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How pileup rejection affects the precision of loss-free counting
Nuclear Instruments and Methods in Physics Research A 432 (1999) 456}470
How pileup rejection a!ects the precision of loss-free counting S. PommeH * SCKzCEN, Belgian Nuclear Research Centre, Boeretang 200, B-2400 Mol, Belgium Received 25 January 1999; received in revised form 29 March 1999
Abstract The mechanism behind the particularly high count scatter observed with loss-free counting (LFC) on spectrometers with pileup rejection is scrutinised. It is shown theoretically and experimentally that, contrary to other loss mechanisms, pileup rejection does not preserve Poisson statistics during the system live-time. Consequently, the statistical uncertainty obtained with loss-free counting increases proportionally to the broadening of the count-yield distribution per live-time inspection period. A new theoretical formula for the LFC uncertainty is presented and predicted values are compared with experimental data obtained on a gamma-ray spectrometer and data from computer simulations of nuclear counting. Additionally, the statistical e!ect of using #oating-point values instead of integers for the LFC correction factors is investigated; the LFC uncertainty shows little change. 1999 Elsevier Science B.V. All rights reserved. PACS: 02.50.Cw; 02.90.#p; 07.85.!m Keywords: Nuclear counting; Loss-free counting; Statistics; c-ray spectrometry; Uncertainty; Neutron activation analysis
1. Introduction Loss-free counting (LFC) is known to be an advanced real-time correction method of counting losses in nuclear pulse spectrometry [1}7]. Count loss is compensated by performing add-n operations to the spectrum, in which the weighting factor n is derived somehow from the amount of loss whenever the system is unable to count. In 1981 Westphal introduced the Virtual Pulse Generator (VPG) technique to evaluate the LFC weighting factor [1,4,5]. Its operation is similar to the
* Tel.: #32-14-332718; fax: #32-14-321056. E-mail address: [email protected] (S. PommeH )
well-known pulser method [1,8], however without introducing pulser signals into the spectrum. As a consequence, the system availability can be checked at a much higher frequency, hence drastically improving the ability to respond to rapid changes in the count rate. Performance tests on c-ray spectrometers have demonstrated that LFC with VPG can provide accurate results (better than 1% uncertainty) over a wide range of incoming count rates (see e.g. Ref. [7]). The LFC method with VPG is well suited for application in e.g. neutron activation analysis, since the commonly higher loss of pulses from short-lived activation products is well taken into account. Such would not be the case with live-time correction techniques or methods supplying one average
0168-9002/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 9 ) 0 0 4 7 5 - 1
S. Pomme& / Nuclear Instruments and Methods in Physics Research A 432 (1999) 456}470
correction factor for the whole spectrum. Another advantage of major importance is its ability to deal with pulse pileup. This is essential, considering that pileup (rejection) is the major pulse-loss mechanism in contemporary fast spectrometry chains. In fact, the combinations of pulse pileup and extending and non-extending deadtime featuring in typical spectrometers present no problem to the powerful LFC principle. The only special precaution is the addition of a pulse evolution time (PET) to each system busy period, to account for loss by leading-edge pileup [1,3}5,7]. In practice, the PET signal width in the LFC module is manually "ne-tuned using the double-source method, to reach optimum loss compensation. (Some additional care is required with reset preampli"ers [7].) At the negative side of the LFC method is the unavoidable modi"cation of counting statistics. Whereas LFC can restore the average count rate of an event train passing through an electronic counter, it is unable to reproduce the original time interval distribution of subsequent events. The distortion of the original exponential interval density distribution, which is characteristic for Poisson processes, will eventually alter the uncertainty on the count integral measured during a "xed realtime. According to Westphal [3,4], the statistical accuracy of loss-free counting can be calculated from
dN 1 dn *$! " 1# N 1n2 (N *$!
(1)
in which N is the number of counts in the *$! considered region of the LFC spectrum, N is the corresponding number of events recorded without loss correction, and dn is the uncertainty on the LFC weighting factor n. Heydorn and Damsgaard [6] were probably the "rst to recognise that the commonly adopted Westphal formula does not always account for the count scatter in LFC spectra. PommeH et al., having scrutinised the statistical e!ects of pileup rejection on nuclear counting [9}12], suggested that the problem of statistical control of loss-free counting was related to pulse pileup rejection [7,13]. The increased count scatter was attributed to the broadening of the statistical distribution of the
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valid events per VPG inspection period. A new LFC uncertainty formula was presented [13]:
p(N ) r p(n) *$! " 1# N 1n2 (N *$!
(2)
with
A 2#A p(Ccounts/IP) +1# f r" 2 1#A (Ccounts/IP in which we have A"1!e\M .#2, A "1!e\D M .#2, and f is the considered fraction of the spectrum ( f"100% is full spectrum). In this work, the theoretical derivation of formula 2 will be described. Its validity will be tested for di!erent experimental situations, such as c-ray spectrometers with a gated-integrator ampli"er or with a classical semi-Gaussian shaping ampli"er. By means of computer simulations, the operation of the LFC principle will be tested on other, hypothetical types of counters. For details on the experimental set-up and the simulation program, the reader is referred to Refs. [7,11,12].
2. Schematic representation of event pulses Rate-related losses are basically caused by nearly synchronous events and the limited time resolution of the counter. Which pulses of an incoming event train are counted eventually depends on the loss mechanism. To visualise the e!ect of deadtime and pileup, a schematic representation of a typical event pulse is shown in Fig. 1. The circle represents the position of a detector pulse, at the point at which the peak is detected. The triangle in front of the peak represents the pulse evolution time; i.e. the time between pulse arrival and peak detection. The rectangle behind the peak represents deadtime; it can correspond to the trailing edge of the pulse, equivalent to extending deadtime [11,12], or to the non-extending deadtime of the analog-to-digital convertor. Below the base line, Virtual Pulse Generator (VPG) time units are shown in which the system is considered `busya by the LFC module. In Fig. 2 an impression is given on how di!erent loss mechanisms interact with a speci"c event train.
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in time zone C can be valid, even though the VPG considers this region as belonging to the extended system busy time. If the second pulse falls within region D, it is valid unless it interacts with a third event falling within its zone A. For a Poisson process, with average count rate o, the probabilities for subsequent pulses falling into a particular zone is calculated from
Fig. 1. Schematic representation of a detector pulse and its associated event loss mechanisms. Time zone A corresponds to pulse pileup, zone B to deadtime, zone C is an extension of the system busy by the LFC device and D is outside the extended system busy time.
A"P(*t)PET)"1!e\M.#2
(3)
B"P(PET(*t)¹ )"e\M.#2!e\M25 5 C"P(¹ (*t)q)"e\M25!e\Ms 5 D"P(*t'q)"e\MO
(4) (5) (6)
with q"PET#¹ . Evidently, the sum of these 5 probabilities, A#B#C#D, is equal to one.
3. Counting statistics in live-time 3.1. Introduction When searching an explanation for the increased loss-free counting uncertainty for systems with pileup rejection [7,13], computer simulations provided an interesting hint; there seemed to be a correlation between p(N ) and the average *$! number of valid counts per `non-voida LFC-VPG inspection period, 1VC/NVIP2. Indeed, one can reproduce the counting uncertainty to a good approximation by Fig. 2. Event train passing through counters having di!erent loss mechanisms; NEDT"non-extending deadtime, EDT"extending deadtime and PU"pileup rejection. The case of combined PU and EDT corresponds to a spectrometer with Gaussian pulse shaping and fast ADC with negligible deadtime. The empty circles correspond to invalid events. The e!ective deadtime is assumed to be equal in all four cases, i.e. q "q "PET#¹ . The tick marks below the baseline rep 5 resent the extended system busy time registered by the VPG.
