- Email: [email protected]
Calculation of the long-range order parameter from atom probe data
Intermetdics
5 (1997) 609-614
0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved PII:
SO966-9795(97)00029-O
0966-9795/97/$17.00
+ 0.00
ELSEVIER
Calculation of the long-range order parameter from atom probe data S. Welzel,“* S. Duval,” E. Camusb & D. Blavette= “Laboratoirede Microscopic Ionique et Electronique, UMR CNRS6634, Universite’de Rouen, F-76821 Mont-Saint-Aignan
Cedex, France
bHahn-iUeitner-Institut Berlin GmbH, Glienockerstr. 100, Bereich NM, D-14109 Berlin, Germany (Received 2 October 1996; accepted 10 March 1997)
In addition to the known diffraction techniques, field-ion microscopy with atom probe is well established for determining the long-range order parameter. The site occupation probabilities of the chemical species on the different sublattices may be estimated from experimental profiles. However, this evaluation method demands that the superstructure planes be identified unequivocally from the data. This condition is not fulfilled in all cases. We propose a new analytical method for which this condition need not be met. 0 1997 Elsevier Science Ltd Keyworcis: A. titanium aluminides, based on TiAI; intermetallics, miscellaneous, B. order/disorder transformation, D. site occupation probability, F. atom probe.
INTRODUCTION
measurement of composition in an ordered alloy, which is described by the long-range order parameter S, along a superstructure direction, reveals the atomic order of the components on different sublattices as an abrupt change in chemical composition.2 The site occupation probabilities pa and pg of the components on the sublattices (r and p of an Lla structure are given as a function of S in Table 1 as an example. A detailed description of the method, which can be used to derive pa and pg from the mean composition of superstructure planes, is given elsewhere.3 The result of an atom probe experiment is a data chain composed of successive collected ions. Ladder diagrams are commonly used for a representation of the measurement. In such a data representation, the cumulative number of detected ions of a given species may be plotted as a function of the total number of detected ions. This number is approximately proportional to the probed depth. Such a diagram is shown in Fig. l(a), where an ordered FeT2Pt2s alloy with L12 structure was analyzed in the < 110 > direction. Taking into account a detector efficiency of 60%, the diameter of the analyzed area is about 1 nm, i.e. 10 ions are expected per layer. The variations of the slope, i.e. the local concentration on such a scale, are sufficient to distinguish between pure iron layers (horizontal
The field-ion microscope with atom probe (FIMAP) enables one to determine the chemical composition of materials on an atomic scale.’ The experimental technique is based on field evaporation. A potential of several kilovolts applied to a specimen needle having a radius of curvature of typically 10 to 100 nm is large enough to extract atoms from the tip surface. This field evaporation process is induced by a nanosecond high voltage pulse which starts a timer. The timer is stopped by any ion reaching the detector. Hence, the field-evaporated ions may be identified by means of time-of-flight spectrometry. As a diaphragm is interposed between specimen and detector, the analysis of a well defined region of about 1 nm in diameter of the specimen may be performed.’ During the experiment, the specimen is continuously field evaporated. In this way, in-depth analysis of the specimen is performed. Under certain conditions, the ultimate depth resolution of one atomic layer can be achieved, allowing a plane by plane investigation of the specimen. The *To whom correspondence should be addressed at: HahnMeitner-Institut GmbH, Bereich NE, Glienickerstr.100, D-14109 Berlin, Germany. 609
S. Welzel
610
Table 1. Site occupation probabilities pa and pp of A and B atoms on the sublattices Q and /3 as a function of the long-range order parameter S in the case of a stoichiometric LIO structure
Sublattice a! A B
Sublattice /l pA,b=f(l-S)
PA.a =+'I +s) pB,a = 2 (1 - s)
PB,B=2(l+S)
II’
I
III
: II
III
I II
II
, 50
at ,
Fe+Pt ions 64 I
I
I
I
Number of Pulses; @I Fig. 1. Typical ladder diagrams for the Lls ordered FersPtzs
alloy measured along the < 110 > direction. The dashed lines mark a sequence of five atomic layers. (a) Number of detected Pt atoms as a function of the cumulative number of detected Fe and Pt atoms. The alternation of Pt-depleted and P&rich layers (1 layer contains approximately 10 atoms) is visible. (b) Corresponding evaporation rate diagram. Ideally, the end of the evaporation of each plane should be revealed by the large number of pulses applied during which no Pt atoms are collected. It is obvious from this diagram that the univocal identification of each layer is not unambigous (see asterisked part).
