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An investigation on the distribution of counts in positron annihilation lifetime spectra
Nuclear Instruments and Methods in Physics Research A 417 (1998) 377—383
An investigation on the distribution of counts in positron annihilation lifetime spectra G. Consolati*, M. Stefanetti Istituto Nazionale di Fisica della Materia, Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci, 23-20133 Milano, Italy Received 13 March 1998
Abstract Positron annihilation lifetime spectra with a high number of counts give useful information for various purposes (like accurate determination of distributions of lifetimes or a comparison among competing models). This requisite, coupled to the need of a good resolution, implies long times of measurement; the consequent instrumental drifts can degrade the quality of the data. In this paper a statistical analysis of spectra of polyethylene terephthalate is presented, which reveals that the distribution of counts displays deviations from the expected Poissonian behaviour, as a consequence of the jitter of the time zero. By means of a simple method for the compensation of such fluctuations it is possible to recover almost ideal statistical properties. 1998 Elsevier Science B.V. All rights reserved. PACS: 71.60.#z; 77.84.Jd; 78.70.Bj Keywords: Positron annihilation; Lifetime spectra; Counts; Polyethylene tetraphthalate
1. Introduction Positron annihilation lifetime spectroscopy is widely used for the investigation and characterization of solid-state materials [1], since the different positron states extracted from a timing spectrum can be correlated to structural defects in metals [2], ceramics [3] and semiconductors [4], as well as to nanovoids forming the free volume in polymers [5]. Furthermore, the presence of phase transitions can be conveniently observed by monitoring the
* Corresponding author. Tel.: #39 2 23996123; #39 2 23996126; e-mail: [email protected].
fax:
positron parameters (lifetime and intensity) as a function of the temperature [6]. In all these cases a total number of annihilation events registered in each spectrum of the order of 10 can be sufficient to obtain the required results. Conversely, a higher statistics must be accumulated to get a distribution of lifetimes with good accuracy, as in the case of ortho-positronium (o-Ps) annihilation in the nanocavities of a macromolecule [7,8]. The comparison among different models is another situation where it is necessary to obtain a high number of counts. As an example, it has been recently pointed out [9] that a possible non-exponential decay of Ps in powders, consequent to its guessed thermalization, would nevertheless produce an
0168-9002/98/$19.00 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 0 7 4 7 - 5
378
G. Consolati, M. Stefanetti/Nucl. Instr. and Meth. in Phys. Res. A 417 (1998) 377—383
acceptable s-test for spectra having a few 10 counts and deconvoluted through classical algorithms using exponential components. In order to observe a clear worsening of the s an increase of the total count over 10 should be needed. The requirement of a high number of counts, combined with a good resolution of the apparatus, can involve long running times (up to several days) for the experiment; the unavoidable long-term drifts of the instrumentation degrade the quality of the data. By using a different technique, the single-photon timecorrelated method, Lami [10] showed that the consequence of such drifts is that the events are distributed according to a non-poissonian statistics; in the case of positron lifetime spectroscopy, this could imply systematic errors in the usual computer deconvolution routines, based on a stest which indeed is supposed to operate with poissonian events. With the aim to probe in a quantitative way this topic we analysed the statistical properties of a set of positron annihilation lifetime spectra of a polymer, polyethylene terephthalate, PET, collected under identical conditions. Anyway, the conclusions we reached are quite general and do not depend on the chosen material. We found that the jitter of the time zero is the main responsible of the deviations from the ideal statistical behaviour. Although it is possible to carry out a digital stabilization of the spectrum, such a procedure is anything but trivial as far as lifetime spectroscopy is concerned and it is not adopted by most of the laboratories. Nevertheless, the phenomenon can be reduced to an acceptable level through simple tricks as well as by means of a straightforward compensation method.