Even though the same e!ective deadtime was considered, di!erent events are counted as valid or invalid. Pulses falling within time zones A and B (see Fig. 1) of another event are always lost. However, in time zone A also the "rst pulse is invalidated by pileup rejection. Pulses falling with-
p(N 1 p(n) *$! + 1#1VC/NVIP2 1n2 N (N *$! (7)
with 1VC/IP2 1VC/NVIP2" P(VC'0)
(8)
in which P(VC'0)"1!P(VC"0) is the probability of having at least one valid count in an inspection period and 1VC/IP2 is the average number of valid counts per inspection period. The remarkable aspect of this quantity is the signi"cant di!erence observed for systems with pileup rejection compared to systems with extending deadtime
S. Pomme& / Nuclear Instruments and Methods in Physics Research A 432 (1999) 456}470
at the same loss rate, i.e. at the same o and with q "PET#¹ . This lead to a study of the count 5 number distribution per inspection period. First consider the case of extending deadtime. Each count arriving in the considered inspection interval creates a certain amount of deadtime in which the detector is unable to count. The VPG of the LFC marks these periods as `busya and prolongs the current inspection period with the same amount of real-time. Consequently, the LFC-VPG conserves the amount of system live-time per inspection period. By removing the `system busya periods one can convert from `counting in realtime using a system with deadtimea to `counting in live-time without lossa. This reduces the size of the inspection period, IP, to the minimum size, IP . (In our LFC apparatus, IP corresponds to 51.2 ls; this is 256 VPG ticks at a frequency of 5 MHz.) For a stationary Poisson process, the arrival time of any event is independent of its `historya, so the same Poisson statistics can be applied in live-time. The average amount of counts per inspection period, 1VC/IP2, is therefore equal to o.IP . Similar considerations are valid for non-extending deadtime, leading to the same average count number per inspection period. The inspection periods, however, are shorter on the average, since 1IP2 "exp(oq ) IP in the case of extending deadtime, and
(9)
1IP2 "1#oq (10) L IP in the case of non-extending deadtime. Now, consider a system with pileup rejection, having an identical e!ective deadtime as the EDT counter, q"PET#¹ "q . Since the throughput 5 is also exponential [12], one expects the same average duration of the inspection periods. Whereas also here 1VC/IP2"o IP , one obtains a signi" cantly higher result for 1VC/NVIP2. This must be due to a higher probability for having zero valid counts in an inspection period, P(VC"0) (see formula 8). Consequently, the number of counts per inspection period is not Poisson distributed when counting with pileup rejection. The underlying rea-
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son is that the LFC extends `system busya beyond the point at which the system is again available for data acquisition, to compensate for the twofold loss of piled-up events. By a detailed comparison between the loss mechanisms of EDT and PU, the necessary adaptations to Poisson probabilities will be made to arrive at the correct counting statistics per inspection period for PU. 3.2. Probabilities for relative count gain and loss Consider a series of closeby events, not more than q apart. With an EDT counter, only the "rst count is valid. Now when switching to a PU counter, the "rst count can possibly become invalid, while other counts in the series could become valid. In this section, the probabilities are calculated for losing a valid count or gaining #k (k"0, 1, 2,2) valid counts when switching from an EDT to a PU system. Di!erences between extending deadtime (EDT) and pileup (PU) systems are shown in Fig. 3 for series of at most two closeby events. Events falling in zone D or B are treated in the same manner as with extending deadtime. However, with PU, the "rst count is lost additionally if the second one falls within zone A. On the other hand, a valid count is gained in zone C. The situation becomes more complex if longer chains of closeby events have to be disentangled.
Fig. 3. Comparison of a counter with only extending deadtime (top) and a counter with symmetrical pulse shaping, applying pileup rejection (bottom). Both represent equal e!ective deadtimes. Both systems yield the same outcome for pulses falling within the B and D zone. Pileup rejection causes the loss of one extra count in the A-zone, while the reverse is true for the C-zone.
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First consider P(*"!1); the following combinations lead to a loss of the valid count, without other counts in the series becoming valid P(*"!1)"AD
The general case of P(*"#k) is easily found in a similar fashion, yielding the following expression: P(*"#k)"+(1#A[N]C)CI ;(1#(B#CA)[N])I>,D.
(16)
#A(A#B#CA)D #A(A#B#CA) (A#B#CA)D #A(A#B#CA) (A#B#CA)
P(*"k#1) P(gain)" P(*"k)
;(A#B#CA)D #2 AD . " 1!(A#B#CA)
The probability for loss, P(loss)" P(*"!1), should be equal to the probability for count gain, P(gain). This is indeed found:
"C(1#(B#CA)[N]) (11)
A " "P(loss). 1#A
(17)
One can de"ne a `neutrala series [N], in which the combinations of gain and loss cancel each other:
The expressions for P(*"#k) can now be simpli"ed.
[N]"[1!(A#B#CA)]\
AC P(*"#0)" 1# (1#A)D
"[(1#A)D]\.
(12)
As a result, the formula for P(*"!1) can be simpli"ed: A P(*"!1)"A[N]D" . 1#A
(13)
The next quantity to assess, is the probability that the series of closeby events yields exactly one valid count, just as much as with the EDT counter: P(*"#0)"D#B[N]D#CA[N]D
A (1#A)C
"[1#A]\"[1!P(loss)]
(18)
P(*"#k)"P(*"#0) P(gain)I AI " . [1#A]I>
(19)
One can verify that the sum of the probabilities is indeed equal to one. 3.3. Distribution of the number of valid counts per inspection period
#A[N]CD#A[N]CB[N]D #A[N]CCA[N]D "+(1#A[N]C) ;(1#(B#CA) [N]),D.
(14)
The probability that one valid count in the EDT system corresponds to two counts in the PU system is then P(*"#1)"+(1#A[N]C)C ;(1#(B#CA)[N]),D.
(15)
Knowing the probabilities for count gain and loss with pileup systems compared to extending deadtime systems, the number of valid counts can be calculated starting from Poisson statistics. By P(k) we denote the Poisson probability of having k valid events per inspection period, possibly followed by a series of invalidated events in their deadtime trail. In the case of a PU system, zero valid counts are found when either no events fall within the inspection period or when all events are invalidated by pileup rejection. The probability for zero valid counts in the inspection interval is hence
S. Pomme& / Nuclear Instruments and Methods in Physics Research A 432 (1999) 456}470
calculated from > P(C"0)" P(k) P(loss)I I > [o IP P(loss)]I " e\M'. k! I o IP "exp ! 2!e\M.#2
(20)
since 1!P(loss)"[2!e\M .#2]\ (see Eq. (18)). The probability for having exactly one valid count is > P(C"1)" P(k) P(*"0)CP(loss)I\ I I > (oIP )I P(loss)I "(o IP )e\M'. P(*"0) k! I o IP . (21) "(o IP ) P(*"0)exp ! 2!e\M .#2
For k'0, one can write as a general expression
L CI\L I\L I o IP A I\ P(C"k)" n! (1#A) 1#A L oIP ;exp ! . (22) 2!e\M'.
461
The validity of formula 22 has been checked by comparing with computer simulation data. An excellent agreement is found for any choice of input parameters. The simulations con"rm that Poisson statistics can be applied in the case of deadtime. However, the probability distributions signi"cantly di!er in the case of pileup. In Fig. 4 the probability distribution of valid counts is compared between a system with pileup and another with deadtime. Whereas in both cases an average of "ve valid counts per IP is taken, one clearly sees a broadening of the probability distribution for counting with pileup rejection. 3.4. Distribution of valid counts for a region of interest In spectrometry, usually only particular regions of interest (ROI) are counted, instead of the fullenergy spectrum. The counting statistics change as a function of the fraction of the spectrum which the ROIs represent. The number of valid counts per inspection period can be derived from the result for the full spectrum, by considering that each count has the same a priori probability of belonging to the considered ROI, being equal to the fraction f. One can readily verify that the distribution of valid
Fig. 4. Theoretical probability distribution of the number of valid counts per "xed live-time period (or inspection period) for counters with a di!erent type of count loss: extending deadtime (PET"0) or pulse pileup (PET"¹ ). In both cases an average of 1VC/IP2"5 5 was taken. The distribution is clearly broader for pileup rejection.
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counts in a ROI can be obtained from > P (VC"k)" P(VC"k#i)CI>G(1!f )G f I D I G
3.5. Quantifying the distribution width
(23)
in which P (VC"k) is the probability of having D k valid counts in a ROI per inspection period. In Fig. 5 an example is shown of the probability distribution of the number of counts for di!erent ROIs of the same spectrum. The smaller the considered part of the spectrum, the more the distribution resembles Poisson statistics.
If the number of valid events counted in an inspection period is Poisson distributed, one can assume that the variance is equal to the expectation value, i.e. p(VC/IP)"1VC/IP2. This can be applied for counters with deadtime, but a new formula is required for counters with pileup rejection. In this work the following approach is suggested:
A 2#A p(VC/IP)+(1VC/IP2 1# f 2 1#A
(24)
with A and A as de"ned in Eq. (2). This is exactly the correction factor applied in the LFC uncertainty formula 2 (and also in Ref. [13]). This formula is a slightly improved version of the "rst, published in Ref. [7], in which a linear relationship as a function of the spectrum fraction f was assumed. In the new version a slight nonlinearity as a function of f was taken into account by introducing the factor A in the formula. The accuracy of formula 24 is demonstrated in Fig. 6, in which its resulting values are compared with the true width of the count distributions shown in Fig. 5. The width converges to (1VC/IP2 when f goes to zero, i.e. for small fractions. The same is true when the factor A approaches zero, i.e. when little or no pileup occurs.
Fig. 5. Probability distribution of the number of valid counts per inspection period for a counter with pileup rejection, valid for di!erent fractions f of the spectrum, as calculated from Eqs. (23) and (22) (oq"1.3, PET"¹ "20 ls and IP "51.2 ls). 5 Deviations from Poisson statistics are more pronounced when considering large fractions of the spectrum.
