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SOME ASPECTS OF THE MEASUREMENT OF COMPOSITION IN THE ATOM PROBE
M. Hetherington, M. Miller
To cite this version:
M. Hetherington, M. Miller. SOME ASPECTS OF THE MEASUREMENT OF COMPOSI- TION IN THE ATOM PROBE. Journal de Physique Colloques, 1989, 50 (C8), pp.C8-535-C8-540.
�10.1051/jphyscol:1989892�. �jpa-00229991�
COLLOQUE DE PHYSIQUE
Colloque C8, Suppl6ment au n0ll, Tome 50, novembre 1989
SOME ASPECTS OF THE MEASUREMENT OF COMPOSITION I N THE ATOM PROBE
M.G. HETHERINGTON* and M.K. MILLER
Metals and Ceramics Division, Oak Ridge National Laboratory, oak Ridge, TN 37831-6376, U.S.A.
Department of Metallurgy and Science of Materials, Oxford University, GB-Oxford OX1 3PH, Great-Britain
A b s t r a c t
-
The measurement of compositions in the atom probe has been studied by considering the phase decomposition of high purity Fe-Cr alloys. Alloys containing 45% Cr were aged for up to 500h at 500'C in order to study the early stages of the decomposition. Typical data representations are considered as fractals.The scaling exponents have been measured for different aging conditions and three different scaling regimes have been identified. The consequences of the fractal nature of the profile are discussed. The results indicate that there is an optimum block size at which the composition and other physical parameters may be reliably measured. The cumulative plot is shown to be a fractional Brownian motion suggesting that the simulation of the microstructures by randomly placed spheres is not consistent with the observed microstructures.
I - INTRODUCTION
The unique capability of the atom probe is the identification of individual atoms and their positions with a precision of less than a nanometer; a description of the atom probe technique is given in Miller and Smith [I]. Analysis on this fine scale raises interesting questions about the meaning of the composition in an alloy. If the composition is calculated by averaging over a scale which is too large then detail about the decomposition will be averaged out: if the distance is too small then the fluctuations in the composition will be due to noise rather than the 'true' variation of the composition.
The ideal scale to use in calculating the composition is of the order of the scale of the fluctuations in the composition due to the phase decomposition. Unfortunately, the determination of this scale is one of the goals of an atom probe experiment and this scale is not therefore known a priori. The Fe-Cr alloys in the present study are expected to decompose spinodally and composition (or more pertinently, the chemical potential) is treated as a continuous variable in current field theoretic treatments o f spinodal decomposition. Estimates of the free energy as a function of composition contain an implicit coarse graining scale over which the energy is computed. The extent of the theoretical spinodal is therefore dependent on this scale and the search for a renormalization group-theoretic type of approach to a correctly scaling kinetic theory is an important aspect of theoretical work in this area. The determination of composition is therefore not only of practical importance to the experimenter but also important to the theoretical modelling of the decomposition.
The key concept is that of scaling; the use of particular microscopical and data analysis techniques may impose a scale on the observations. However, the magnitude of a physical effect should be independent of the means used to measure.
In the next three sections the scaling behaviour of typical representations of atom probe data is considered. The first section considers the scaling of the composition profile as a function of the number of atoms used to calculate the composition (block size), the second section considers the fractal behaviour of the composition plot with a sample size of one atom and the third section considers the fractal behaviour of the cumulative plot. It is unlikely that a scaling behaviour will be exhibited over all length ranges and even the most universal scaling will have a lower cut-off of order of one atom and an upper cut-off of order of the sample size.
The experiments were a11 carried in the ORNt atom probe out at a temperature o f 50K and a specimen voltage of IOkV.
Under these conditions approximately 25 atoms were detected from each atomic layer removed from the specimen.
These conditions were chosen to facilitate the comparison between samples from different aging times and are not necessarily the optimum conditions for measuring composition; the optimum conditions for the very earliest stages have been discussed in a previous paper [2].
2
-
SCALING (A)A typical composition profile for the Fe-45%Cr alloy aged for 500 h at 500'C is shown in Fig. 1. This profile is calculated with a block size of 100 which is a typical block size for representation of atom probe data. The study of the scaling behaviour of this profile requires the definition of a typical length. This length was chosen as (IC(t, N)
-
C(t+l, N)I) where C(t, N) is the composition of the tfh block of atoms with a block size of N and the0
brackets indicate that the mean value is calculated; the length thus represents the mean value of the composition
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989892
difference between adjacent blocks. The variation of this length as function of block size for the Fe-45%Cr alloy in the as-quenched condition and at three different aging times is shown in Fig. 2 on a log-log plot. The as-quenched specimens display a monotonic decrease in the length. This is because as the sample size increases the variance of the composition profile about the mean composition, p, of the alloy decreases as p(1-p)/N for a random solid solution and the steps between adjacent blocks therefore decrease. The shortest aging time displays a similar behaviour though possibly with a different slope at large blocksizes.