2. Experimental The positron source consisted of a droplet of Na from a carrier-free neutral solution, dried onto two identical Kapton foils (thickness 1.08 mg cm\), which were afterwards glued together by a thin layer of a cyanoacrylate glue (Loctite 407 by Loctite Corporation). Care was taken to prevent the glue from reaching the region involved in annihilation. The activity of the source was about 0.3 MBq. Positron spectra were collected through
a conventional fast—fast coincidence apparatus: the c-rays corresponding to the emission of the positron (“start”) and to the annihilation event (“stop”) were detected by two NE111 plastic scintillators, whose dimensions (1 ;1 and 1 ;0.5 , for the start and the stop channels, respectively) ensured a reasonable compromise between resolution and efficiency of the spectrometer. The scintillators were coupled to Philips XP2020 photomultiplier tubes. The resolution of the apparatus resulted to be about 200 ps; it was obtained through the deconvolution of timing spectra of a Bi source, at Na energy windows settings. Positron spectra extended over 2048 channels; conversion * was 10 ps/ch. Polyethylene terephthalate (PET) was used as the test material in the present investigation; it was purchased from Aldrich as pellets, which were compression moulded at 550 K in the form of discs (diameter 20 mm, height 2 mm). The density of the sample was 1.375 g cm\ the degree of crystallinity was estimated to be about 60%. Measurements were carried out at room temperature, where the polymer is glassy (¹ "356 K). The temperature of the laboratory was stable within 1°. 100 spectra were collected under identical conditions; each one contained exactly 10 counts. They were analysed through the Positronfit program [11].
3. Results and discussion In a positron annihilation lifetime spectrum the number of counts contained in the kth channel is expected to be a realization of a Poisson process with mean value 1X(k)2"p(k)N, where p(k) is the probability to find a count in the interval [k*, (k#1)*], * is the amplitude of the channel and N is the total number of counts. Such a property was checked on our data through a s-test; in order to explain the method we consider first of all the background, since in the time interval where only the background is present we can regard the counts as identical repetitions of the same statistical process. Fig. 1 shows the frequencies of the counts measured in the 500 channels containing the background in a single spectrum of PET (open circles), with mean value k"7.33; the dots are the
G. Consolati, M. Stefanetti/Nucl. Instr. and Meth. in Phys. Res. A 417 (1998) 377—383
Fig. 1. Measured frequencies (open circles) of the counts in the channels corresponding to the background (average value k"7.33). Dots: expected values of a Poisson distribution with mean value k. The continuous line is an interpolation of the Poisson distribution. The vertical lines correspond to the six intervals chosen for the s-test.
expected values of a Poisson distribution with the same mean k. The s-test was carried out by classifying the data within six intervals, whose extreme were [0, k!2(k, k!(k, k, k#(k, k# 2(k,R] [12]; they correspond to the vertical lines in Fig. 1. It resulted in s"1.5, corresponding to a probability P(s51.5)"0.2; we conclude that the distribution of counts in the interval containing the background is Poissonian, in a very good approximation. The same test has been performed for the channels containing the lifetime spectrum of PET, by taking into account the “ensemble” of the 100 spectra, having the same number of counts N"10. The 100 data corresponding to the same channel supply an average value k, which obviously depends on the channel, and are distributed into the six intervals defined above. In stationary conditions we should expect 100 repetitions of the same Poisson process in each channel; ideally, the counts in different channels should be independent. Fig. 2 shows the s-test for each channel (full squares): it is clear that in the two regions surrounding the peak s 1. Therefore, the distribution in these channels is not Poissonian. An alternative method for the check of the distribution of counts consists of comparing the ratio R between standard deviation and mean value of the counts in each channel [13]: for
379
Fig. 2. s-test for each channel of the 100 PET spectra. Full squares: original (non-shifted) spectra; open circles: shifted spectra. Only the first 500 channels are reported.
a Poisson distribution it should result R"1. This analysis led to a conclusion identical to that obtained with the s and it is not reported here. The investigation on the reasons of deviations of part of the data from the Poissonian behaviour is conveniently carried out by means of the correlation matrix [10]: S HI , R " HI (S S HH II where
(1)
1 L S " [X (k)!1X(k)2][X ( j)!1X( j)2] HI n!1 G G G (2) is the covariance matrix. In Eq. (2) X (k) is the G random variable corresponding to the kth channel of the ith spectrum, 1X(k)2" L X (k)/n, G G 1[X(k)!1X(k)2]2" L [X(k)!1X(k)2]/(n G !1), and n"100. In stationary conditions and in the absence of fluctuations it is expected that R "d : indeed, owing to the independence of the HI HI processes X( j) and X(k), the signs of the terms appearing in (Eq. (2)) are positive and negative with the same likelihood and their contribution becomes smaller, by increasing n. The result for the PET spectra is displayed in Fig. 3a, where different grays correspond to different degrees of correlation (white: #1, black: !1). Large deviations from the ideal behaviour occur in the regions contiguous to
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G. Consolati, M. Stefanetti/Nucl. Instr. and Meth. in Phys. Res. A 417 (1998) 377—383
Fig. 4. A schematic example showing the fluctuation of t . X: average spectrum; X : a particular spectrum. It turns out that I R and R '0, R and R (0. GH FI HF GI
Fig. 3. Correlation matrix for the 100 spectra of PET: (a) nonshifted, (b) shifted spectra. The degree of correlation is shown as different grays (white: #1, black: !1).