Fig. 6. Relative width of the probability distribution of the number of valid counts per inspection period for a counter with pileup rejection, as a function of the fraction f of the spectrum. The true width, as calculated from Eqs. (23) and (22) (oq"1.3, PET"¹ "20 ls and IP "51.2 ls), is well reproduced by 5 formula 24.
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3.6. Experimental counting uncertainty at xxed live-time The theoretical considerations depicted above enable the prediction of the counting uncertainty in an uncorrected spectrum measured at "xed livetime. In the case of a counter with pileup rejection one can expect a deviation from Poisson statistics, following p(N)"r(N
(25)
in which N is the number of counted events and r the correction factor de"ned in Eqs. (2) and (24). The validity of this premise has been veri"ed experimentally on the HPGe c-ray spectrometer set-up `Panoramixa, equipped with a pileup rejection circuit (see Refs. [7,12]). Series of measurements ('2000) of c-rays from a Cs source were repeated at di!erent source}detector distances. The shaping ampli"er is the main bottleneck of the counting device, since the ultra-fast ADC causes no additional pulse loss. Semi-Gaussian pulse shaping was applied with a timing constant of 4 ls. This corresponds to an average pulse width of 22 ls and a pulse evolution time of 12 ls, hence an e!ective deadtime of 34 ls (see Ref. [12]). Fig. 7 shows the standard deviation of the number of counted events in spectra acquired at a "xed live-time of 0.2 s, relative to (N. The deviation of the counting uncertainty from Poisson statistics clearly grows with increasing incoming count rate. The actual experimental values are slightly higher than theoretically expected (dashed curve). Better agreement is obtained with a hypothetical value of PET"14 ls, instead of 12 ls. At extremely high count rates (oq'1.8, with more than 80% count loss) the discrepancy between theory and experiment is more signi"cant, probably due to the imperfection of the actual pileup rejection system. Also the dependency of counting statistics on the considered fraction of the c-ray energy spectrum has been investigated. Whereas Fig. 7 corresponds to the full spectrum, Fig. 8 shows the standard deviation for fractions of the spectrum ranging from 0.01% up to 100%, at a "xed total incoming count rate of 25.6;10 s\. The nearly
Fig. 7. Statistical uncertainty of the number of counts in a full c-ray spectrum as a function of the incoming count rate, as observed experimentally with a HPGe c-ray detector with pileup rejection and semi-Gaussian pulse shaping, using a "xed live-time of 0.2 s. The full line corresponds to the theoretical values from Eqs. (25) and (24), assuming PET"14 ls. The dashed curve corresponds to a more realistic value of PET"12 ls.
Fig. 8. Statistical uncertainty of nuclear counting as a function of the spectrum fraction, observed experimentally with a HPGe c-ray spectrometer with pileup rejection and semi-Gaussian pulse shaping, at an incoming count rate of o"25.6;10 s\ and using a "xed live-time of 0.2 s. The full line corresponds to the theoretical values from Eqs. (25) and (24), assuming PET"14 ls.
linear behaviour of the relative uncertainty is well reproduced by theory. Convergence towards Poisson statistics is demonstrated for small fractions.
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4. Statistics of loss-free counting 4.1. Validity range of the Westphal formula The Westphal formula (see Eq. (1)) seems to be directly applicable to the counting uncertainty of small fractions of the LFC spectra. Indeed, then the number of counts N can be interpreted as the *$! product of two independent entities: the number of processed events N and the average weighting factor 1n2. Their relative uncertainties being, respectively, p(N)/N"1/(N and p(1n2)/1n2"p(n)/ (N1n2, one easily "nds the Westphal formula as the square root of the sum of the variances. In spite of some fundamental objections to applying this reasoning also to larger parts of the spectrum [11], the extrapolation of loss-free counting statistics to large regions of interest appears to be straightforward in practice. In Figs. 9 and 10 the standard deviation of the complete LFC spectrum is shown as a function of the incoming count rate for a hypothetical counter with extending and non-extending deadtime respectively; the data are obtained from computer simulations. The graphs clearly show that the standard deviation is nicely reproduced by the Westphal formula (Eq. (1)) as well as by the formula presented in this work
Fig. 10. Same as Fig. 9, for a counter with a non-extending deadtime of q "34.8 ls.
(Eq. (2)). Indeed, it turns out that the counting statistics for counters with deadtime show no signi"cant dependence on the considered fraction of the spectrum. This can be veri"ed from Fig. 11, in which counting statistics of the LFC spectrum and the corresponding uncorrected spectrum are shown at oq"1 as a function of the spectrum fraction f. The LFC uncertainty remains practically constant near the level of the theoretical value, in spite of the drastic change of the variance on the corresponding number of real events counted in the uncorrected spectrum. 4.2. Counters with pileup rejection
Fig. 9. `Relativea uncertainty of loss-free counting as a function of incoming count rate using a hypothetical counter with an extending deadtime of q "34.8 ls and a VPG with a frequency of 5 MHz (IP "51.2 ls). The data points are obtained from computer simulations and the theoretical values from Eq. (1) (dashed line) and Eq. (2) (full line).
The Wesphal formula for LFC uncertainty fails for systems in which counts are lost by pileup rejection. As an example, the default set-up of the Panoramix c-ray spectrometer (semi-Gaussian pulses with 4 ls shaping constant) was tested by experiment and by computer simulation. The standard deviation on the count integral of the LFC spectra is presented as a function of incoming count rate in Fig. 12. One can see an excellent agreement between theory and experimental data up to at least o"1/q, corresponding to 63% count loss. Slight deviations at higher count rates are attributed to the "nite pulse pair resolution of the pileup rejection system, allowing an increasing number of compound pulses to slip through. The simulation data con"rm that Eq. (2) is in principle
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Fig. 12. Standard deviation of the count integral of measured LFC spectra, compared to theoretical expectations and computer simulation data. Experimental data correspond to typically 2000 c-ray spectra of Co taken on HPGe detection set-up `Panoramixa (q"PET#¹ "12.1#22.7 ls). The the5 oretical data are calculated from Eq. (1) (dashed line) and Eq. (2) (full line), for which the relevant p(n)-values were derived from the simulations.
Fig. 11. Counting statistics obtained by computer simulation in an LFC spectrum and the corresponding uncorrected spectrum for counters with extending deadtime and non-extending deadtime, as a function of the considered fraction of the spectrum and at a "xed incoming count rate of oq"1. Theoretical values for LFC spectra are calculated from Eq. (1) or Eq. (2), and for the uncorrected spectrum from a formula for EDT counters in Ref. [12].
correct up to extremely high count rates, even close to 100% count loss. Similar results were obtained on another detector set-up with a transistor reset preampli"er instead of the classical RC feedback type. The occasional reset times of the preampli"er are apparently insu$cient to change the counting statistics in a signi"cant way. The new formula also works for a spectrometer equipped with a gated-integrator ampli"er, which produces pulses in the shape of a cumulative Gaussian. The trailing edge is extremely short, and as a result, the pulse evolution corresponds to the duration of almost the entire pulse, i.e. PET+¹ 5 and q+2PET. For experimental tests, the Panoramix main ampli"er was temporarily replaced by a gated-integrator ampli"er. The observed loss-free
Fig. 13. Standard deviation of the count integral of full LFC spectra of Cs c-rays measured by a detector with gatedintegrator ampli"er, for which the pulse evolution time is about equal to the total pulse width (q/2"PET"¹ "15 ls). 5
counting statistics for c-ray spectra of Cs are presented in Fig. 13. Again, the theoretical and experimental values agree very well up to oq"1. In fact, computer simulations demonstrate that statistical control is achieved for any ratio between the pulse evolution time and the total pulse width, PET/¹ , going from 0 up to 1. 5
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Fig. 14. Standard deviation of the number of counts in a region of interest of LFC spectra on c-ray detector `Panoramixa (o"32 kBq), compared to theoretical expectations (formulas 1 and 2).
More importantly, also the counting statistics of a random selection of the event spectrum can be determined accurately. The LFC uncertainty formula varies according to the fraction of the spectrum represented by the selected region of interest, as shown by experiment in Fig. 14 for oq"1.1 and PET"q/2.8. The variation of p(N ) with f can *$! be well approximated by a linear interpolation between the Eqs. (1) (for f"100%) and (2), but the slightly bent curve is better taken into account by the improved representation of the correction factor r (see Eq. (2)). Also here the Westphal formula is an excellent approximation when considering small fractions of the spectrum, like e.g. one c-ray peak in a complex activation spectrum.
