STATISTICAL CORRECTION FOR PILE-UP IN THE ATOM-PROBE DETECTOR SYSTEM

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STATISTICAL CORRECTION FOR PILE-UP IN THE ATOM-PROBE DETECTOR SYSTEM

U. Rolander, H.-O. Andrén

To cite this version:

U. Rolander, H.-O. Andrén. STATISTICAL CORRECTION FOR PILE-UP IN THE ATOM- PROBE DETECTOR SYSTEM. Journal de Physique Colloques, 1989, 50 (C8), pp.C8-529-C8-534.

�10.1051/jphyscol:1989891�. �jpa-00229990�

COLLOQUE DE PHYSIQUE

Colloque C 8 , Suppl6ment au n o l l , Tome 50, novembre 1989

STATISTICAL CORRECTION FOR PILE-UP I N THE ATOM-PROBE DETECTOR SYSTEM U. ROLANDER and H. -0. A N D R ~ N

Department of Physics Chalmers University o f Technology, S-412 96 Gbteborg, Sweden

Abstract - A procedure is described which can be used to correct atom-probe data for losses due to the detector dead time. The accuracy of the procedure is verified by comparing the experimental isotope distributions of Fe and C with tabulated values. It is also shown that a tilted detector can be used to increase the multi-hit detectivity.

Introduction

Modern atom-probes, which are equipped with energy compensators, exhibit time resolutions of the same order of magnitude as the multi-hit dead time of the electronics used in the time-of- flight measurements. Since there is a finite probability that more than one ion with a certain mass-to-charge ratio is evaporated by the same pulse, this necessarily means that ions are systematically lost due to pile-up in the detector system.

The problem is well known and three different solutions have been proposed previously.

Blavette et a1 [I] have chosen to increase the multi-hit detectivity by the use of four parallel detectors. Tsong et a1 [2] and Cerezo et a1 [3] instead propose procedures for statistical correction of atom-probe analyses.

In this work a method for statistical correction is proposed. The method has the advantage that all species (the term is used to denote all isotopes of the same element and charged state) are corrected independently. It is also shown that a 30' tilted detector can be used to introduce an artificial spread in flight time, and thus increase the multi-hit detectivity, in a controlled manner.

Experimental

The atom-probe used in this investigation has been described in detail previously [4,51. A Poschenrieder type energy compensator is used to increase the mass resolution of the system.

Flight times are measured with 5 ns resolution over a flight distance of 1.6 m, using four parallel counters. The flight times are converted into equivalent mass to charge ratios and stored with a resolution of 0.01 u (atomic mass units). The multi-hit dead time of the detection system has been measured previously [ 6 ] . It was shown that two ions which generate pulses with a time difference of less than 9 ns are detected as one ion.

Statistical correction procedure

In general, the probability for simultaneous evaporation of two ions of a specific element and charged state is not a function simply of the abundance of those ions in the acquired spectrum and the mean field evaporation rate. Instead, parameters such as the local composition and electric field at each instance during the analysis, and the chemical nature and field evaporation behavior of that element may play an important role (See e.g. Wagner and Seidman [7]).

However, given that two ions have been evaporated simultaneously and have generated two pulses in the detector, the probability that the time difference between these two pulses is larger than the detector dead time, and thus both ions detected, can be calculated. This is done using the acquired mass spectrum as an experimentally determined statistical distribution.

Consider an atom-probe analysis where the ions are stored with a resolution of 0.01 atomic mass units. From this analysis a mass spectrum with 100 channels per atomic mass unit can be constructed. Now consider only one section of this spectrum, ranging from channel m,i, to

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989891

channel mma,, which contains all isotopes of one species and no other ions. If one ion has generated a pulse in the detector, the probability , Pi, that it will arrive in channel "i" is,

where ni is the number of ions in channel "i" and N is the total number of ions in the range m,in to mmax. The probability for two ions to arrive in channels "i" andj" respectively is,

where Am is the mass to charge equivalent of the detector dead time, At.