the peak, as already found by means of the s-test; in particular, positive correlation for a pair of channels preceding the peak or subsequent to it, and negative correlation for channels lying on opposite sides with respect to the peak. We can attribute the correlations to fluctuations of the time zero t ; indeed, let us suppose, for instance, that the kth spectrum X is shifted with respect to the average I spectrum X as in Fig. 4. Then, the pairs of channels (i, j) and (h, k) give a positive contribution to the correlation matrix; on the other hand, it is evident Eqs. (2) and (3) that R and R (0; the same GF HI results are obtained when the shift has opposite sign. We conclude that random fluctuations of t give rise to correlations, whose signs are in agree ment with our results. The higher the difference
"X (t)!X(t)", the more significant the effect, which I is maximum in the spectral regions near to the peak, where the slope is highest. Time-zero fluctuations have been monitored in a period of about two days by using as “start” signal the second annihilation photon, so as to obtain “prompt curves” with high counting rate; their centroids could be evaluated in less than 500 s. Then, the data have been averaged on a time interval of 2 h, which corresponds to the time necessary to collect a single PET spectrum, in order to obtain the standard deviations. These are shown in Fig. 5, and are contained within 10 ps (1 channel), which is of the same order of the thermal drifts of the time-to-amplitude converter (10 ps/°C). From the above discussion it follows that a compensation of t fluctuations is essential in order to restore a poissonian statistics — or, at least, to approach to it. This has been obtained as follows. Each spectrum has been deconvoluted into three components, and the t has been obtained as a fit ting parameter; one of them has been chosen as a reference, t. Then, each spectrum has been shif ted (when necessary) by an integer number of channels so as to carry the time zeros within t$0.5 channels. In such a way, fluctuations of t have been contained within one channel; in this connection, we remark that the statistics of the single spectrum (10) has been chosen as a compromise between the duration of the measurement and the precision necessary to estimate the t . It is impor tant to point out that shifts of a fraction of channel could report all the time zeros to the value t, by
G. Consolati, M. Stefanetti/Nucl. Instr. and Meth. in Phys. Res. A 417 (1998) 377—383
381
Fig. 6. An example showing the effect of a decrease of the conversion *. X, average spectrum; X , a particular spectrum. I R , R and R are all positive. GH FI GI Fig. 5. Centroids of prompt curves registered as a function of time. The points refer to averages on 15 data.
virtually eliminating the fluctuations among the spectra; however, the statistical feature of the data should be altered. Indeed, if X and ½ are two Poissonian distributions with mean values x and y, respectively, the variable Z"aX#b½ is no longer Poissonian if a#b"1 and 0(a, b(1, since E[Z],z"ax#by'ax#by"Var[Z]. A fit on Z would produce a completely deceptive good s, since usually such a test assumes the mean value as estimation of the variance of the data: 1 A [Z (k)!Z(k)] s" (3) c!l!1 Z(k) I where c is the number of degree of freedom (the number of channels, in the case of a timing spectrum) and l is the number of fitting parameters. Fig. 3b shows the correlation matrix obtained from the same set of spectra, but shifted through the above-mentioned method. There is a strong reduction of the correlation among the channels with respect to Fig. 3a. The residual correlation, which can be reasonably attributed to the uncomplete compensation of the t fluctuations, has the same signs of that displayed in Fig. 3a: this allows us to infer that possible correlations induced by variations of * are negligible [10]. Indeed, the effect of these last manifests itself in a compression or a stretching of the spectrum and the deformation remains after the correction of the t shift.