5. Variance of the LFC weighting factor 5.1. The need for an expression on p(n) When applying the LFC uncertainty formulas 1 and 2, use was made of simulation results for the standard deviation of the VPG weighting factor, p(n). Of course, a more practical way to assess p(n) is needed for everyday c-ray spectrometry. Up to now, literature provided no in-depth theoretical analysis of this complex matter. An alternative approach is to determine p(n) directly from the measured LFC spectrum and corresponding un-
corrected spectrum, by sampling the n-values in about 20 energy channels around the considered c-ray peak. However, aside from perhaps an insu$cient accuracy of this method one has to consider here the danger that p(n) can be severely overestimated if local variations of n-values occur because di!erent peaks and continua in the spectrum correspond to radionuclides with di!erent half-lives, having experienced a di!erent average amount of count loss during the measurement. In fact, one needs an expression which is related to the n-value of the considered region of interest alone. In this work we have attempted to develop a semi-empirical formula to calculate p(n) with suf"cient precision in a wide range of measurement conditions. A di!erent result was found for counters with non-extending deadtime on one hand and for counters with extending deadtime or pileup rejection on the other. 5.2. Non-extending deadtime In the case of non-extending deadtime the distribution of the LFC weighting factor n can readily be predicted. Consider that the length of the VPG inspection period IP is the sum of its minimum value IP and the amount of system busy time during the inspection period, which in this case is simply the number of valid counts multiplied by the characteristic deadtime q . The LFC weighting factor n is de"ned as the ratio between the length of the inspection period IP and the minimum length of an inspection period IP . As only integer values are accepted, the fractional part of IP/IP is kept and added to the IP-value of the next inspection period(s). (For simplicity, it is assumed that the remainder varies in a continuous and unpreferential way between 0 and 1.) Now assume that the deadtime per event is smaller than IP , with a ratio of a"q /IP . The probability for n to take the value x, P(n"x), is then I P(n"x)" P(k)(ka!(x!2)) II I # P(k)(x!ka) II
(26)
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for which we de"ne
k "#oor
n!2 n!1 , k "#oor , a a
n!1 k "ceil a
n and k "#oor , a
in which `ceila and `#oora signify rounding to the closest upper or lower integer respectively and P(k) is the Poisson probability for having k valid events during the inspection period, considering that the expectation value is equal to oIP . The validity of Eq. (26) has been succesfully demonstrated by comparison with simulation results. Whereas the variance of n can be calculated from the theoretical distribution from Eq. (26), a more direct and fast method is desirable. One can show mathematically that, for systems with non-extending deadtime and a"q /IP (1, a good approxi mation for the standard deviation on the LFC weighting factor is obtained from p(n)+(oq !(oq )(1!a) (27) in which oq follows directly from the correspond ing 1n2-value, through oq "1n2!1. In Fig. 15 a direct comparison is made between the approximate values from Eq. (27) with the true standard deviations derived from Eq. (26) as a function of 1n2 for a counter with a"0.68. In the considered
467
example, the relative error of Eq. (27) remained under 5% for 1n2-values lower than 4. 5.3. Extending deadtime and pileup In the case of non-extending deadtime a certain amount of valid pulses in an inspection period can lead to at most two di!erent n-values. With extending deadtime and pileup, however, there are more possibilities. One valid pulse can hide several other pulses which stretch the system busy time and hence can induce larger n-values. In the case of pileup rejection, the weighting factor can exceed the value of 1 even if no valid count was registered during the inspection period. An in-depth theoretical analysis of the distribution of the weighting factors was not attempted here. A semi-empirical approach was followed instead. The computer simulation program was used to produce a data set of p(n)-values for di!erent circumstances; free parameters being the ratios PET/¹ (PET"0 corresponds to extending dead5 time [11,12]), a"q/IP , and the count rate o. Fortunately, identical p(n)-values were found for extending deadtime as for pileup rejection with the same `characteristic deadtimea q"PET#¹ ; 5 one formula su$ces for any ratio of leading edge to total pulse width, PET/¹ . 5 The following semi-empirical formula seems to work satisfactorily in almost any conceivable practical circumstance
p(n)+ (n!1)
n#0.45a@>D 1#0.45a@
(28)
in which we de"ne a"q/IP , D"0.6(1!e\L\) and b"!2.6. If the value of a is close to 1, one can also consider using this simple expression:
p(n)+
Fig. 15. Standard deviation of the LFC weighting factor as a function of the average value 1n2 for a counter with nonextending deadtime (q /IP "0.68). The squares represent exact L values derived from Eq. (26) and the circles represent approximate values from Eq. (27).
n!1 . 2
(29)
The parameter values of D and b are chosen in a way to render a good over-all performance, even though sometimes better accuracy could be obtained from locally adjusted values. In general, the approximating formula works the best for 0.5(a(1. However, even at a"0.16 (the lowest
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a-value tested, giving also the worst result) the error stays within 6% for 1n2-values below 2 ("50% count loss). Moreover, there is considerable error reduction when applying this semi-empirical formula in the LFC uncertainty formula 2, in particular for a(0.5 due to the relatively lower absolute values of p(n). The performance of Eq. (28) is demonstrated in Fig. 16, showing the standard deviation of the LFC weighting factor for a"0.68 at di!erent count rates. In this particular case the relative error remained well below 5% for 1n2-values lower than 20. The "gure also includes the induced error on the LFC uncertainty by using the approximating formula 28, which in the present case is less than 1% for 1n2 below 10. From numerical tests, one can conclude that the combination of Eqs. (2) and (28) or (27) yields an excellent estimate of the true LFC uncertainty, the systematic error being less than 2% for incoming count rates up to oq"1. The required input is the weighting factor n, the type of count loss, and the e!ective deadtime relative to the minimum VPG inspection period.
5.4. A persistent myth concerning LFC statistics It is assumed by many users of the loss-free counting method that the `statistical problem with LFCa is mainly a consequence of applying discrete, integer values for the weighting factor. Indeed, as nuclear spectra are still recorded as an array of integers, the LFC performs add-n operations only with integer values of n. The fractional part obtained for the current inspection period is saved and added to the n-value of the following inspection period. The sum of the fractional parts eventually lead to a full integer increase of n during one inspection period. It seems to be a widely entertained opinion that this procedure leads to a signi"cant increase of the variance of loss-free counting, and that counting statistics would bene"t from the use of #oating-point values for n and hence also the entire spectrum. However, this premise is contested in this work. First consider the case of non-extending deadtime. If the deadtime is shorter than the minimum inspection period length, one needs more than one valid count to increase n by one unit. Assuming that e.g. a"q/IP "0.7, one obtains n"2 for two valid pulses and n"3 for 3 valid pulses, together with a remaining fractional part of, respectively, 0.4 and 0.1. Alternatively, one could apply the true fractional representation of the weighting factors. Then one, two or three valid counts would yield, respectively, n"1.7, 2.4 and 3.1. The average weighting factor is the same in both scenarios. However, the distribution of the #oating-point factors is now calculated from P(n"1#ka)"P(k)
Fig. 16. (bottom) Standard deviation of the LFC weighting factor as a function of the average value 1n2 for a counter with extending deadtime or pileup rejection (q/IP "0.68). The squares represent data from computer simulations and the circles represent approximate values from Eq. (28). (top). The relative error on the standard deviation of loss-free counting as a consequence of using an approximation for p(n).
(30)
in which P(k) is again the Poisson probability for k valid events in one inspection period. In Fig. 17 a typical distribution of n is shown for a counter with non-extending deadtime, for the integer as well as for the #oating-point representation. A "rst remarkable fact is that the distribution of n is still restricted to a discrete number of values; the application of the true fractional representation does not imply a continuous distribution of n. Secondly, the n-distribution does not appear to be signi"cantly narrower than for an integer representation. In fact, numerical tests for a"0.68 and
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Fig. 17. Probability distribution of the LFC weighting factor for a counter with non-extending deadtime in the case of a #oatingpoint representation of n (upper part) (Eq. (30)) and an integer representation (lower part) (Eq. (26)). This example corresponds to an incoming count rate of oq"1.74 and a"0.68. Both representations yield the same average value of 1n2"2.77 and a relative width of p(n)/1n2"39% (#oating point) and 42% (integer).
IP "51.2 ls show that the resulting in#uence on the LFC uncertainty is nearly negligible, i.e. the standard deviation p(N ) is lowered by 2% or less. *$! Then there is also the case of extending deadtime and pulse pileup. Again these loss mechanisms give rise to similar probability distributions of the weighting factor. Fig. 18 shows a nice example, obtained by computer simulation. The graph still shows some discrete character at low n-values, but gradually evolves to a continuous distribution. Yet the width is still comparable with the corresponding integer representation, and consequently also here only minute changes, in the order of 0}2%, are found in the LFC uncertainty.