et al. parts) and mixed Fe-Pt layers (inclined parts). In Fig. l(b), the steplike behavior of the arrival rate reveals that a plane-by-plane investigation was carried out. The site occupation numbers and the order parameter S are deduced from the mean composition of superstructure planes,3p4 or graphically from counting antisite defects in the pure iron layer. ’ To obtain reliable results using such a method, the following conditions must be fulfilled. (i) The atom probe analysis must be carried out layer by layer on a superstructure pole, while ensuring that each layer is completely field evaporated before detection of the next takes place.3 (ii) Each layer must be identified univocally from the data chain. The latter was until now done in one of two ways: (a) by identifying the end of each evaporating plane from the modulations of the arrival rate, as described above. As the evaporation of the layers is irregular - see for example the part of Fig. l(b) marked with an asterisk - the localization of the end of each evaporating plane is not always possible everywhere; (b) by recording interactively the end of each layer using FIM images during the whole experiment. This was done recently by Wesemann et al. in a video-controlled atom probe investigation.5 However, the method of analysis is hardly practicable for the long profiles required to achieve sufficient statistical accuracy. (iii) Any severe preferential evaporation or retention effects must not occur for any of the atom species.6 In the present contribution, we propose an analytical method to evaluate the order parameter, in which there is no need to identify each individual layer from the measured data chain. The calculation may be performed automatically without any tedious and time-consuming hand treatment of atom probe data. We simply count the number of sequences containing a fixed number of consecutively detected atoms of one given species. We show that it is enough to count sequences containing two atoms to evaluate the long-range order parameter. In the theoretical model presented in the following, we shall assume that the ions do evaporate randomly within each layer.*
*As noted by Miller and Smith,’ an experimental fact has to be taken into account. Atom probes are equipped with detectors that can detect up to eight ions on a single pulse. Due to the time-of-flight principle of the measurement, the lighter ions will reach the detector first. This leads to a bias in the randomness of the data chain. To overcome this problem, we have rearranged these multiple ions by computing random permutation before treating the data chain.
Calculation of long-range order parameter
611
PROBABILISTIC DESCRIPTION OF DATA CHAINS If the atoms within each individual plane of a plane-by-plane analysis are assumed to evaporate randomly, the data chain of A and B atoms of a disordered material is a random chain. Given a certain degree of long-range order, longer chains of consecutive detection of A and B atoms are expected. The question is to evaluate the probability P(Ak) to detect (k - 1) A ions after an A ion has been detected. In the case of a random distribution, it is obvious that this probability equals Ci-‘, where CA is the mean concentration of A atoms. If long-range order occurs in the alloy, then P(Ak) > Ci-‘. To find a measure for this enhancement, we define wk = -P(Ak) ck- 1
(1)
A
where Ak is a sequence of k consecutive A atoms. The parameters Ok, which depend on the degree of order, are one if the material is disordered and are greater than one when some order is present. Let us first consider only chains of length k = 2, for the sake of simplicity. For this special case, eqn (1) reads w2
=
p(A2) CA
(2)
where A2 is a sequence of two successively collected A atoms. Let us now consider the case of a plane-by-plane analysis of the L10 ordered structure along the < 001 > tetragonal axis. Thus, the mean composition of each type of plane is equal to the occupation frequencies of a! and jI sites (P_,+,PA,B). We consider first the case where S = 1. Figure 2 shows an idealized ladder diagram of the fully ordered Ll,, structure, obtained by such plane-byplane analysis. It is assumed that each plane contains a constant number N of atoms. The validity of this assumption will be discussed in the next section. This idealized ladder diagram is divided into four parts: parts 1 and 3 are these where either a pure A or a pure B layer is detected, and parts 2 and 4 are regions where the transition from an A to a B layer or vice versa takes place. Let us define the probability P(A2) of finding an A ion when the previous ion is also an A ion. It is
A ions Fig. 2. Idealized ladder diagram for an Lla fully ordered A5eB5s alloy (S= 1). A ions are closed circles and B ions are open circles. The number N of ions detected per layer is assumed to be constant. The two successive detected layers are divided into four parts to take into account the different probabilities of detecting A atoms within the layers and at the interface between the layers.
the weighted sum of the partial probabilities PAJ in the parts i, i = 1 to 4. It reads: 5
P(A2)
nA,i
‘pA,i
= ‘=I 4 g
(3) n.‘Q
where nA,i is the number of A ions involved in the respective parts. As noted above, the probability P(A2) equals the mean concentration (CA = 4) for the case of a random alloy. For a fully ordered structure of stoichiometric composition, as shown in Fig. 2, the site occupation probabilities are equal to one or zero and the numbers of ions involved A nA,j are summarized in Table 2. w;! is expressed as
w2 > w&S= 1) =7=2-k=
@?.,max
(4)
A
where W,max is the maximum value of w2 and depends only on the number N of ions collected per layer. The general case of partly ordered L10 structure with the order parameter S smaller than unity may be deduced directly from eqn (3) by introducing the probabilities PA,i as given in Table 1. This results in the following expression for the parameter w:
S. Welzel et
612
Table 2. Number “Aj of A atoms involved and probability PM of lkling an A atom after an A atom had been detected, for parts 1 to 4 as shown in Fig. 2 for a fally ordered alloy Parts i
n.4,i
PA&
Note that, analogous to eqn (5), the general form for a k sequence &(k -c N - 1) can be deduced as k-l wk+m&+P~,g)+2
al.