Fig. 6 shows the case of a decrease of * for the kth spectrum: R , R and R are all positive, in conGH FI GF trast with the black regions of Fig. 3b, which correspond to R (0. The behaviour of the s is also GJ indicative of the improvement produced by the adopted compensation (Fig. 2, open circles): s41.5 for t't, except a small region of about 50 ps where sK2. Therefore, even in the absence of digital stabilization, a high number of counts can be accumulated in a timing spectrum; however, it is necessary to operate a compensation of the fluctuations which otherwise can influence the statistics of the data. As an example of this procedure the PET spectra have been summed so as to obtain a spectrum with a total number of counts N"10, which was analyzed in terms of discrete or distributed components (in this case the program LT by Kansy [14] was used). Information contained in the spectrum allows one to resolve, in principle, six components [15]. The results are displayed in Table 1. According to the common interpretation, the spectrum was first of all deconvoluted into three discrete components, obtaining a s"1.33. Then, a deconvolution into four discrete components was carried out, leading to a much better s (1.13). Therefore, such a high number of counts allows us to discriminate between the two models on the basis of the s-test. We also deconvoluted the spectrum into five components; however, the analysis was not significant, since one lifetime diverges and the
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G. Consolati, M. Stefanetti/Nucl. Instr. and Meth. in Phys. Res. A 417 (1998) 377—383
Table 1 Results of deconvolutions of the positron lifetime spectrum in PET having 100 Mcounts. q: lifetime (ns); I: intensity (%), p: standard deviation of the distribution (ns). 3D: analysis into three discrete components; 3C: analysis into three distributed components; 4D: analysis into four discrete components. Errors on the data are shown in parentheses Model
q
I
p
q
I
p
q
I
p
3D 3C 4D
0.201 (6) 0.118 (5) 0.129 (4)
26.8 (8) 7.5 (2) 9.9 (2)
0.02 (3)
0.407 (7) 0.362 (4) 0.316 (4)
57.3 (9) 76.8 (8) 54 (1)
0.096 (3)
1.69 (2) 1.70 (2) 0.52 (1)
15.9 (2) 15.7 (2) 21.1 (5)
0.189 (3)
corresponding intensity becomes negligible: the other lifetimes reproduce the four-component fit. Deconvolution into three distributed components supplies a narrow distribution as far as the longest lifetime is concerned, which corresponds to o-Ps trapped in the nanocavities of the PET; this is reasonable, since in glassy polymers the distribution of lifetimes is narrower than in the rubbery state. The shortest component is comparable with that obtained from the deconvolution into four discrete components and the standard deviation of the distribution is zero, within the errors; on the other hand, the middle component (q , I ) is prob ably a weighted mean of the two components (q , I and q , I ) in the four-discrete components analysis, rather than to be a real distribution. Deconvolution into four distributions resulted unstable, with the standard deviations greater than the values of the parameters. Anyway, analyses with distributions do not give a significantly better s with respect to the four-component discrete analysis. Finally, we observe that the resolution factor, that is, the ratio of the closest lifetimes that can be resolved, is about 2, for our spectrum [15]; this can explain the result of the five-component deconvolution, since it should not be possible to extract a further lifetime among the components already found.
4. Conclusion In this paper we showed that jitter of the time zero can strongly influence the distribution of counts in a lifetime spectrum, by introducing correlations among channels, in particular in the
q
1.72 (2)
I
15.2 (2)
s 1.33 1.12 1.13
initial part of it. The consequence is a deviation from the expected (Poissonian) statistics. However, a simple compensation method allows a reduction of the fluctuations: such a procedure is recommended if the goal of a high number of counts must be achieved, together with a good time resolution. Furthermore, the method can be used to keep controlled the stability of the experimental apparatus. The accuracy depends on the conversion * (ps/ch) adopted, as well as on the stability of the time-zero. The only pitfall to be avoided is the shift of a non-integer number of channels, which leads to an incorrectly good s because of a non-Poissonian distribution of counts.
Acknowledgements This work has been jointly supported by Istituto Nazionale di Fisica della Materia and by “Fondo di Ricerca di Ateneo”, Politecnico di Milano.
References [1] W. Brandt, A. Dupasquier, Positron Solid State Physics (Eds.), North-Holland, Amsterdam, 1983. [2] M.J. Puska, R.M. Nieminen, Rev. Mod. Phys. 66 (1994) 841. [3] Xin Guo, Run-Zhang Yuan, Solid State Ionics 80 (1995) 159. [4] C. Corbel, P. Hautojarvi, in: A. Dupasquier, A.P Mills, Jr. (Eds.), Positron Spectroscopy of Solids, IOS Press, Amsterdam, 1993, p. 533. [5] Y.C. Yean, in: Y.C. Yean (Eds.), Positron Annihilation, World Scientific, Singapore, 1990, p. 1. [6] L. Xie, D.W. Gidley, H.A. Hristov, A.F. Yee, J. Polym. Sci. B 33 (1995) 77.