Now the probability, I'det, of detecting both of two ions which have generated two pulses in the detector can be written,

In order to use this equation, an expression for Am also'has to be derived. At a certain mass to charge ratio, m, and field evaporation voltage, V, energy conservation gives,

where e is the electronic charge, 1 is the flight distance and t is the flight time. This can be rewritten as,

Consider a small time difference At and an equivalent mass to charge difference Am, 2 e V

m + A m = - ( t + ~ t ) ~

l2 (6)

Using equations (5) and (6) Am can be written,

For all values of interest here (1 u I m 1300 u, 1 kV I V I 30 kV, At=9ns, 1=1.6m) the first term in equation (7) is at least 164 times larger than the second term. Therefore, the approximate relation,

may be used. Since Am varies rather slowly with m, the mean mass to charge ratio of the ions in the selected range, mmin to mmax, can be used to estimate m. For the same reason, for moderate changes in voltage, the mean voltage of the analysis can be used instead of V. For large increments in voltage the analysis can always be divided into smaller parts and Pdet found for each part separately. If this is done, however, it is important that each part is sufficiently large so that it can still be thought of as a statistically significant distribution.

Once Pdet is known, the total number of events, n, where two ions of a certain species have generated two pulses in the detector can be estimated using,

where ndet is the detected number of such events and s is one standard deviation for a binomial distribution,

The number of lost ions, nlost is then,

By repeating the procedure for each species in the spectrum the whole analysis can be corrected.

In some cases, such as when the isotope distribution must be used to deconvolute overlaps, it is not sufficient to know the total number of lost ions for a certain species. Instead one would like to know how many ions were lost for each isotope. In these cases it is of course possible to make the correction for each peak separately. Usually, however, more accurate results are obtained by using the total number of events, n, in equation (9) and the assumption that all events involving two ions of different isotopes are detected. With this assumption the total number of events concerning two ions from peak "a", n,, is

where Pa is the natural abundance of isotope "a" if no overlap exists in any of the peaks.

Otherwise Pa is the fraction of all ions in the range mmin to mmax which belongs to peak "a".

Each peak from the same species can then be corrected separately using an expression similar to equation (11).

So far, only events involving two ions have been considered. Equation (31, however, can easily be generalized to concern three or more ions. This has not been done for two reasons:

i) the detected number of such events has been found to be so small that they can not be considered to be a reliable statistical population; ii) the computing times will increase drastically.

Events involving more than two ions of the same species have instead been taken into account by thinking of three ions as two pairs and four ions as three pairs. Since the probability of detecting two or three pairs is larger than the probability of detecting events with three or four ions respectively, the importance of such events will be underestimated but the correction will go in the right direction.

One generalization which has been made concerns situations where two species have isotope distributions which are intertwined. In such cases values of i and j in equation (3) which correspond to ions of the other species must be excluded from the sums. If this is done all species can still be corrected separately.

Effect of a 30" tilted detector

In some cases, for instance investigations involving elements with only one isotope, the probability of detecting a pair from a specific species will be rather low. As seen from equation (9) this will lead to large statistical errors. One way to improve matters is to decrease the resolution of the atom-probe in a controlled manner. This can be done by tilting the detector as shown in figure 1. The Poschenrieder type energy compensator used here behaves as if it were slightly astigmatic. The effect of this is that ions arriving at the detector can be focused into anything from a vertical line via a small disc and to a horizontal line, by changing the flight distance of the ions through the energy compensator.

The use of a 30° tilted detector and a 2 cm vertical line focus will introduce a spread in flight distance of 1 cm. At a field evaporation voltage of 10 kV and a mass to charge ratio of 24 u this will lead to a spread in flight time of 35.3 ns, which is equivalent to a spread in mass to charge of 0.30 u according to equations (5) and (8). The resolution obtained in this manner is often fully acceptable. If a better resolution is needed, it can be increased continuously by simply decreasing the length of the line focus. If the ions are focused into a small disc or a horizontal line the tilting of the detector has little or no effect on the resolution.