6. Conclusions The statistical uncertainty of loss-free counting of a stationary Poisson process was scrutinised for di!erent types of count loss, using data from com-
469
Fig. 18. Probability distribution of the LFC weighting factor for a counter with extending deadtime in the case of a #oating-point representation of n (upper part) and an integer representation (lower part). (The distributions were subdivided into bins of 0.1.) This example corresponds to an incoming count rate of oq"1.39 and a"0.68. Both representations yield the same average value of 1n2"4.1 and a relative width of p(n)/1n2"64.65% (#oating point) and 65.0% (integer). A similar result would be obtained for a counter with pileup rejection having the same e!ective deadtime.
puter simulations of nuclear counting, experimental tests on a HPGe c-ray detection set-up, and mathematical models. Counting statistics are dependent on the amount and type of non-random loss in the counter. For counters with deadtime only, the relative LFC uncertainty is independent of the size of the considered fraction of the LFC spectrum, even though this is not the case in the corresponding uncorrected spectrum. For counters with pileup rejection, the statistical uncertainty is relatively higher and there is a clear dependency on the spectrum fraction, in anti-correlation with the statistics of the uncorrected spectrum. The mechanism behind the increased count scatter observed on spectrometers with pileup rejection is studied theoretically. Contrary to other loss mechanisms, pileup rejection does not preserve Poisson statistics during the system live-time. The countyield distribution per LFC inspection period is signi"cantly broadened and the standard deviation of
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the count integral of LFC spectra increases proportionally. Such broadening was also observed in "xed live-time spectra taken with a HPGe detector. A new formula (Eq. (2)) was presented to account for the statistical uncertainty of Loss-Free Counting of a Poisson process. It replaces the existing Westphal formula (Eq. (1)), which is valid for pulseloss by extending or non-extending deadtime, but fails in the case of pileup. In addition useful formulas (Eqs. (27) and (28)) are presented to calculate the variance of the LFC-VPG weighting factor, which is an indispensable component of LFC statistics. The combination of Eqs. (2) and (27) or (28) predicts the standard deviation of LFC very well, the error generally being less than 2%. It accounts for the uncertainty of any spectrum fraction, including the full LFC spectrum. The required input parameters can easily be deduced from the measured dual LFC spectra and from basic knowledge concerning the counting device. Furthermore, proof is presented that the LFC uncertainty is hardly in#uenced by the fact that integer values are used for the VPG weighting factors, instead of #oating-point values. On the other hand, it does depend on the ratio between the characteristic deadtime of the counter and the inspection interval of the virtual pulse generator checking the system's availability for counting.
References [1] G. Gilmore, J. Hemingway, Practical Gamma-Ray Spectrometry, Wiley, Chichester, England, 1995. [2] J. Harms, Nucl. Instr. Meth. 53 (1967) 192. [3] G.P. Westphal, Nucl. Instr. and Meth. 163 (1979) 189. [4] G.P. Westphal, J. Radioanal. Chem. 70 (1982) 387. [5] G.P. Westphal, Nucl. Instr. and Meth. B10/11 (1985) 1047}50. [6] K. Heydorn, E. Damsgaard, J. Radioanal. Nucl. Chem. 215 (1997) 157. [7] S. PommeH , J.-P. Alzetta, J. Uyttenhove, B. Denecke, G. Arana, P. Robouch, Nucl. Instr. and Meth. A422 (1999) 388}394. [8] K. Debertin, R. Helmer, Gamma- and X-ray Spectrometry with Semiconductor Detectors, Elsevier Science Publishers BV, Amsterdam, The Netherlands, New York, USA, 1988. [9] S. PommeH , Proceedings of the 2nd International k Users Workshop, Ljubljana, Slovenia, 1996, Jozef Stefan Institute, 1997, p.11}14. [10] S. PommeH , F. Hardeman, P. Robouch, N. Etxebarria, G. Arana, Neutron activation analysis with k -standar disation: general formalism and procedure, Internal Report SCKzCEN, BLG-700, pp. 105. [11] S. PommeH , Appl. Rad. Isotop. 49 (1998) 1213. [12] S. PommeH , B. Denecke, J.-P. Alzetta, Nucl. Instr. and Meth. A 426 (1999) 564. [13] S. PommeH , Is loss-free counting under statistical control?, presented at International Conference on Nuclear Analytical Methods in The Life Sciences, Beijing, China, 1998, Biological Trace Element Research, in press.
How pileup rejection a!ects the precision of loss-free counting S. PommeH * SCKzCEN, Belgian Nuclear Research Centre, Boeretang 200, B-2400 Mol, Belgium Received 25 January 1999; received in revised form 29 March 1999
Abstract The mechanism behind the particularly high count scatter observed with loss-free counting (LFC) on spectrometers with pileup rejection is scrutinised. It is shown theoretically and experimentally that, contrary to other loss mechanisms, pileup rejection does not preserve Poisson statistics during the system live-time. Consequently, the statistical uncertainty obtained with loss-free counting increases proportionally to the broadening of the count-yield distribution per live-time inspection period. A new theoretical formula for the LFC uncertainty is presented and predicted values are compared with experimental data obtained on a gamma-ray spectrometer and data from computer simulations of nuclear counting. Additionally, the statistical e!ect of using #oating-point values instead of integers for the LFC correction factors is investigated; the LFC uncertainty shows little change. 1999 Elsevier Science B.V. All rights reserved. PACS: 02.50.Cw; 02.90.#p; 07.85.!m Keywords: Nuclear counting; Loss-free counting; Statistics; c-ray spectrometry; Uncertainty; Neutron activation analysis
1. Introduction Loss-free counting (LFC) is known to be an advanced real-time correction method of counting losses in nuclear pulse spectrometry [1}7]. Count loss is compensated by performing add-n operations to the spectrum, in which the weighting factor n is derived somehow from the amount of loss whenever the system is unable to count. In 1981 Westphal introduced the Virtual Pulse Generator (VPG) technique to evaluate the LFC weighting factor [1,4,5]. Its operation is similar to the
* Tel.: #32-14-332718; fax: #32-14-321056. E-mail address: [email protected] (S. PommeH )
well-known pulser method [1,8], however without introducing pulser signals into the spectrum. As a consequence, the system availability can be checked at a much higher frequency, hence drastically improving the ability to respond to rapid changes in the count rate. Performance tests on c-ray spectrometers have demonstrated that LFC with VPG can provide accurate results (better than 1% uncertainty) over a wide range of incoming count rates (see e.g. Ref. [7]). The LFC method with VPG is well suited for application in e.g. neutron activation analysis, since the commonly higher loss of pulses from short-lived activation products is well taken into account. Such would not be the case with live-time correction techniques or methods supplying one average
0168-9002/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 9 ) 0 0 4 7 5 - 1
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correction factor for the whole spectrum. Another advantage of major importance is its ability to deal with pulse pileup. This is essential, considering that pileup (rejection) is the major pulse-loss mechanism in contemporary fast spectrometry chains. In fact, the combinations of pulse pileup and extending and non-extending deadtime featuring in typical spectrometers present no problem to the powerful LFC principle. The only special precaution is the addition of a pulse evolution time (PET) to each system busy period, to account for loss by leading-edge pileup [1,3}5,7]. In practice, the PET signal width in the LFC module is manually "ne-tuned using the double-source method, to reach optimum loss compensation. (Some additional care is required with reset preampli"ers [7].) At the negative side of the LFC method is the unavoidable modi"cation of counting statistics. Whereas LFC can restore the average count rate of an event train passing through an electronic counter, it is unable to reproduce the original time interval distribution of subsequent events. The distortion of the original exponential interval density distribution, which is characteristic for Poisson processes, will eventually alter the uncertainty on the count integral measured during a "xed realtime. According to Westphal [3,4], the statistical accuracy of loss-free counting can be calculated from
dN 1 dn *$! " 1# N 1n2 (N *$!
(1)
in which N is the number of counts in the *$! considered region of the LFC spectrum, N is the corresponding number of events recorded without loss correction, and dn is the uncertainty on the LFC weighting factor n. Heydorn and Damsgaard [6] were probably the "rst to recognise that the commonly adopted Westphal formula does not always account for the count scatter in LFC spectra. PommeH et al., having scrutinised the statistical e!ects of pileup rejection on nuclear counting [9}12], suggested that the problem of statistical control of loss-free counting was related to pulse pileup rejection [7,13]. The increased count scatter was attributed to the broadening of the statistical distribution of the
457
valid events per VPG inspection period. A new LFC uncertainty formula was presented [13]:
p(N ) r p(n) *$! " 1# N 1n2 (N *$!
(2)
with
A 2#A p(Ccounts/IP) +1# f r" 2 1#A (Ccounts/IP in which we have A"1!e\M .#2, A "1!e\D M .#2, and f is the considered fraction of the spectrum ( f"100% is full spectrum). In this work, the theoretical derivation of formula 2 will be described. Its validity will be tested for di!erent experimental situations, such as c-ray spectrometers with a gated-integrator ampli"er or with a classical semi-Gaussian shaping ampli"er. By means of computer simulations, the operation of the LFC principle will be tested on other, hypothetical types of counters. For details on the experimental set-up and the simulation program, the reader is referred to Refs. [7,11,12].