G. Consolati, M. Stefanetti/Nucl. Instr. and Meth. in Phys. Res. A 417 (1998) 377—383 [7] C.L. Wang, S.J. Wang, Z.N. QI, J. Polym. Sci. B 34 (1996) 193. [8] K. Okamoto, K. Tanaka, M. Ito, H. Kita, Y. Ito, Mat. Sci. Forum 743 (1995) 175. [9] C. Dauwe, G. Consolati, T. VanHoecke, D. Segers, Nucl. Instr. and Meth. A 371 (1996) 497. [10] H. Lami, E. Piemont, Chem. Phys. 163 (1992) 149. [11] P. Kirkegaard, N.J. Pedersen, M. Eldrup, in: L. DorikensVanpraet, M. Dorikens, D. Segers (Eds.), Positron Annihilation, World Scientific, Singapore, 1988, p. 642.
383
[12] J.R. Taylor, An Introduction to Error Analysis, the Study of Uncertainties in Physical Measurements, Ch. 12, University Science Books, 1982. [13] J. Wawryszczuk, B. Jasinska, W. Gorniak, T. Goworek, Nucl. Instr. and Meth. A 363 (1995) 574. [14] J. Kansy, Nucl. Instr. and Meth. A 374 (1996) 235. [15] D.M. Schrader, S.G. Usmar, in: S. Sharma (Ed.), Positron Annihilation Studies of Fluids, World Scientific, Singapore, 1988, p. 215.
An investigation on the distribution of counts in positron annihilation lifetime spectra G. Consolati*, M. Stefanetti Istituto Nazionale di Fisica della Materia, Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci, 23-20133 Milano, Italy Received 13 March 1998
Abstract Positron annihilation lifetime spectra with a high number of counts give useful information for various purposes (like accurate determination of distributions of lifetimes or a comparison among competing models). This requisite, coupled to the need of a good resolution, implies long times of measurement; the consequent instrumental drifts can degrade the quality of the data. In this paper a statistical analysis of spectra of polyethylene terephthalate is presented, which reveals that the distribution of counts displays deviations from the expected Poissonian behaviour, as a consequence of the jitter of the time zero. By means of a simple method for the compensation of such fluctuations it is possible to recover almost ideal statistical properties. 1998 Elsevier Science B.V. All rights reserved. PACS: 71.60.#z; 77.84.Jd; 78.70.Bj Keywords: Positron annihilation; Lifetime spectra; Counts; Polyethylene tetraphthalate
1. Introduction Positron annihilation lifetime spectroscopy is widely used for the investigation and characterization of solid-state materials [1], since the different positron states extracted from a timing spectrum can be correlated to structural defects in metals [2], ceramics [3] and semiconductors [4], as well as to nanovoids forming the free volume in polymers [5]. Furthermore, the presence of phase transitions can be conveniently observed by monitoring the
* Corresponding author. Tel.: #39 2 23996123; #39 2 23996126; e-mail: [email protected].
fax:
positron parameters (lifetime and intensity) as a function of the temperature [6]. In all these cases a total number of annihilation events registered in each spectrum of the order of 10 can be sufficient to obtain the required results. Conversely, a higher statistics must be accumulated to get a distribution of lifetimes with good accuracy, as in the case of ortho-positronium (o-Ps) annihilation in the nanocavities of a macromolecule [7,8]. The comparison among different models is another situation where it is necessary to obtain a high number of counts. As an example, it has been recently pointed out [9] that a possible non-exponential decay of Ps in powders, consequent to its guessed thermalization, would nevertheless produce an
0168-9002/98/$19.00 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 0 7 4 7 - 5
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G. Consolati, M. Stefanetti/Nucl. Instr. and Meth. in Phys. Res. A 417 (1998) 377—383
acceptable s-test for spectra having a few 10 counts and deconvoluted through classical algorithms using exponential components. In order to observe a clear worsening of the s an increase of the total count over 10 should be needed. The requirement of a high number of counts, combined with a good resolution of the apparatus, can involve long running times (up to several days) for the experiment; the unavoidable long-term drifts of the instrumentation degrade the quality of the data. By using a different technique, the single-photon timecorrelated method, Lami [10] showed that the consequence of such drifts is that the events are distributed according to a non-poissonian statistics; in the case of positron lifetime spectroscopy, this could imply systematic errors in the usual computer deconvolution routines, based on a stest which indeed is supposed to operate with poissonian events. With the aim to probe in a quantitative way this topic we analysed the statistical properties of a set of positron annihilation lifetime spectra of a polymer, polyethylene terephthalate, PET, collected under identical conditions. Anyway, the conclusions we reached are quite general and do not depend on the chosen material. We found that the jitter of the time zero is the main responsible of the deviations from the ideal statistical behaviour. Although it is possible to carry out a digital stabilization of the spectrum, such a procedure is anything but trivial as far as lifetime spectroscopy is concerned and it is not adopted by most of the laboratories. Nevertheless, the phenomenon can be reduced to an acceptable level through simple tricks as well as by means of a straightforward compensation method.