alloys with L12 structure the main point is that we are dealing with a similar situation to that in Llo structures: an alternating sequence of layers. But in this case of AsB we have to take into account probabilities PA,~ and pA,m, for pure and mixed layers, of finding an A atom in the pure or in the mixed plane; these are calculated from the site-occupation probabilities for L12. Insert PAp and pA,m in expression (3) for P(A2) and eqn (7) follows with respect to eqn (4) and eqn (9). The value of o,, is presented in eqn (9). In particular, for the case of an off-stoichoimetric L12 ordered structure of type Fes_XPtl+, (which is treated in the next section) measured in the < 100 > or direction, it holds that:
p&$htdk-i
ok =
w
N*(PA,a +pA,~)k-l. c;-'
,max=
1 + ($)‘.(J lo-z. 9
9N
- &J
forspeciespt for species Fe
(6)
By inserting the site occupation probabilities ~A,(Y and PA,@as given in Table 1, into eqn (S), the order parameter can be derived from 02 as follows:
J
co.2 -
1
s(w)= *,_ -
(7)
1
Hence, S(m) depends monotonically on q and ranges, as expected, between 0 and 1. 02 is obtained from experiment7 and q,max can be calculated from eqn (4). This can be done for A and B atoms. As a consequence the long-range order parameter is measured out of AA or BB sequences. The results summarized in Tables 3 and 4 take this into account. In the case of an off-stoichiometric Llo alloy At+xBr-x, eqn (7) remains valid with the following expression for the maximum value of ~02:
(9) DISCUSSION
Two assumptions were made to derive eqn (7): (i) the number N of atoms per layer is constant for any layer, and (ii) removal of the atoms from the layer does not change the composition of the remaining layer. Simulations are an appropriate tool to test the relevance of these assumptions. As noted in the previous section, N is not constant if the efficiency of the detector is non-ideal. In that case the number N = nA + ns of atoms detected per layer is distributed binomially. lo We have simulated ladder diagrams by generating random deviations from a multinomial distribution: P(nA, OB) =
NM! IZA!FZB!(N,,- TZA - ~zB)!
x (QPA)""(QPB)"~.
1 + (fi)2. W,max =
2-i
(1 -h)
forspeciesA
(1 - Q)(N~x-nA-nB)
(10)
for species B
(8) where x > 0 is the deviation from the stoichiometry. To derive these equations, we have supposed that (i) the maximum value of the order parameter is one for all compositions8 and that (ii) no constitutional vacancies are created. We will show in a forthcoming paper that the general form of eqn (7) holds also for other ordered structures.9 For the expansion of our model to other structures like A3B
where N,, is the maximum number of atoms per layer, Q the detector efficiency, nA and nB the numbers of A and B atoms detected in a given layer, and PA,B the probability of detecting an A or B atom in a given layer. In the case of an analysis of the fully ordered Llo structure along the < 001 > direction, the two sublattices IXand fi are investigated successively. Hence the probabilities PA and PB are given directly by the site occupation probabilities in the respective sublattices, as given in Table 1.
Calculationof long-range order parameter
The simulations showed that the mean value of the long-range order parameter S is properly estimated by eqn (7). Hence, the fact that N is binomially distributed experimentally need not be considered for the theoretical calculation of 02. This can be understood as follows: 0.~2is estimated basically by counting the total number of AA sequences in the ladder diagram, and not by dividing the data chain into smaller blocks and analyzing the composition of these blocks, as is done for example in the evaluation of correlation coefficients. In the latter case, the bimonial distribution as given in eqn (10) must be considered explicitly. ’ l As the probabilitiespA,i were taken as constant in eqn (3), which means that removal of the atoms was considered to be non-exhaustive, we have investigated the influence of the number of atoms per layer, N max, on the accuracy of the calculated order parameter. It turns out that the latter is estimated correctly for values of Nmaxas low as Nmax = 5. Let us now apply the model to measured ladder diagrams. An L12 ordered FeTzPtzs alloy was analyzed in the < 110 > direction. The homogenized and water-quenched samples were annealed for 1 h at 1173 K and then cooled down to 723 K at a cooling rate of 1 Kh-‘. The results for the determination of the long-range order parameter are summarized in Table 3. For comparison, the order parameter was determined by means of conventional X-ray diffraction techniques.12
613
Table 3. Values of the long-range order parameter S for the ordered alloy Fe&‘t~,~~ obtahd using the model pmented in this article (eqn (7)) couuting FeFe aud PtPt sequemw of length k =2, (this gives L&e and SR, as equal witbin error), comparison with S by means of X-ray diffractometry tecbniques’4 in the same sample SFeFe
Ordered Fer2Ptzs Ordered Fe7zPt2s
SP,P,
0.81 zko.13 0.76hO.13 S = 0.90 f 0.10 by X-rays
The given errors are estimates of the 95% confidence interval.
