical extrapolation of the IR(k) ]-values greater t%an 20 as shown in figs. 5, 7 and 9.

IV

-

COMPARISON OF INPUT AND OUTPUT PARAMETERS

With the combination of computer model and autocorrelation analysis we have now the possibility to compare input and output parameters even for "difficult" concentration profiles. Three points have been of major interest.

a) Diameter distribution

In order to investigate the influence of the diameter distribution on the autocorrelation analysis we took as an example the parameters of a Cu-1.9 at% Ti alloy /I/. Fig. 3 shows the simulated concentration profile. In this case, the diameter distribution is relatively sharp (o = 0.14 d). Mean diameter and mean distance (48 a/2) are in good agreement to the input parameters (fig. 4). If the d distribution is broadened (6 = 0.64 d) the k -value is still obtained quite well whereas a k,-value cannot be fixed (fig. 5). O

b) Concentration difference between particle and matrix

Further, we are interested in the minimal concentration amplitude which is necessary to give significant additions to ~ ( k ) to determine the mean diameter. For this purpose we reduced the amplitude from 20 at% to 1 at%. Figs. 6 and 7 show the concentration profile and the related autocorrelogram. At the first view the concentration profile seems to be a pure noisy one. The standard deviation due to the limited number of detected atoms is 3.7 at% whereas the concentration difference between matrix and particle which we want to detect is only slightly higher:

4 at%. On the other hand the autocorrelation proves the existence of a second phase. The diameter of the precipitates is obtained fairly well (input 30 a/2, output between 24 and 32 a/2). A mean distance between particles cannot be determined. Smaller concentration amplitudes cannot be resolved and do not seem to be

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C7-492 JOURNAL DE PHYSIQUE

of metallurgical interest.

c) Particle diameter

To determine the lowest diameter of a particle yet clearly resolved in the autocorrelogram we started with greater diameters than the aperture diameter which is then reduced gradually. The number density, the concentration c and c and the standard deviation of the particle diameter were kept constant. Big. 8 'shows the concentration profile at the lowest detectable diameter d = 0.75 d

.

The profile does not clearly resolve the cuteed particle but the aut~corre'fg~ram, fig. 9, does, The mean diameter is found exactly (ko=d) and the mean distance between two particles, kl, too.

V

-

CONCLUSIONS

By computer simulations of AP concentration prof ilea it could be demonstrated that the autocorrelation analysis of random area concentration profiles is an appropriate method to investigate homogeneous decomposition of alloys even in the very early stages. Despite of the cutting of precipitates and the noise in the concentration profiles the existence and the mean diameter of precipitates can be determined correctly even under critical circumstances, i.e. if the precipitate diameter is small, the diameter distribution is broad, or if the concentration difference between precipitate and matrix is small. In decomposed alloys where the investigations are less difficult concerning AP analysis it could be shown that the problem of cutting precipitates does not impede correct values of precipitate diameters determined by autocorrelation. It should be noted that the accurate depth scale inherent in FIM-AP concentration profiles is absolutely necessary for the investigation of decomposition kinetics.

Acknowledgements

We are grateful to R. Klapproth and Dr. R. Wagner for fruitful discussions.

References

1 L v. Alvensleben, R. Wagner, in: Decomposition of Alloys: the early stages, Proc. 2nd Acta-Scripta Met. Conf., Sonnenberg, 1983, ed. by P. Haasen, V. Gerold, R. Wagner, M. F. Ashby, Pergamon Press, (1984) p.143.

/ 2 / R. Griine, P. Haasen, J. de Physique

5

(1986) C2-259.

/3/ M. K. Miller, S. S. Brenner, P. P. Camus, J. Piller, W. A.

Soffa, Proc. 29th Int. Field Emission Symp., GSteborg, 1982, ed. by H.-0. Andren, H. Norden, Almqvist and Wiksell, (1982) p.489.

/4/ D. Blavette, S. Chambreland, J. de Physique 47 (1986) C2-227 151 F. Zhu

,

^H. Wendt, P. Haasen, Scripta Met. 7 6 (1982) 1175.

/ 6 / R. Griine, A. Hiitten, L. v. Alvensleben, thiTvolume.

/7/ T. T. Tsong, S. B. McLane, Jr., M. Ahmad, C. S. Wu, J. Appl.

Phys. 53 (1982) 4180.

/8/ J. ~ i m e r , H. Wendt, Proc. 29th Int. Field Emission Symp., Goteborg, 1982, ed. by H.-0. Andren, H. Norden, Almqvist and Wiksell, (1982) p.265.

/9/ G. E. P. Box, G. M. Jenkins, Time Series Analysis, Forecasting and Control, Holden Day, San Francisco, (1970).

/lo/ M. Oehring, P. Haasen, this volume.

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Figs. 1 , 2

-

Computer simulated concentration profile (fig. 1 ) obtained with input parameters which are well suited for random area AP analysis and the related autocorrelogram (fig. 2). The mean diameter of the precipitates d corresponds to k and 4 indicates their mean distance.

-

0

Fig.

r, = 1.9 a l I 'Y

c, = 2 0 a l l DL

c, = 0.2 a l I

-

^d = 22 a12 C

100 200 300 400

-

3

Desorbed Layers

Fig

"

-1

M

. 4 O

40 80 120

Correlat;on Length k

Figs. 3,4

-

Computer simulated concentration profile (fig. 3) obtained with a sharp precipitate diameter distribution and the related autocorrelogram (fig. 4). The input parameters are given in the figs.

-

DL 2 0 .at X

Y = 0.2 a l X

C = 22 a 0

0.5 6 l d l = 14 a12

d,, = 16 a12 N, = 2.8*102' mC3

Fig. 5

-

Autocorrelogram of a computer simulated concentration profile obtained with a broad

g-0.5/

_w

, , , , !

; , , , , , , , , , , , , , , , , , ,

1

L

precipitate diameter distribution. L 0 -1

The other input parameters given in 0 m 4 0 60 80 l o o the fig. are the same as in figs.

3,4. Fig. 5

Correlat ;on Length k

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J O U R N A L DE PHYSIQUE

50

-

⁴⁰ ^d = 30 a l l

W

..-

_m ^Gldl^{= 5 a12}

-

C 0 30

-

Y (0

+ L 20 C

(U U C

0 '0

0

Fig. 6

Desorbed Layers

~ f g . 7

Correlation Length k

Figs. 6 , 7

-

Computer simulated concentration profile (fig. 6) obtained with a small concentration amplitude of 4 at% and the related autocorrelogram (fig. 7). The input parameters are given in the figs.

Fig.

100 200 300 400 500

8

Desorbed Layers

^Fig.

d, , = 16 a/2

l a , ~ ' 1 ' ' ' ' 1 ~ ' ' ' : ~ ' ~ ' 1 1 ' ' 1 1

0 10 20 30 40 50

Correlat;on Length k

Figs. 8,9

-

Computer simulated concentration profile (fig. 8) obtained with an aperture diameter which is by a factor 413 greater than the mean precipitate diameter and the related autocorrelogram (fig. 9 ) . The input parameters are given in the figs.