Fig. 1. Composition profile in atomic % of Cr for Fe-45%Cr aged for 500h at 500°C and with a block size of 100 atoms.
However, the lOOh and 500 h aged specimens have t h r e e distinct regimes. Initially the slope decreases, followed by a plateau region in the 100 h specimen and an increasing region in the 500h specimen, finally this reaches a peak and starts to decrease again. This behaviour occurs because after these two longer aging treatments significant phase separation has occurred: the initial decrease is because the noise decreases as the sample size is increased. At the minimum, the sample size is of the same order as the scale of the composition fluctuations in the alloy and therefore if C(t, N) is a Cr rich region there is an increasing probability that C(t+ 1, N) is an Fe rich region and vice- versa (i.e. aliasing occurs).
I 10 100 1000 10000 1 10 100 1000 10000
block size block size
Fig. 2 The function AC = ([C(t, N)
-
C(t+ 1, N) 1) asa
function of block size for four different aging conditions As the block size increases this probability increases until the block size becomes larger than the scale of the phase decomposition. At this point, a block C(t, N) now averages over both Fe and Cr rich regions and the value (IC(t, N)-
C(t+l, N)]) starts to decrease again. The region between the two turning points is therefore an acceptable range of block sizes to measure composition. At the first minimum the function (IC(t, N)-
C(t+l, N)I) is to first order independent of the block size and therefore this is the optimum value.The autocorrelation function is often used for estimating the scale of decompositions. The autocorrelation function, Rk, at a lag k is defined as
The first minimum in the autocorrelation function (as shown in Fig. 3), which is often interpreted as the scale of a decomposition, does not estimate the appropriate scale for measuring the composition variations since the first minimum in the autocorrelation function is biased to the larger size fluctuations. Spooner and Miller [3] pointed out that the autocorrelation function in small angle neutron scattering has a delta function at lag=O due to incoherent scattering. The
measured value of
R1
gives an estimate of the height of this delta function and is plotted against sample size in Fig. 4 for the 500h aged alloy; this indicates that for atom probe data, the size of the delta function is a function of the sample size. As is expected from the scaling considered above, the maximum value of R1 occurs at the minimum in the function of (IC(t, N)-
C(t+l, N)I). For smaller values the noise increases and reduces the correlation while for higher values the correlation decreases because a lag of one corresponds to an ever increasing absolute step and the correlation is thus reduced.lag k block size
Fig. 3 Autocorrelation plot for Fe-45%Cr alloy aged Fig. 4. The value of the first lag a s a function of the
for 500h. block size for the Fe-45%Cr alloy aged for 500h.
The amplitude, Pa, of a spinodal may be calculated using a maximum likelihood technique [4] or mean estimator method [5]. The value of Pa /o(Pa) where o(Pa) is the error on the estimate of Pa is plotted in Fig 5. Again it is consistent with the other scaling considerations that the optimum value for estimating Pa is at the minimum in the function (IC(t, N)
-
C(t+ 1, N)I).Fig. 5. The amplitude o f the spinodal, Pa, divided by Fig. 6 Part of the composition profile of the aged to 500h the error, o, as a function of the block size for specimen with a block size of 1.
aged for 500h at 500°C.