2. Experimental The positron source consisted of a droplet of Na from a carrier-free neutral solution, dried onto two identical Kapton foils (thickness 1.08 mg cm\), which were afterwards glued together by a thin layer of a cyanoacrylate glue (Loctite 407 by Loctite Corporation). Care was taken to prevent the glue from reaching the region involved in annihilation. The activity of the source was about 0.3 MBq. Positron spectra were collected through
a conventional fast—fast coincidence apparatus: the c-rays corresponding to the emission of the positron (“start”) and to the annihilation event (“stop”) were detected by two NE111 plastic scintillators, whose dimensions (1 ;1 and 1 ;0.5 , for the start and the stop channels, respectively) ensured a reasonable compromise between resolution and efficiency of the spectrometer. The scintillators were coupled to Philips XP2020 photomultiplier tubes. The resolution of the apparatus resulted to be about 200 ps; it was obtained through the deconvolution of timing spectra of a Bi source, at Na energy windows settings. Positron spectra extended over 2048 channels; conversion * was 10 ps/ch. Polyethylene terephthalate (PET) was used as the test material in the present investigation; it was purchased from Aldrich as pellets, which were compression moulded at 550 K in the form of discs (diameter 20 mm, height 2 mm). The density of the sample was 1.375 g cm\ the degree of crystallinity was estimated to be about 60%. Measurements were carried out at room temperature, where the polymer is glassy (¹ "356 K). The temperature of the laboratory was stable within 1°. 100 spectra were collected under identical conditions; each one contained exactly 10 counts. They were analysed through the Positronfit program [11].
3. Results and discussion In a positron annihilation lifetime spectrum the number of counts contained in the kth channel is expected to be a realization of a Poisson process with mean value 1X(k)2"p(k)N, where p(k) is the probability to find a count in the interval [k*, (k#1)*], * is the amplitude of the channel and N is the total number of counts. Such a property was checked on our data through a s-test; in order to explain the method we consider first of all the background, since in the time interval where only the background is present we can regard the counts as identical repetitions of the same statistical process. Fig. 1 shows the frequencies of the counts measured in the 500 channels containing the background in a single spectrum of PET (open circles), with mean value k"7.33; the dots are the
G. Consolati, M. Stefanetti/Nucl. Instr. and Meth. in Phys. Res. A 417 (1998) 377—383
Fig. 1. Measured frequencies (open circles) of the counts in the channels corresponding to the background (average value k"7.33). Dots: expected values of a Poisson distribution with mean value k. The continuous line is an interpolation of the Poisson distribution. The vertical lines correspond to the six intervals chosen for the s-test.
expected values of a Poisson distribution with the same mean k. The s-test was carried out by classifying the data within six intervals, whose extreme were [0, k!2(k, k!(k, k, k#(k, k# 2(k,R] [12]; they correspond to the vertical lines in Fig. 1. It resulted in s"1.5, corresponding to a probability P(s51.5)"0.2; we conclude that the distribution of counts in the interval containing the background is Poissonian, in a very good approximation. The same test has been performed for the channels containing the lifetime spectrum of PET, by taking into account the “ensemble” of the 100 spectra, having the same number of counts N"10. The 100 data corresponding to the same channel supply an average value k, which obviously depends on the channel, and are distributed into the six intervals defined above. In stationary conditions we should expect 100 repetitions of the same Poisson process in each channel; ideally, the counts in different channels should be independent. Fig. 2 shows the s-test for each channel (full squares): it is clear that in the two regions surrounding the peak s 1. Therefore, the distribution in these channels is not Poissonian. An alternative method for the check of the distribution of counts consists of comparing the ratio R between standard deviation and mean value of the counts in each channel [13]: for
379
Fig. 2. s-test for each channel of the 100 PET spectra. Full squares: original (non-shifted) spectra; open circles: shifted spectra. Only the first 500 channels are reported.