The results demonstrate the applicability of our analytical method and confirm that no pronounced preferential evaporation effects occur in that alloy. The error limits given in Table 3 are estimates of the 95% confidence interval.13 Next, two near-stoichiometric AlTi alloys were analyzed. Experiments were carried out in the yphase, which has an Ll,, ordered structure.5 Fig. 3 shows an FIM picture of an A1s2Ti4s alloy. The ordered poles may be easily recognized as they appear in bright contrast on the pictures.14 No constitutional vacancies are created in near-stoichiometrical AlTi alloys. l5 An as-cast A154Ti46 alloy was measured, while the A1s2Ti4s alloy was homogenized, quenched, and heat treated for 24 h at 1523 K, below the ordering temperature of about 1733 K of the y-phase. Part of a ladder diagram of this state is shown in Fig. 4. In Table 4, the results of the statistical analysis are summarized. The degree of long-range order was evaluated for aluminium and titanium separately. The error
Al ions Fig. 3. FIM picture of the A152Ti4s alloy (homogenized, quenched, and heat treated for 24 h at 1523 K, below the ordering temperature of about 1733 K of the y-phase). The ordered poles appear in bright contrast on the pictures (Ttip = 5&60 K, imaging gas is neon).
Fig. 4. Part of a ladder diagram of the A1s2T& alloy (homogenized, quenched, and heat treated for 24 h at 1523 K, below the ordering temperature of about 1733 K of y-phase). The long-range order parameter evaluated from the analytical model is S= 0.87 f 0.06.
S. Welzel et al.
614
Table 4. Values of the long-range order parameter obtained using the model presented in this article on the cast alloys AlsTii and AlszTii (homogenized, quenched, annealed at 1573 K for 24 II) &Al
A154T& Als2Ti4s
0~75ItO~lO 0.86 f 0.07
different lengths. elsewhere.9
We will show this procedure
ACKNOWLEDGEMENTS
STiTi
0.75 Ito. 0.87 f 0.07
As in Table 3 sequences of length k = 2 are used for both species. The given errors are estimates of the 95% confidence interval.
limits given in Table 4 are estimates of the 95% confidence interval.13
CONCLUSION
The examples demonstrate the applicability of the proposed method as a convenient tool to determine the long-range order parameter from AP data. Nevertheless it has to be pointed out once again that the presented calculation of the long-range order parameter is valid only under certain conditions. These conditions (see (i) to (iii) in the introduction) have to be checked before every application to any experimental data. As a natural extension, histograms of Ok versus chain length (see eqn (6)) may be calculated and fitted to the actual experimental distribution by means of an appropriate statistical procedure. In that way, the long-range order parameter is deduced from sequences having
We are grateful to J. Jimenez, Max-Planck-Institut Dusseldorf, for providing the value of the order parameter, determined by means of a diffraction technique. We express our thanks to the HMIBerlin-FIM-Group where this work was initiated.
REFERENCES 1. Miller, M. K. and Smith, G. D. W., Atom Probe Microanalysis: Principles and Application to Materials Problems. Materials Research Society, Pittsburgh, 1989. 2. Schulze, E. R., Metallphysk. Akademie-Verlag, Berlin, 1967. 3. Blavette, D., Bostel, A. and Sarrau, .I. M., Metallurgical Transactions A, 1985, 16A, 1703. 4. Duval, S., Chambreland, S., Caron, P. and Blavette, D., Acta Metall. Mater., 1994, 42, 185. 5. Wesemann, J., Frommeyer, G. and Kreuss, M., Appl. Surf Sci., 1995,87/88, 179. 6. Blavette, D., Surf Sci., 1992, 266, 299. 7. Johnson, A. and Klotz, J. H., Technometrics, 1974,16,483. 8. Fowler, R and Guggenheim, E. A., Statistical Thermodynamics. Cambridge Universal Press, Cambridge, 1956. 9. Welzel, S., (in preparation). 10. Macrander, A. T., Yamamoto, M., Seidman, D. N. and Brenner, S., Rev. Sci. Znstrum., 1983, 54(9), 1077. 11. Camus, E. and Abromeit, C., Journal of Applied Physics, 1994,75,2373. 12. Jimenez, J., private communication. 13. Klotz, J. H., The Annals of Statistics, 1973, 1, 373. 14. Liu, J. H., Frommeyer, G. and Kreuss, M., Phys. Stat. Sol. (A), 1992, 131,495. 15. Brossmann, U., Wtirschum, R., Badura, K. and Schafer, H. E., Physics Review B, 1994,49, 6457.