2. Schematic representation of event pulses Rate-related losses are basically caused by nearly synchronous events and the limited time resolution of the counter. Which pulses of an incoming event train are counted eventually depends on the loss mechanism. To visualise the e!ect of deadtime and pileup, a schematic representation of a typical event pulse is shown in Fig. 1. The circle represents the position of a detector pulse, at the point at which the peak is detected. The triangle in front of the peak represents the pulse evolution time; i.e. the time between pulse arrival and peak detection. The rectangle behind the peak represents deadtime; it can correspond to the trailing edge of the pulse, equivalent to extending deadtime [11,12], or to the non-extending deadtime of the analog-to-digital convertor. Below the base line, Virtual Pulse Generator (VPG) time units are shown in which the system is considered `busya by the LFC module. In Fig. 2 an impression is given on how di!erent loss mechanisms interact with a speci"c event train.
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in time zone C can be valid, even though the VPG considers this region as belonging to the extended system busy time. If the second pulse falls within region D, it is valid unless it interacts with a third event falling within its zone A. For a Poisson process, with average count rate o, the probabilities for subsequent pulses falling into a particular zone is calculated from
Fig. 1. Schematic representation of a detector pulse and its associated event loss mechanisms. Time zone A corresponds to pulse pileup, zone B to deadtime, zone C is an extension of the system busy by the LFC device and D is outside the extended system busy time.
A"P(*t)PET)"1!e\M.#2
(3)
B"P(PET(*t)¹ )"e\M.#2!e\M25 5 C"P(¹ (*t)q)"e\M25!e\Ms 5 D"P(*t'q)"e\MO
(4) (5) (6)
with q"PET#¹ . Evidently, the sum of these 5 probabilities, A#B#C#D, is equal to one.
3. Counting statistics in live-time 3.1. Introduction When searching an explanation for the increased loss-free counting uncertainty for systems with pileup rejection [7,13], computer simulations provided an interesting hint; there seemed to be a correlation between p(N ) and the average *$! number of valid counts per `non-voida LFC-VPG inspection period, 1VC/NVIP2. Indeed, one can reproduce the counting uncertainty to a good approximation by Fig. 2. Event train passing through counters having di!erent loss mechanisms; NEDT"non-extending deadtime, EDT"extending deadtime and PU"pileup rejection. The case of combined PU and EDT corresponds to a spectrometer with Gaussian pulse shaping and fast ADC with negligible deadtime. The empty circles correspond to invalid events. The e!ective deadtime is assumed to be equal in all four cases, i.e. q "q "PET#¹ . The tick marks below the baseline rep 5 resent the extended system busy time registered by the VPG.
Even though the same e!ective deadtime was considered, di!erent events are counted as valid or invalid. Pulses falling within time zones A and B (see Fig. 1) of another event are always lost. However, in time zone A also the "rst pulse is invalidated by pileup rejection. Pulses falling with-
p(N 1 p(n) *$! + 1#1VC/NVIP2 1n2 N (N *$! (7)
with 1VC/IP2 1VC/NVIP2" P(VC'0)
(8)
in which P(VC'0)"1!P(VC"0) is the probability of having at least one valid count in an inspection period and 1VC/IP2 is the average number of valid counts per inspection period. The remarkable aspect of this quantity is the signi"cant di!erence observed for systems with pileup rejection compared to systems with extending deadtime
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at the same loss rate, i.e. at the same o and with q "PET#¹ . This lead to a study of the count 5 number distribution per inspection period. First consider the case of extending deadtime. Each count arriving in the considered inspection interval creates a certain amount of deadtime in which the detector is unable to count. The VPG of the LFC marks these periods as `busya and prolongs the current inspection period with the same amount of real-time. Consequently, the LFC-VPG conserves the amount of system live-time per inspection period. By removing the `system busya periods one can convert from `counting in realtime using a system with deadtimea to `counting in live-time without lossa. This reduces the size of the inspection period, IP, to the minimum size, IP . (In our LFC apparatus, IP corresponds to 51.2 ls; this is 256 VPG ticks at a frequency of 5 MHz.) For a stationary Poisson process, the arrival time of any event is independent of its `historya, so the same Poisson statistics can be applied in live-time. The average amount of counts per inspection period, 1VC/IP2, is therefore equal to o.IP . Similar considerations are valid for non-extending deadtime, leading to the same average count number per inspection period. The inspection periods, however, are shorter on the average, since 1IP2 "exp(oq ) IP in the case of extending deadtime, and
(9)
1IP2 "1#oq (10) L IP in the case of non-extending deadtime. Now, consider a system with pileup rejection, having an identical e!ective deadtime as the EDT counter, q"PET#¹ "q . Since the throughput 5 is also exponential [12], one expects the same average duration of the inspection periods. Whereas also here 1VC/IP2"o IP , one obtains a signi" cantly higher result for 1VC/NVIP2. This must be due to a higher probability for having zero valid counts in an inspection period, P(VC"0) (see formula 8). Consequently, the number of counts per inspection period is not Poisson distributed when counting with pileup rejection. The underlying rea-
459
son is that the LFC extends `system busya beyond the point at which the system is again available for data acquisition, to compensate for the twofold loss of piled-up events. By a detailed comparison between the loss mechanisms of EDT and PU, the necessary adaptations to Poisson probabilities will be made to arrive at the correct counting statistics per inspection period for PU. 3.2. Probabilities for relative count gain and loss Consider a series of closeby events, not more than q apart. With an EDT counter, only the "rst count is valid. Now when switching to a PU counter, the "rst count can possibly become invalid, while other counts in the series could become valid. In this section, the probabilities are calculated for losing a valid count or gaining #k (k"0, 1, 2,2) valid counts when switching from an EDT to a PU system. Di!erences between extending deadtime (EDT) and pileup (PU) systems are shown in Fig. 3 for series of at most two closeby events. Events falling in zone D or B are treated in the same manner as with extending deadtime. However, with PU, the "rst count is lost additionally if the second one falls within zone A. On the other hand, a valid count is gained in zone C. The situation becomes more complex if longer chains of closeby events have to be disentangled.
Fig. 3. Comparison of a counter with only extending deadtime (top) and a counter with symmetrical pulse shaping, applying pileup rejection (bottom). Both represent equal e!ective deadtimes. Both systems yield the same outcome for pulses falling within the B and D zone. Pileup rejection causes the loss of one extra count in the A-zone, while the reverse is true for the C-zone.
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460
First consider P(*"!1); the following combinations lead to a loss of the valid count, without other counts in the series becoming valid P(*"!1)"AD
The general case of P(*"#k) is easily found in a similar fashion, yielding the following expression: P(*"#k)"+(1#A[N]C)CI ;(1#(B#CA)[N])I>,D.
(16)
#A(A#B#CA)D #A(A#B#CA) (A#B#CA)D #A(A#B#CA) (A#B#CA)
P(*"k#1) P(gain)" P(*"k)
;(A#B#CA)D #2 AD . " 1!(A#B#CA)
The probability for loss, P(loss)" P(*"!1), should be equal to the probability for count gain, P(gain). This is indeed found:
"C(1#(B#CA)[N]) (11)
A " "P(loss). 1#A
(17)
One can de"ne a `neutrala series [N], in which the combinations of gain and loss cancel each other:
The expressions for P(*"#k) can now be simpli"ed.
[N]"[1!(A#B#CA)]\
AC P(*"#0)" 1# (1#A)D
"[(1#A)D]\.
(12)
As a result, the formula for P(*"!1) can be simpli"ed: A P(*"!1)"A[N]D" . 1#A
(13)
The next quantity to assess, is the probability that the series of closeby events yields exactly one valid count, just as much as with the EDT counter: P(*"#0)"D#B[N]D#CA[N]D
A (1#A)C
"[1#A]\"[1!P(loss)]
(18)
P(*"#k)"P(*"#0) P(gain)I AI " . [1#A]I>
(19)
One can verify that the sum of the probabilities is indeed equal to one. 3.3. Distribution of the number of valid counts per inspection period
#A[N]CD#A[N]CB[N]D #A[N]CCA[N]D "+(1#A[N]C) ;(1#(B#CA) [N]),D.
(14)
The probability that one valid count in the EDT system corresponds to two counts in the PU system is then P(*"#1)"+(1#A[N]C)C ;(1#(B#CA)[N]),D.
(15)
Knowing the probabilities for count gain and loss with pileup systems compared to extending deadtime systems, the number of valid counts can be calculated starting from Poisson statistics. By P(k) we denote the Poisson probability of having k valid events per inspection period, possibly followed by a series of invalidated events in their deadtime trail. In the case of a PU system, zero valid counts are found when either no events fall within the inspection period or when all events are invalidated by pileup rejection. The probability for zero valid counts in the inspection interval is hence
S. Pomme& / Nuclear Instruments and Methods in Physics Research A 432 (1999) 456}470
calculated from > P(C"0)" P(k) P(loss)I I > [o IP P(loss)]I " e\M'. k! I o IP "exp ! 2!e\M.#2
(20)
since 1!P(loss)"[2!e\M .#2]\ (see Eq. (18)). The probability for having exactly one valid count is > P(C"1)" P(k) P(*"0)CP(loss)I\ I I > (oIP )I P(loss)I "(o IP )e\M'. P(*"0) k! I o IP . (21) "(o IP ) P(*"0)exp ! 2!e\M .#2
For k'0, one can write as a general expression
L CI\L I\L I o IP A I\ P(C"k)" n! (1#A) 1#A L oIP ;exp ! . (22) 2!e\M'.