a Poisson distribution it should result R"1. This analysis led to a conclusion identical to that obtained with the s and it is not reported here. The investigation on the reasons of deviations of part of the data from the Poissonian behaviour is conveniently carried out by means of the correlation matrix [10]: S HI , R " HI (S S HH II where
(1)
1 L S " [X (k)!1X(k)2][X ( j)!1X( j)2] HI n!1 G G G (2) is the covariance matrix. In Eq. (2) X (k) is the G random variable corresponding to the kth channel of the ith spectrum, 1X(k)2" L X (k)/n, G G 1[X(k)!1X(k)2]2" L [X(k)!1X(k)2]/(n G !1), and n"100. In stationary conditions and in the absence of fluctuations it is expected that R "d : indeed, owing to the independence of the HI HI processes X( j) and X(k), the signs of the terms appearing in (Eq. (2)) are positive and negative with the same likelihood and their contribution becomes smaller, by increasing n. The result for the PET spectra is displayed in Fig. 3a, where different grays correspond to different degrees of correlation (white: #1, black: !1). Large deviations from the ideal behaviour occur in the regions contiguous to
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G. Consolati, M. Stefanetti/Nucl. Instr. and Meth. in Phys. Res. A 417 (1998) 377—383
Fig. 4. A schematic example showing the fluctuation of t . X: average spectrum; X : a particular spectrum. It turns out that I R and R '0, R and R (0. GH FI HF GI
Fig. 3. Correlation matrix for the 100 spectra of PET: (a) nonshifted, (b) shifted spectra. The degree of correlation is shown as different grays (white: #1, black: !1).
the peak, as already found by means of the s-test; in particular, positive correlation for a pair of channels preceding the peak or subsequent to it, and negative correlation for channels lying on opposite sides with respect to the peak. We can attribute the correlations to fluctuations of the time zero t ; indeed, let us suppose, for instance, that the kth spectrum X is shifted with respect to the average I spectrum X as in Fig. 4. Then, the pairs of channels (i, j) and (h, k) give a positive contribution to the correlation matrix; on the other hand, it is evident Eqs. (2) and (3) that R and R (0; the same GF HI results are obtained when the shift has opposite sign. We conclude that random fluctuations of t give rise to correlations, whose signs are in agree ment with our results. The higher the difference
"X (t)!X(t)", the more significant the effect, which I is maximum in the spectral regions near to the peak, where the slope is highest. Time-zero fluctuations have been monitored in a period of about two days by using as “start” signal the second annihilation photon, so as to obtain “prompt curves” with high counting rate; their centroids could be evaluated in less than 500 s. Then, the data have been averaged on a time interval of 2 h, which corresponds to the time necessary to collect a single PET spectrum, in order to obtain the standard deviations. These are shown in Fig. 5, and are contained within 10 ps (1 channel), which is of the same order of the thermal drifts of the time-to-amplitude converter (10 ps/°C). From the above discussion it follows that a compensation of t fluctuations is essential in order to restore a poissonian statistics — or, at least, to approach to it. This has been obtained as follows. Each spectrum has been deconvoluted into three components, and the t has been obtained as a fit ting parameter; one of them has been chosen as a reference, t. Then, each spectrum has been shif ted (when necessary) by an integer number of channels so as to carry the time zeros within t$0.5 channels. In such a way, fluctuations of t have been contained within one channel; in this connection, we remark that the statistics of the single spectrum (10) has been chosen as a compromise between the duration of the measurement and the precision necessary to estimate the t . It is impor tant to point out that shifts of a fraction of channel could report all the time zeros to the value t, by
G. Consolati, M. Stefanetti/Nucl. Instr. and Meth. in Phys. Res. A 417 (1998) 377—383
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Fig. 6. An example showing the effect of a decrease of the conversion *. X, average spectrum; X , a particular spectrum. I R , R and R are all positive. GH FI GI Fig. 5. Centroids of prompt curves registered as a function of time. The points refer to averages on 15 data.
virtually eliminating the fluctuations among the spectra; however, the statistical feature of the data should be altered. Indeed, if X and ½ are two Poissonian distributions with mean values x and y, respectively, the variable Z"aX#b½ is no longer Poissonian if a#b"1 and 0(a, b(1, since E[Z],z"ax#by'ax#by"Var[Z]. A fit on Z would produce a completely deceptive good s, since usually such a test assumes the mean value as estimation of the variance of the data: 1 A [Z (k)!Z(k)] s" (3) c!l!1 Z(k) I where c is the number of degree of freedom (the number of channels, in the case of a timing spectrum) and l is the number of fitting parameters. Fig. 3b shows the correlation matrix obtained from the same set of spectra, but shifted through the above-mentioned method. There is a strong reduction of the correlation among the channels with respect to Fig. 3a. The residual correlation, which can be reasonably attributed to the uncomplete compensation of the t fluctuations, has the same signs of that displayed in Fig. 3a: this allows us to infer that possible correlations induced by variations of * are negligible [10]. Indeed, the effect of these last manifests itself in a compression or a stretching of the spectrum and the deformation remains after the correction of the t shift.