The scaling of (IC(t, N)
-
C(t+ 1, N)I), the autocorrelation function and the Pa values all suggest that there is an optimum block size for longer-aged specimens. However, for the early stages it is not possible to define a suitable scale and there appears to be only one scaling regime. Calculating the composition over any block size greater than order 1 in the earliest stages of decomposition may therefore be averaging out significant composition variations.3- SCALING (B)
In this section, the scaling of the function AC = (IC(t, 1)
-
C(t+k, 1)l) is considered as a function of k. A block size of 1 is used so that information in the short time aged material is preserved. A section of the composition profile for the material aged for 500h is shown in Fig. 6.The values of I n ( l ~ ( t , 1)
-
C(t+k, 1)l) are plotted against In k in Fig. 7. (IC(t, 1)-
C(t+k, 1)l) counts the number of steps in the composition plots as a function of k and therefore the slope of this plot, H, is related to a fractal dimension of the concentration profile. The fractal dimension of curves such as this must be considered carefully since the curve is not self-similar but self-affine 161 as the two coordinates are independent and therefore scale with different factors. Ifthe length scales along the two axes are chosen such that t << (IC(t, 1)
-
C(t+k, 1)l) then displacements along the t axis can be ignored and the fractal dimension of the curve will be D=1/(1-H). The 500h plots as well as the lOOh plot (not shown) have three regions: an initial flat part; a region with positive slope; and then a region with negative slope.On the other hand, the as-quenched and 8h aged specimens have only one linear region where the slope is nearly constant and equal to zero. This behaviour is similar to that described for the composition curves in the previous section and occurs for the same reasons. The fractal dimension of a random solid solution should be D = l (H=O) since the probability of a step is independent of the value of k. A model of this process can be developed by considering the series of atoms as a Markov chain. Johnson and Klotz [7] defined an order parameter 0 such that the conditional probability for the transition from a Cr atom to a Cr atom is p2 - 0p (pI2 is the probability of a Cr following an Fe in the sequence to Cr, etc.) and all of the other values pij can be cafcilated from normalization and Bayes theorem [8].
The transition probabilities can be written as a probability matrix P where q = (1
-
p) andIf the state of a system is considered as either an Fe or a Cr atom then the state probabilities after trial r can be represented by a vector a r where a' = ( I , 0) if the expectation of the Ifh atom is an Fe atom, and a r = (0, 1) if the expectation is a Cr atom and in general a' = (ar], a;) with a i + a i = 1. The state probabilities on the trial following the state ar are therefore given by ar+I = a P . Similarly, the state probabilities on the n+2fitrial will be a*2 = a r + l p =
a P P = a F 2 . The elements of the matrix P2 are therefore the probabilities of the transition from state si to state sj in two steps summed over all the possible paths. By induction, the elements of the matrix p k are the probabilities of the state transitions in k steps. With some restrictions, aOPk -t u in the limit k -+ it can be shown that,where u is the stationary distribution and u P = u . Using the transition matrix P it is straightforward to calculate
k k k
(IC(t, 1) - C(t+k, 1)l) since it is equal to ulp12+ u2p2] where p.. are the components of the m a t r i x ~ k .
1J
The maximum likelihood estimators of 8 are 1.0 1 for the as-quenched specimen (within error of the value I), 1.02 for 8h, 1.071 for lOOh and 1.32 for 500h. The theoretical curve of In (lC(t, 1)
-
C(t+k, I)[) for these alloys is plotted against In k in Fig. 8 for values of 8 of 1.02 and 1.32. The scaling behaviour for the 8h is consistent with the theoretical plot whereas there is no similarity between the theoretical and measured curves for the 500h treatment. The clustering at 8h is therefore similar to the short range order described by the Johnson and Klotz model. However, although it is possible to calculate the Johnson and Klotz order parameter, 0, after long aging times, this parameter cannot correctly describe the scaling behaviour, and is therefore a heuristic guide to the degree of clustering rather than providing an accurate description of the microstructure.The application of the Markov chain technique is not limited to ordering or clustering. A simple example shows the possible extensions of this idea. Consider a material containing cubic precipitates whose size is 5 units. Furthermore, assume that these precipitates are randomly spaced with the additional constraints that they do not overlap, their faces are perpendicular to the direction of the sampled volume and their size is much larger than the area selected by the probe aperture so that there is no simultaneous sampling of the particles and the matrix. For these conditions
In this representation, the state s l is a matrix composition layer, s2 is the first layer of a particle, s3 is second layer,?! is the third and so on. In addition, there must be at least one matrix layer between particles and p'(=l-q) is the probablllty of encountering a particle in the matrix (the generalization to different particle sizes is trivial). The particles are assumed to have a composition of 100% solute and the matrix contains 0% solute, so that the mean composition of the system is p = 5p'/(l + 5p'). Attempts to fit this model to the scaling behaviour, however, have not been found to be consistent with any of the experimental data. Further extensions of this model to more realistic size distributions is obviously straightforward, although the number of parameters becomes rather unwieldy and, as is discussed in the next section, this approach may be necessarily inconsistent with the data because it is not stationary.
Fig. 7 AC = (IC(t, 1)
-
C(t+k, 1)l) for the specimens Fig. 8. The scaling of AC calculated from theaged for 8h and 500h. Johnson and KIotz parameter. Upper curve for 8h
(0= 1.02) lower curve 500h (0=1.32).