5 (1997) 609-614
0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved PII:
SO966-9795(97)00029-O
0966-9795/97/$17.00
+ 0.00
ELSEVIER
Calculation of the long-range order parameter from atom probe data S. Welzel,“* S. Duval,” E. Camusb & D. Blavette= “Laboratoirede Microscopic Ionique et Electronique, UMR CNRS6634, Universite’de Rouen, F-76821 Mont-Saint-Aignan
Cedex, France
bHahn-iUeitner-Institut Berlin GmbH, Glienockerstr. 100, Bereich NM, D-14109 Berlin, Germany (Received 2 October 1996; accepted 10 March 1997)
In addition to the known diffraction techniques, field-ion microscopy with atom probe is well established for determining the long-range order parameter. The site occupation probabilities of the chemical species on the different sublattices may be estimated from experimental profiles. However, this evaluation method demands that the superstructure planes be identified unequivocally from the data. This condition is not fulfilled in all cases. We propose a new analytical method for which this condition need not be met. 0 1997 Elsevier Science Ltd Keyworcis: A. titanium aluminides, based on TiAI; intermetallics, miscellaneous, B. order/disorder transformation, D. site occupation probability, F. atom probe.
INTRODUCTION
measurement of composition in an ordered alloy, which is described by the long-range order parameter S, along a superstructure direction, reveals the atomic order of the components on different sublattices as an abrupt change in chemical composition.2 The site occupation probabilities pa and pg of the components on the sublattices (r and p of an Lla structure are given as a function of S in Table 1 as an example. A detailed description of the method, which can be used to derive pa and pg from the mean composition of superstructure planes, is given elsewhere.3 The result of an atom probe experiment is a data chain composed of successive collected ions. Ladder diagrams are commonly used for a representation of the measurement. In such a data representation, the cumulative number of detected ions of a given species may be plotted as a function of the total number of detected ions. This number is approximately proportional to the probed depth. Such a diagram is shown in Fig. l(a), where an ordered FeT2Pt2s alloy with L12 structure was analyzed in the < 110 > direction. Taking into account a detector efficiency of 60%, the diameter of the analyzed area is about 1 nm, i.e. 10 ions are expected per layer. The variations of the slope, i.e. the local concentration on such a scale, are sufficient to distinguish between pure iron layers (horizontal
The field-ion microscope with atom probe (FIMAP) enables one to determine the chemical composition of materials on an atomic scale.’ The experimental technique is based on field evaporation. A potential of several kilovolts applied to a specimen needle having a radius of curvature of typically 10 to 100 nm is large enough to extract atoms from the tip surface. This field evaporation process is induced by a nanosecond high voltage pulse which starts a timer. The timer is stopped by any ion reaching the detector. Hence, the field-evaporated ions may be identified by means of time-of-flight spectrometry. As a diaphragm is interposed between specimen and detector, the analysis of a well defined region of about 1 nm in diameter of the specimen may be performed.’ During the experiment, the specimen is continuously field evaporated. In this way, in-depth analysis of the specimen is performed. Under certain conditions, the ultimate depth resolution of one atomic layer can be achieved, allowing a plane by plane investigation of the specimen. The *To whom correspondence should be addressed at: HahnMeitner-Institut GmbH, Bereich NE, Glienickerstr.100, D-14109 Berlin, Germany. 609
S. Welzel
610
Table 1. Site occupation probabilities pa and pp of A and B atoms on the sublattices Q and /3 as a function of the long-range order parameter S in the case of a stoichiometric LIO structure
Sublattice a! A B
Sublattice /l pA,b=f(l-S)
PA.a =+'I +s) pB,a = 2 (1 - s)
PB,B=2(l+S)
II’
I
III
: II
III
I II
II
, 50
at ,
Fe+Pt ions 64 I
I
I
I
Number of Pulses; @I Fig. 1. Typical ladder diagrams for the Lls ordered FersPtzs
alloy measured along the < 110 > direction. The dashed lines mark a sequence of five atomic layers. (a) Number of detected Pt atoms as a function of the cumulative number of detected Fe and Pt atoms. The alternation of Pt-depleted and P&rich layers (1 layer contains approximately 10 atoms) is visible. (b) Corresponding evaporation rate diagram. Ideally, the end of the evaporation of each plane should be revealed by the large number of pulses applied during which no Pt atoms are collected. It is obvious from this diagram that the univocal identification of each layer is not unambigous (see asterisked part).