461
The validity of formula 22 has been checked by comparing with computer simulation data. An excellent agreement is found for any choice of input parameters. The simulations con"rm that Poisson statistics can be applied in the case of deadtime. However, the probability distributions signi"cantly di!er in the case of pileup. In Fig. 4 the probability distribution of valid counts is compared between a system with pileup and another with deadtime. Whereas in both cases an average of "ve valid counts per IP is taken, one clearly sees a broadening of the probability distribution for counting with pileup rejection. 3.4. Distribution of valid counts for a region of interest In spectrometry, usually only particular regions of interest (ROI) are counted, instead of the fullenergy spectrum. The counting statistics change as a function of the fraction of the spectrum which the ROIs represent. The number of valid counts per inspection period can be derived from the result for the full spectrum, by considering that each count has the same a priori probability of belonging to the considered ROI, being equal to the fraction f. One can readily verify that the distribution of valid
Fig. 4. Theoretical probability distribution of the number of valid counts per "xed live-time period (or inspection period) for counters with a di!erent type of count loss: extending deadtime (PET"0) or pulse pileup (PET"¹ ). In both cases an average of 1VC/IP2"5 5 was taken. The distribution is clearly broader for pileup rejection.
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counts in a ROI can be obtained from > P (VC"k)" P(VC"k#i)CI>G(1!f )G f I D I G
3.5. Quantifying the distribution width
(23)
in which P (VC"k) is the probability of having D k valid counts in a ROI per inspection period. In Fig. 5 an example is shown of the probability distribution of the number of counts for di!erent ROIs of the same spectrum. The smaller the considered part of the spectrum, the more the distribution resembles Poisson statistics.
If the number of valid events counted in an inspection period is Poisson distributed, one can assume that the variance is equal to the expectation value, i.e. p(VC/IP)"1VC/IP2. This can be applied for counters with deadtime, but a new formula is required for counters with pileup rejection. In this work the following approach is suggested:
A 2#A p(VC/IP)+(1VC/IP2 1# f 2 1#A
(24)
with A and A as de"ned in Eq. (2). This is exactly the correction factor applied in the LFC uncertainty formula 2 (and also in Ref. [13]). This formula is a slightly improved version of the "rst, published in Ref. [7], in which a linear relationship as a function of the spectrum fraction f was assumed. In the new version a slight nonlinearity as a function of f was taken into account by introducing the factor A in the formula. The accuracy of formula 24 is demonstrated in Fig. 6, in which its resulting values are compared with the true width of the count distributions shown in Fig. 5. The width converges to (1VC/IP2 when f goes to zero, i.e. for small fractions. The same is true when the factor A approaches zero, i.e. when little or no pileup occurs.
Fig. 5. Probability distribution of the number of valid counts per inspection period for a counter with pileup rejection, valid for di!erent fractions f of the spectrum, as calculated from Eqs. (23) and (22) (oq"1.3, PET"¹ "20 ls and IP "51.2 ls). 5 Deviations from Poisson statistics are more pronounced when considering large fractions of the spectrum.
Fig. 6. Relative width of the probability distribution of the number of valid counts per inspection period for a counter with pileup rejection, as a function of the fraction f of the spectrum. The true width, as calculated from Eqs. (23) and (22) (oq"1.3, PET"¹ "20 ls and IP "51.2 ls), is well reproduced by 5 formula 24.
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3.6. Experimental counting uncertainty at xxed live-time The theoretical considerations depicted above enable the prediction of the counting uncertainty in an uncorrected spectrum measured at "xed livetime. In the case of a counter with pileup rejection one can expect a deviation from Poisson statistics, following p(N)"r(N
(25)
in which N is the number of counted events and r the correction factor de"ned in Eqs. (2) and (24). The validity of this premise has been veri"ed experimentally on the HPGe c-ray spectrometer set-up `Panoramixa, equipped with a pileup rejection circuit (see Refs. [7,12]). Series of measurements ('2000) of c-rays from a Cs source were repeated at di!erent source}detector distances. The shaping ampli"er is the main bottleneck of the counting device, since the ultra-fast ADC causes no additional pulse loss. Semi-Gaussian pulse shaping was applied with a timing constant of 4 ls. This corresponds to an average pulse width of 22 ls and a pulse evolution time of 12 ls, hence an e!ective deadtime of 34 ls (see Ref. [12]). Fig. 7 shows the standard deviation of the number of counted events in spectra acquired at a "xed live-time of 0.2 s, relative to (N. The deviation of the counting uncertainty from Poisson statistics clearly grows with increasing incoming count rate. The actual experimental values are slightly higher than theoretically expected (dashed curve). Better agreement is obtained with a hypothetical value of PET"14 ls, instead of 12 ls. At extremely high count rates (oq'1.8, with more than 80% count loss) the discrepancy between theory and experiment is more signi"cant, probably due to the imperfection of the actual pileup rejection system. Also the dependency of counting statistics on the considered fraction of the c-ray energy spectrum has been investigated. Whereas Fig. 7 corresponds to the full spectrum, Fig. 8 shows the standard deviation for fractions of the spectrum ranging from 0.01% up to 100%, at a "xed total incoming count rate of 25.6;10 s\. The nearly
Fig. 7. Statistical uncertainty of the number of counts in a full c-ray spectrum as a function of the incoming count rate, as observed experimentally with a HPGe c-ray detector with pileup rejection and semi-Gaussian pulse shaping, using a "xed live-time of 0.2 s. The full line corresponds to the theoretical values from Eqs. (25) and (24), assuming PET"14 ls. The dashed curve corresponds to a more realistic value of PET"12 ls.
Fig. 8. Statistical uncertainty of nuclear counting as a function of the spectrum fraction, observed experimentally with a HPGe c-ray spectrometer with pileup rejection and semi-Gaussian pulse shaping, at an incoming count rate of o"25.6;10 s\ and using a "xed live-time of 0.2 s. The full line corresponds to the theoretical values from Eqs. (25) and (24), assuming PET"14 ls.
linear behaviour of the relative uncertainty is well reproduced by theory. Convergence towards Poisson statistics is demonstrated for small fractions.
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4. Statistics of loss-free counting 4.1. Validity range of the Westphal formula The Westphal formula (see Eq. (1)) seems to be directly applicable to the counting uncertainty of small fractions of the LFC spectra. Indeed, then the number of counts N can be interpreted as the *$! product of two independent entities: the number of processed events N and the average weighting factor 1n2. Their relative uncertainties being, respectively, p(N)/N"1/(N and p(1n2)/1n2"p(n)/ (N1n2, one easily "nds the Westphal formula as the square root of the sum of the variances. In spite of some fundamental objections to applying this reasoning also to larger parts of the spectrum [11], the extrapolation of loss-free counting statistics to large regions of interest appears to be straightforward in practice. In Figs. 9 and 10 the standard deviation of the complete LFC spectrum is shown as a function of the incoming count rate for a hypothetical counter with extending and non-extending deadtime respectively; the data are obtained from computer simulations. The graphs clearly show that the standard deviation is nicely reproduced by the Westphal formula (Eq. (1)) as well as by the formula presented in this work
Fig. 10. Same as Fig. 9, for a counter with a non-extending deadtime of q "34.8 ls.
(Eq. (2)). Indeed, it turns out that the counting statistics for counters with deadtime show no signi"cant dependence on the considered fraction of the spectrum. This can be veri"ed from Fig. 11, in which counting statistics of the LFC spectrum and the corresponding uncorrected spectrum are shown at oq"1 as a function of the spectrum fraction f. The LFC uncertainty remains practically constant near the level of the theoretical value, in spite of the drastic change of the variance on the corresponding number of real events counted in the uncorrected spectrum. 4.2. Counters with pileup rejection
Fig. 9. `Relativea uncertainty of loss-free counting as a function of incoming count rate using a hypothetical counter with an extending deadtime of q "34.8 ls and a VPG with a frequency of 5 MHz (IP "51.2 ls). The data points are obtained from computer simulations and the theoretical values from Eq. (1) (dashed line) and Eq. (2) (full line).
The Wesphal formula for LFC uncertainty fails for systems in which counts are lost by pileup rejection. As an example, the default set-up of the Panoramix c-ray spectrometer (semi-Gaussian pulses with 4 ls shaping constant) was tested by experiment and by computer simulation. The standard deviation on the count integral of the LFC spectra is presented as a function of incoming count rate in Fig. 12. One can see an excellent agreement between theory and experimental data up to at least o"1/q, corresponding to 63% count loss. Slight deviations at higher count rates are attributed to the "nite pulse pair resolution of the pileup rejection system, allowing an increasing number of compound pulses to slip through. The simulation data con"rm that Eq. (2) is in principle
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Fig. 12. Standard deviation of the count integral of measured LFC spectra, compared to theoretical expectations and computer simulation data. Experimental data correspond to typically 2000 c-ray spectra of Co taken on HPGe detection set-up `Panoramixa (q"PET#¹ "12.1#22.7 ls). The the5 oretical data are calculated from Eq. (1) (dashed line) and Eq. (2) (full line), for which the relevant p(n)-values were derived from the simulations.