In this final section, the scaling of the function V(t), the cumulative total of Cr atoms observed as a function of the total number of atoms observed, t. This function is known as the cumulative plot in atom probe analysis and an example for the 500h heat treatment is shown in Fig. 10. This curve is a fractional Brownian motion if V(t)
-
V(t+k) has a Gaussian distribution with variancevariance = (W(t)
-
V(t+k)l2) = k2HFor the case H=0.5 the curve is the familiar Brownian motion. If H is greater than 0.5 then displacements are positively correlated and if H is less than 0.5 then displacements are negatively correlated. Fig. 10 shows a log-log plot of the mean square increments in V against k for the as-quenched and 500h heat treatment conditions. These plots for all have extensive linear regions from which the slope, H, can be measured. The value of H increases with time as expected since clustering is occurring in the alloy and this increase is shown in Fig. 11 as a plot of the In H against In(aging time) (this allows the scaling dimension of the scaling exponent to be determined). Like the composition profile in the previous section, the cumulative plot is self-affine and therefore by making similar assumptions it is possible to relate H to a fiactal dimension, D.
Fig. 9. A
ort ti on
of the cumulative plot for the Fe- Fig 10 A plot o f ~ V = (IV(t)-
V(t+k)12) as a function of k45%Cr aged for 500h. for different aging times.
A curve displaying fractional Brownian motion is stationary and isotropic. The existence of a scaling regime , as shown in Fig. 10, suggests that, within this linear region, the atoms detected in an atom probe experiment form a stationary time-series. The linear region for the lOOh and 500h specimens extends over the k corresponding to the scale of the microstructure. A stationary time series on this scale cannot be simulated by placing distributions of particles randomly in a matrix as performed in the simulations of the Gottingen group [9], Blavette et a1 [ l o ] or the analytical calculations outlined in the previous section since this will necessarily be non-stationary. Scaling analysis therefore reveals something about the nature of the microstructure and indicates future directions for modelling, and furthermore
is in agreement with studies of Cerezo and Hetherington [ 111 which suggest that a spinodal microstructure may itself be a fractal. The value of H quantifies the degree of correlation in the same way that the Johnson and Klotz parameter quantifies the short-range ordering. However, the parameter H appears to be more generally applicable since it can be applied to all of the aging conditions; whereas the Johnson and Klotz parameter is strictly limited to the very earliest stages.
1 2 3 4 5 6 7
In
timeFig. 11. The scaling exponent of the cumulative profile plotted against In(aging time) in hours.
5
-
CONCLUSIONSThe division of this paper into three main sections is somewhat artificial since future work in this area will surely show that all of the scaling procedures are extremely closely related. However, as well as representation techniques, several reasons for exploring scaling have been discussed: the first section shows that scaling can be used to determine the optimum block size for the computation of composition; the second shows that scaling can be used to test the validity of models; and the third that scaling exponents can be used to parameterize the data and hence the microstructure. The simplicity of the concept of scaling belies its power in the analysis of data from atom probe experiments.
Acknowledgments
This research was sponsored by the Division of Materials Sciences, U.S. Department of Energy, under contract DE- AC05-840R21400 with Martin Marietta Energy Systems, Inc and MGH gratefully acknowledges funding as an SERC Fellow. Thanks are due to K.F. Russell for technical assistance
.
References
[ l ] M.K. Miller and G.D.W. Smith "Atom Probe Microanalysis: Principles and Applications to Materials Problems", (1989), publ. Materials Research Society, Pittsburgh.
121 M.G. Hetherington and M.K. Miller, J. de Physique, (1988), 49, C6-427 [3] S. Spooner and M.K. Miller, J. de Physique, (1988), 49, C6-405.
[4] T.J. Godfrey, M.G. Hethetington. J. Sasson and G.D.W. Smith, J. de Physique, (1988), 49, p. C6-411.
[S] P. Auger, A. Menand and D. Blavette, J. de Physique, (1988), 49, p. C6-439.
[ 6 ] B.B. Mandelbrot, "The Fractal Geometry of Nature", (1977), publ. W.H. Freeman and Co., New York.
[7] C.A. Johnson and J.H. Klotz, Technometrics, (1974), 16, p.483.
[8] P.E. Pfeiffer, "Concepts of Probability Theory", (1965), publ. Dover Publications, New York [9] M. Oehring and L. v. Alvensleben, J. de Physique, (1988), 49, p. C6-415.
[ l o ] D. Blavette, A. Menand and A. Bostel, J. de Physique, (1987), 48, p. C7-495.
[ I I] A. Cerezo and M.G. Hetherington, J. de Physique, (1 989), [this volume]