et al. parts) and mixed Fe-Pt layers (inclined parts). In Fig. l(b), the steplike behavior of the arrival rate reveals that a plane-by-plane investigation was carried out. The site occupation numbers and the order parameter S are deduced from the mean composition of superstructure planes,3p4 or graphically from counting antisite defects in the pure iron layer. ’ To obtain reliable results using such a method, the following conditions must be fulfilled. (i) The atom probe analysis must be carried out layer by layer on a superstructure pole, while ensuring that each layer is completely field evaporated before detection of the next takes place.3 (ii) Each layer must be identified univocally from the data chain. The latter was until now done in one of two ways: (a) by identifying the end of each evaporating plane from the modulations of the arrival rate, as described above. As the evaporation of the layers is irregular - see for example the part of Fig. l(b) marked with an asterisk - the localization of the end of each evaporating plane is not always possible everywhere; (b) by recording interactively the end of each layer using FIM images during the whole experiment. This was done recently by Wesemann et al. in a video-controlled atom probe investigation.5 However, the method of analysis is hardly practicable for the long profiles required to achieve sufficient statistical accuracy. (iii) Any severe preferential evaporation or retention effects must not occur for any of the atom species.6 In the present contribution, we propose an analytical method to evaluate the order parameter, in which there is no need to identify each individual layer from the measured data chain. The calculation may be performed automatically without any tedious and time-consuming hand treatment of atom probe data. We simply count the number of sequences containing a fixed number of consecutively detected atoms of one given species. We show that it is enough to count sequences containing two atoms to evaluate the long-range order parameter. In the theoretical model presented in the following, we shall assume that the ions do evaporate randomly within each layer.*
*As noted by Miller and Smith,’ an experimental fact has to be taken into account. Atom probes are equipped with detectors that can detect up to eight ions on a single pulse. Due to the time-of-flight principle of the measurement, the lighter ions will reach the detector first. This leads to a bias in the randomness of the data chain. To overcome this problem, we have rearranged these multiple ions by computing random permutation before treating the data chain.
Calculation of long-range order parameter
611
PROBABILISTIC DESCRIPTION OF DATA CHAINS If the atoms within each individual plane of a plane-by-plane analysis are assumed to evaporate randomly, the data chain of A and B atoms of a disordered material is a random chain. Given a certain degree of long-range order, longer chains of consecutive detection of A and B atoms are expected. The question is to evaluate the probability P(Ak) to detect (k - 1) A ions after an A ion has been detected. In the case of a random distribution, it is obvious that this probability equals Ci-‘, where CA is the mean concentration of A atoms. If long-range order occurs in the alloy, then P(Ak) > Ci-‘. To find a measure for this enhancement, we define wk = -P(Ak) ck- 1
(1)
A
where Ak is a sequence of k consecutive A atoms. The parameters Ok, which depend on the degree of order, are one if the material is disordered and are greater than one when some order is present. Let us first consider only chains of length k = 2, for the sake of simplicity. For this special case, eqn (1) reads w2
=
p(A2) CA
(2)
where A2 is a sequence of two successively collected A atoms. Let us now consider the case of a plane-by-plane analysis of the L10 ordered structure along the < 001 > tetragonal axis. Thus, the mean composition of each type of plane is equal to the occupation frequencies of a! and jI sites (P_,+,PA,B). We consider first the case where S = 1. Figure 2 shows an idealized ladder diagram of the fully ordered Ll,, structure, obtained by such plane-byplane analysis. It is assumed that each plane contains a constant number N of atoms. The validity of this assumption will be discussed in the next section. This idealized ladder diagram is divided into four parts: parts 1 and 3 are these where either a pure A or a pure B layer is detected, and parts 2 and 4 are regions where the transition from an A to a B layer or vice versa takes place. Let us define the probability P(A2) of finding an A ion when the previous ion is also an A ion. It is
A ions Fig. 2. Idealized ladder diagram for an Lla fully ordered A5eB5s alloy (S= 1). A ions are closed circles and B ions are open circles. The number N of ions detected per layer is assumed to be constant. The two successive detected layers are divided into four parts to take into account the different probabilities of detecting A atoms within the layers and at the interface between the layers.
the weighted sum of the partial probabilities PAJ in the parts i, i = 1 to 4. It reads: 5
P(A2)
nA,i
‘pA,i
= ‘=I 4 g
(3) n.‘Q
where nA,i is the number of A ions involved in the respective parts. As noted above, the probability P(A2) equals the mean concentration (CA = 4) for the case of a random alloy. For a fully ordered structure of stoichiometric composition, as shown in Fig. 2, the site occupation probabilities are equal to one or zero and the numbers of ions involved A nA,j are summarized in Table 2. w;! is expressed as
w2 > w&S= 1) =7=2-k=
@?.,max
(4)
A
where W,max is the maximum value of w2 and depends only on the number N of ions collected per layer. The general case of partly ordered L10 structure with the order parameter S smaller than unity may be deduced directly from eqn (3) by introducing the probabilities PA,i as given in Table 1. This results in the following expression for the parameter w:
S. Welzel et
612
Table 2. Number “Aj of A atoms involved and probability PM of lkling an A atom after an A atom had been detected, for parts 1 to 4 as shown in Fig. 2 for a fally ordered alloy Parts i
n.4,i
PA&
Note that, analogous to eqn (5), the general form for a k sequence &(k -c N - 1) can be deduced as k-l wk+m&+P~,g)+2
al.