Fig. 6 shows the case of a decrease of * for the kth spectrum: R , R and R are all positive, in conGH FI GF trast with the black regions of Fig. 3b, which correspond to R (0. The behaviour of the s is also GJ indicative of the improvement produced by the adopted compensation (Fig. 2, open circles): s41.5 for t't, except a small region of about 50 ps where sK2. Therefore, even in the absence of digital stabilization, a high number of counts can be accumulated in a timing spectrum; however, it is necessary to operate a compensation of the fluctuations which otherwise can influence the statistics of the data. As an example of this procedure the PET spectra have been summed so as to obtain a spectrum with a total number of counts N"10, which was analyzed in terms of discrete or distributed components (in this case the program LT by Kansy [14] was used). Information contained in the spectrum allows one to resolve, in principle, six components [15]. The results are displayed in Table 1. According to the common interpretation, the spectrum was first of all deconvoluted into three discrete components, obtaining a s"1.33. Then, a deconvolution into four discrete components was carried out, leading to a much better s (1.13). Therefore, such a high number of counts allows us to discriminate between the two models on the basis of the s-test. We also deconvoluted the spectrum into five components; however, the analysis was not significant, since one lifetime diverges and the
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G. Consolati, M. Stefanetti/Nucl. Instr. and Meth. in Phys. Res. A 417 (1998) 377—383
Table 1 Results of deconvolutions of the positron lifetime spectrum in PET having 100 Mcounts. q: lifetime (ns); I: intensity (%), p: standard deviation of the distribution (ns). 3D: analysis into three discrete components; 3C: analysis into three distributed components; 4D: analysis into four discrete components. Errors on the data are shown in parentheses Model
q
I
p
q
I
p
q
I
p
3D 3C 4D
0.201 (6) 0.118 (5) 0.129 (4)
26.8 (8) 7.5 (2) 9.9 (2)
0.02 (3)
0.407 (7) 0.362 (4) 0.316 (4)
57.3 (9) 76.8 (8) 54 (1)
0.096 (3)
1.69 (2) 1.70 (2) 0.52 (1)
15.9 (2) 15.7 (2) 21.1 (5)
0.189 (3)
corresponding intensity becomes negligible: the other lifetimes reproduce the four-component fit. Deconvolution into three distributed components supplies a narrow distribution as far as the longest lifetime is concerned, which corresponds to o-Ps trapped in the nanocavities of the PET; this is reasonable, since in glassy polymers the distribution of lifetimes is narrower than in the rubbery state. The shortest component is comparable with that obtained from the deconvolution into four discrete components and the standard deviation of the distribution is zero, within the errors; on the other hand, the middle component (q , I ) is prob ably a weighted mean of the two components (q , I and q , I ) in the four-discrete components analysis, rather than to be a real distribution. Deconvolution into four distributions resulted unstable, with the standard deviations greater than the values of the parameters. Anyway, analyses with distributions do not give a significantly better s with respect to the four-component discrete analysis. Finally, we observe that the resolution factor, that is, the ratio of the closest lifetimes that can be resolved, is about 2, for our spectrum [15]; this can explain the result of the five-component deconvolution, since it should not be possible to extract a further lifetime among the components already found.
4. Conclusion In this paper we showed that jitter of the time zero can strongly influence the distribution of counts in a lifetime spectrum, by introducing correlations among channels, in particular in the
q
1.72 (2)
I
15.2 (2)
s 1.33 1.12 1.13
initial part of it. The consequence is a deviation from the expected (Poissonian) statistics. However, a simple compensation method allows a reduction of the fluctuations: such a procedure is recommended if the goal of a high number of counts must be achieved, together with a good time resolution. Furthermore, the method can be used to keep controlled the stability of the experimental apparatus. The accuracy depends on the conversion * (ps/ch) adopted, as well as on the stability of the time-zero. The only pitfall to be avoided is the shift of a non-integer number of channels, which leads to an incorrectly good s because of a non-Poissonian distribution of counts.
Acknowledgements This work has been jointly supported by Istituto Nazionale di Fisica della Materia and by “Fondo di Ricerca di Ateneo”, Politecnico di Milano.
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