Figure 2 shows Ti2+ peaks as detected with (a) O0 tilt, (b) 30° tilt and the ions focused into a small disc and (c) 30' tilt and the ions focused into a 2 cm vertical line. The peaks are fully resolved in all cases but they are considerably broader in (c) than in (a) and (b). The probability of detecting pairs (Pdet ) for the Ti3+ and Ti2+ peaks in the three cases are listed in table 1. As can be seen Pdet increases significantly when the line focus is used. If only one peak is considered the relative increase in multi-hit detectivity is especially large. Thus, as expected, the effect of the tilted detector is more prominent when elements with only one isotope or one dominating isotope are analysed.

Effect of the statistical correction procedure

The effect of the statistical correction procedure on the isotope distribution of Fez+ is shown in table 2. In the uncorrected case the largest isotope is significantly too small while in the corrected case the isotope distribution is very close to the correct one. The errors given are maximum errors, that is the sum of the usual statistical error and the error of the coryection procedure for each peak. The analysis was done using a O0 tilted detector. The probability of detecting a pair was 23.38 % and 230 pairs were detected.

The accuracy of the correction procedure was also verified by adding all carbon ions from several analyses of titanium carbide and comparing the obtained result with the natural abundance of 13C. Without correction, 1.22 % of the ions were detected as 13C. When all analyses were corrected and then added, this figure decreased to 1.09 %. This actually differed by less than one ion from the correct answer (1.10 %).

Conclusions

A method for statistical correction of atom-probe analyses for losses due to the detector dead time has been developed.

The method has the advantage that all species are corrected independently.

A comparison of the corrected isotope distributions and the natural abundances of Fe and C isotopes shows that the method leads to correct results.

A tilted detector can be used to increase the multi-hit detectivity in a controlled manner.

The effect of the tilted detector is more promiilent for elements with only one isotope or one dominant isotope.

Acknowledgements

This work has been financially supported by the National Swedish Board for Technical Development (STU) and AB Sandvik Coromant. Bertil Josefsson is thanked for providing the Fez+ spectrum.

References

[I] D. Blavette, A. Bostel and J. M. Sarrau, J. Phys. (Paris), Colloq. C2,47 (1986) 473 [2] T. T. Tsong, Yee S. Ng and S. V. Krishnaswamy, Appl. Phys. Lett. 32 (1978) 778 [3] A. Cerezo, G. D. W. Smith and A. R. Waugh, J. Phys. (Paris), Colloq. C9,45 (1984) 329 [4] H. 0. Andr6n and H. Nordkn, Scand. J. Met., 8 (1979) 147

[5] H. 0. Andrbn, J. Phys. (Paris), Colloq. C7,47 (1986) 483

[6] U. Rolander and H. 0. Andrkn, J. Phys. (Paris), Colloq. C6,49 (1988) 299 A. Wagner and D. N. Seidman, J. Phys. (Paris), Colloq. C2,47 (1986) 415

Table 1. The effect of a 30" tilted detector on the multi hit detectivity of Ti.

Table 2. The effect of the correction procedure on the isotope distribution of Fe2+.

Pdct = 23.38 %, ndet = 230 pairs.

masslcharge

I

I

I

I

Detector tilt

0°

30"

30"

Pdet ( Ti3+ )

shape 3 mm disc 2 cm line

all isotopes 50.27 % 58.04 % 66.83 %

mean voltage

8.38 kV 7.59 kV 10.40 kV main isotope

14.59 % 16.97 % 39.55 %

Pdet ( Ti2+ )

all isotopes 51.04 ^% 59.77 % 73.97 %

main isotope 17.10 ^% 26.37 ^% 53.30 %

Figure 1. Schematic diagram of the tilted microchannel plate ion detector.

mass to charge

Figure 2. Ti2+ peaks as detected with (a) O0 tilted detector, (b) 30° tilt and the ions focused into a small disc and (c) 30° tilt and the ions focused into a 2 cm vertical line. The spectra contain approximately 500 ions each.