alloys with L12 structure the main point is that we are dealing with a similar situation to that in Llo structures: an alternating sequence of layers. But in this case of AsB we have to take into account probabilities PA,~ and pA,m, for pure and mixed layers, of finding an A atom in the pure or in the mixed plane; these are calculated from the site-occupation probabilities for L12. Insert PAp and pA,m in expression (3) for P(A2) and eqn (7) follows with respect to eqn (4) and eqn (9). The value of o,, is presented in eqn (9). In particular, for the case of an off-stoichoimetric L12 ordered structure of type Fes_XPtl+, (which is treated in the next section) measured in the < 100 > or direction, it holds that:
p&$htdk-i
ok =
w
N*(PA,a +pA,~)k-l. c;-'
,max=
1 + ($)‘.(J lo-z. 9
9N
- &J
forspeciespt for species Fe
(6)
By inserting the site occupation probabilities ~A,(Y and PA,@as given in Table 1, into eqn (S), the order parameter can be derived from 02 as follows:
J
co.2 -
1
s(w)= *,_ -
(7)
1
Hence, S(m) depends monotonically on q and ranges, as expected, between 0 and 1. 02 is obtained from experiment7 and q,max can be calculated from eqn (4). This can be done for A and B atoms. As a consequence the long-range order parameter is measured out of AA or BB sequences. The results summarized in Tables 3 and 4 take this into account. In the case of an off-stoichiometric Llo alloy At+xBr-x, eqn (7) remains valid with the following expression for the maximum value of ~02:
(9) DISCUSSION
Two assumptions were made to derive eqn (7): (i) the number N of atoms per layer is constant for any layer, and (ii) removal of the atoms from the layer does not change the composition of the remaining layer. Simulations are an appropriate tool to test the relevance of these assumptions. As noted in the previous section, N is not constant if the efficiency of the detector is non-ideal. In that case the number N = nA + ns of atoms detected per layer is distributed binomially. lo We have simulated ladder diagrams by generating random deviations from a multinomial distribution: P(nA, OB) =
NM! IZA!FZB!(N,,- TZA - ~zB)!
x (QPA)""(QPB)"~.
1 + (fi)2. W,max =
2-i
(1 -h)
forspeciesA
(1 - Q)(N~x-nA-nB)
(10)
for species B
(8) where x > 0 is the deviation from the stoichiometry. To derive these equations, we have supposed that (i) the maximum value of the order parameter is one for all compositions8 and that (ii) no constitutional vacancies are created. We will show in a forthcoming paper that the general form of eqn (7) holds also for other ordered structures.9 For the expansion of our model to other structures like A3B
where N,, is the maximum number of atoms per layer, Q the detector efficiency, nA and nB the numbers of A and B atoms detected in a given layer, and PA,B the probability of detecting an A or B atom in a given layer. In the case of an analysis of the fully ordered Llo structure along the < 001 > direction, the two sublattices IXand fi are investigated successively. Hence the probabilities PA and PB are given directly by the site occupation probabilities in the respective sublattices, as given in Table 1.
Calculationof long-range order parameter
The simulations showed that the mean value of the long-range order parameter S is properly estimated by eqn (7). Hence, the fact that N is binomially distributed experimentally need not be considered for the theoretical calculation of 02. This can be understood as follows: 0.~2is estimated basically by counting the total number of AA sequences in the ladder diagram, and not by dividing the data chain into smaller blocks and analyzing the composition of these blocks, as is done for example in the evaluation of correlation coefficients. In the latter case, the bimonial distribution as given in eqn (10) must be considered explicitly. ’ l As the probabilitiespA,i were taken as constant in eqn (3), which means that removal of the atoms was considered to be non-exhaustive, we have investigated the influence of the number of atoms per layer, N max, on the accuracy of the calculated order parameter. It turns out that the latter is estimated correctly for values of Nmaxas low as Nmax = 5. Let us now apply the model to measured ladder diagrams. An L12 ordered FeTzPtzs alloy was analyzed in the < 110 > direction. The homogenized and water-quenched samples were annealed for 1 h at 1173 K and then cooled down to 723 K at a cooling rate of 1 Kh-‘. The results for the determination of the long-range order parameter are summarized in Table 3. For comparison, the order parameter was determined by means of conventional X-ray diffraction techniques.12
613
Table 3. Values of the long-range order parameter S for the ordered alloy Fe&‘t~,~~ obtahd using the model pmented in this article (eqn (7)) couuting FeFe aud PtPt sequemw of length k =2, (this gives L&e and SR, as equal witbin error), comparison with S by means of X-ray diffractometry tecbniques’4 in the same sample SFeFe
Ordered Fer2Ptzs Ordered Fe7zPt2s
SP,P,
0.81 zko.13 0.76hO.13 S = 0.90 f 0.10 by X-rays
The given errors are estimates of the 95% confidence interval.
The results demonstrate the applicability of our analytical method and confirm that no pronounced preferential evaporation effects occur in that alloy. The error limits given in Table 3 are estimates of the 95% confidence interval.13 Next, two near-stoichiometric AlTi alloys were analyzed. Experiments were carried out in the yphase, which has an Ll,, ordered structure.5 Fig. 3 shows an FIM picture of an A1s2Ti4s alloy. The ordered poles may be easily recognized as they appear in bright contrast on the pictures.14 No constitutional vacancies are created in near-stoichiometrical AlTi alloys. l5 An as-cast A154Ti46 alloy was measured, while the A1s2Ti4s alloy was homogenized, quenched, and heat treated for 24 h at 1523 K, below the ordering temperature of about 1733 K of the y-phase. Part of a ladder diagram of this state is shown in Fig. 4. In Table 4, the results of the statistical analysis are summarized. The degree of long-range order was evaluated for aluminium and titanium separately. The error
Al ions Fig. 3. FIM picture of the A152Ti4s alloy (homogenized, quenched, and heat treated for 24 h at 1523 K, below the ordering temperature of about 1733 K of the y-phase). The ordered poles appear in bright contrast on the pictures (Ttip = 5&60 K, imaging gas is neon).
Fig. 4. Part of a ladder diagram of the A1s2T& alloy (homogenized, quenched, and heat treated for 24 h at 1523 K, below the ordering temperature of about 1733 K of y-phase). The long-range order parameter evaluated from the analytical model is S= 0.87 f 0.06.
S. Welzel et al.
614
Table 4. Values of the long-range order parameter obtained using the model presented in this article on the cast alloys AlsTii and AlszTii (homogenized, quenched, annealed at 1573 K for 24 II) &Al
A154T& Als2Ti4s
0~75ItO~lO 0.86 f 0.07
different lengths. elsewhere.9
We will show this procedure
ACKNOWLEDGEMENTS
STiTi
0.75 Ito. 0.87 f 0.07
As in Table 3 sequences of length k = 2 are used for both species. The given errors are estimates of the 95% confidence interval.
limits given in Table 4 are estimates of the 95% confidence interval.13
CONCLUSION
The examples demonstrate the applicability of the proposed method as a convenient tool to determine the long-range order parameter from AP data. Nevertheless it has to be pointed out once again that the presented calculation of the long-range order parameter is valid only under certain conditions. These conditions (see (i) to (iii) in the introduction) have to be checked before every application to any experimental data. As a natural extension, histograms of Ok versus chain length (see eqn (6)) may be calculated and fitted to the actual experimental distribution by means of an appropriate statistical procedure. In that way, the long-range order parameter is deduced from sequences having
We are grateful to J. Jimenez, Max-Planck-Institut Dusseldorf, for providing the value of the order parameter, determined by means of a diffraction technique. We express our thanks to the HMIBerlin-FIM-Group where this work was initiated.
REFERENCES 1. Miller, M. K. and Smith, G. D. W., Atom Probe Microanalysis: Principles and Application to Materials Problems. Materials Research Society, Pittsburgh, 1989. 2. Schulze, E. R., Metallphysk. Akademie-Verlag, Berlin, 1967. 3. Blavette, D., Bostel, A. and Sarrau, .I. M., Metallurgical Transactions A, 1985, 16A, 1703. 4. Duval, S., Chambreland, S., Caron, P. and Blavette, D., Acta Metall. Mater., 1994, 42, 185. 5. Wesemann, J., Frommeyer, G. and Kreuss, M., Appl. Surf Sci., 1995,87/88, 179. 6. Blavette, D., Surf Sci., 1992, 266, 299. 7. Johnson, A. and Klotz, J. H., Technometrics, 1974,16,483. 8. Fowler, R and Guggenheim, E. A., Statistical Thermodynamics. Cambridge Universal Press, Cambridge, 1956. 9. Welzel, S., (in preparation). 10. Macrander, A. T., Yamamoto, M., Seidman, D. N. and Brenner, S., Rev. Sci. Znstrum., 1983, 54(9), 1077. 11. Camus, E. and Abromeit, C., Journal of Applied Physics, 1994,75,2373. 12. Jimenez, J., private communication. 13. Klotz, J. H., The Annals of Statistics, 1973, 1, 373. 14. Liu, J. H., Frommeyer, G. and Kreuss, M., Phys. Stat. Sol. (A), 1992, 131,495. 15. Brossmann, U., Wtirschum, R., Badura, K. and Schafer, H. E., Physics Review B, 1994,49, 6457.