Fig. 11. Counting statistics obtained by computer simulation in an LFC spectrum and the corresponding uncorrected spectrum for counters with extending deadtime and non-extending deadtime, as a function of the considered fraction of the spectrum and at a "xed incoming count rate of oq"1. Theoretical values for LFC spectra are calculated from Eq. (1) or Eq. (2), and for the uncorrected spectrum from a formula for EDT counters in Ref. [12].
correct up to extremely high count rates, even close to 100% count loss. Similar results were obtained on another detector set-up with a transistor reset preampli"er instead of the classical RC feedback type. The occasional reset times of the preampli"er are apparently insu$cient to change the counting statistics in a signi"cant way. The new formula also works for a spectrometer equipped with a gated-integrator ampli"er, which produces pulses in the shape of a cumulative Gaussian. The trailing edge is extremely short, and as a result, the pulse evolution corresponds to the duration of almost the entire pulse, i.e. PET+¹ 5 and q+2PET. For experimental tests, the Panoramix main ampli"er was temporarily replaced by a gated-integrator ampli"er. The observed loss-free
Fig. 13. Standard deviation of the count integral of full LFC spectra of Cs c-rays measured by a detector with gatedintegrator ampli"er, for which the pulse evolution time is about equal to the total pulse width (q/2"PET"¹ "15 ls). 5
counting statistics for c-ray spectra of Cs are presented in Fig. 13. Again, the theoretical and experimental values agree very well up to oq"1. In fact, computer simulations demonstrate that statistical control is achieved for any ratio between the pulse evolution time and the total pulse width, PET/¹ , going from 0 up to 1. 5
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Fig. 14. Standard deviation of the number of counts in a region of interest of LFC spectra on c-ray detector `Panoramixa (o"32 kBq), compared to theoretical expectations (formulas 1 and 2).
More importantly, also the counting statistics of a random selection of the event spectrum can be determined accurately. The LFC uncertainty formula varies according to the fraction of the spectrum represented by the selected region of interest, as shown by experiment in Fig. 14 for oq"1.1 and PET"q/2.8. The variation of p(N ) with f can *$! be well approximated by a linear interpolation between the Eqs. (1) (for f"100%) and (2), but the slightly bent curve is better taken into account by the improved representation of the correction factor r (see Eq. (2)). Also here the Westphal formula is an excellent approximation when considering small fractions of the spectrum, like e.g. one c-ray peak in a complex activation spectrum.
5. Variance of the LFC weighting factor 5.1. The need for an expression on p(n) When applying the LFC uncertainty formulas 1 and 2, use was made of simulation results for the standard deviation of the VPG weighting factor, p(n). Of course, a more practical way to assess p(n) is needed for everyday c-ray spectrometry. Up to now, literature provided no in-depth theoretical analysis of this complex matter. An alternative approach is to determine p(n) directly from the measured LFC spectrum and corresponding un-
corrected spectrum, by sampling the n-values in about 20 energy channels around the considered c-ray peak. However, aside from perhaps an insu$cient accuracy of this method one has to consider here the danger that p(n) can be severely overestimated if local variations of n-values occur because di!erent peaks and continua in the spectrum correspond to radionuclides with di!erent half-lives, having experienced a di!erent average amount of count loss during the measurement. In fact, one needs an expression which is related to the n-value of the considered region of interest alone. In this work we have attempted to develop a semi-empirical formula to calculate p(n) with suf"cient precision in a wide range of measurement conditions. A di!erent result was found for counters with non-extending deadtime on one hand and for counters with extending deadtime or pileup rejection on the other. 5.2. Non-extending deadtime In the case of non-extending deadtime the distribution of the LFC weighting factor n can readily be predicted. Consider that the length of the VPG inspection period IP is the sum of its minimum value IP and the amount of system busy time during the inspection period, which in this case is simply the number of valid counts multiplied by the characteristic deadtime q . The LFC weighting factor n is de"ned as the ratio between the length of the inspection period IP and the minimum length of an inspection period IP . As only integer values are accepted, the fractional part of IP/IP is kept and added to the IP-value of the next inspection period(s). (For simplicity, it is assumed that the remainder varies in a continuous and unpreferential way between 0 and 1.) Now assume that the deadtime per event is smaller than IP , with a ratio of a"q /IP . The probability for n to take the value x, P(n"x), is then I P(n"x)" P(k)(ka!(x!2)) II I # P(k)(x!ka) II
(26)
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for which we de"ne
k "#oor
n!2 n!1 , k "#oor , a a
n!1 k "ceil a
n and k "#oor , a
in which `ceila and `#oora signify rounding to the closest upper or lower integer respectively and P(k) is the Poisson probability for having k valid events during the inspection period, considering that the expectation value is equal to oIP . The validity of Eq. (26) has been succesfully demonstrated by comparison with simulation results. Whereas the variance of n can be calculated from the theoretical distribution from Eq. (26), a more direct and fast method is desirable. One can show mathematically that, for systems with non-extending deadtime and a"q /IP (1, a good approxi mation for the standard deviation on the LFC weighting factor is obtained from p(n)+(oq !(oq )(1!a) (27) in which oq follows directly from the correspond ing 1n2-value, through oq "1n2!1. In Fig. 15 a direct comparison is made between the approximate values from Eq. (27) with the true standard deviations derived from Eq. (26) as a function of 1n2 for a counter with a"0.68. In the considered
467
example, the relative error of Eq. (27) remained under 5% for 1n2-values lower than 4. 5.3. Extending deadtime and pileup In the case of non-extending deadtime a certain amount of valid pulses in an inspection period can lead to at most two di!erent n-values. With extending deadtime and pileup, however, there are more possibilities. One valid pulse can hide several other pulses which stretch the system busy time and hence can induce larger n-values. In the case of pileup rejection, the weighting factor can exceed the value of 1 even if no valid count was registered during the inspection period. An in-depth theoretical analysis of the distribution of the weighting factors was not attempted here. A semi-empirical approach was followed instead. The computer simulation program was used to produce a data set of p(n)-values for di!erent circumstances; free parameters being the ratios PET/¹ (PET"0 corresponds to extending dead5 time [11,12]), a"q/IP , and the count rate o. Fortunately, identical p(n)-values were found for extending deadtime as for pileup rejection with the same `characteristic deadtimea q"PET#¹ ; 5 one formula su$ces for any ratio of leading edge to total pulse width, PET/¹ . 5 The following semi-empirical formula seems to work satisfactorily in almost any conceivable practical circumstance
p(n)+ (n!1)
n#0.45a@>D 1#0.45a@
(28)
in which we de"ne a"q/IP , D"0.6(1!e\L\) and b"!2.6. If the value of a is close to 1, one can also consider using this simple expression:
p(n)+
Fig. 15. Standard deviation of the LFC weighting factor as a function of the average value 1n2 for a counter with nonextending deadtime (q /IP "0.68). The squares represent exact L values derived from Eq. (26) and the circles represent approximate values from Eq. (27).
n!1 . 2
(29)
The parameter values of D and b are chosen in a way to render a good over-all performance, even though sometimes better accuracy could be obtained from locally adjusted values. In general, the approximating formula works the best for 0.5(a(1. However, even at a"0.16 (the lowest
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a-value tested, giving also the worst result) the error stays within 6% for 1n2-values below 2 ("50% count loss). Moreover, there is considerable error reduction when applying this semi-empirical formula in the LFC uncertainty formula 2, in particular for a(0.5 due to the relatively lower absolute values of p(n). The performance of Eq. (28) is demonstrated in Fig. 16, showing the standard deviation of the LFC weighting factor for a"0.68 at di!erent count rates. In this particular case the relative error remained well below 5% for 1n2-values lower than 20. The "gure also includes the induced error on the LFC uncertainty by using the approximating formula 28, which in the present case is less than 1% for 1n2 below 10. From numerical tests, one can conclude that the combination of Eqs. (2) and (28) or (27) yields an excellent estimate of the true LFC uncertainty, the systematic error being less than 2% for incoming count rates up to oq"1. The required input is the weighting factor n, the type of count loss, and the e!ective deadtime relative to the minimum VPG inspection period.
5.4. A persistent myth concerning LFC statistics It is assumed by many users of the loss-free counting method that the `statistical problem with LFCa is mainly a consequence of applying discrete, integer values for the weighting factor. Indeed, as nuclear spectra are still recorded as an array of integers, the LFC performs add-n operations only with integer values of n. The fractional part obtained for the current inspection period is saved and added to the n-value of the following inspection period. The sum of the fractional parts eventually lead to a full integer increase of n during one inspection period. It seems to be a widely entertained opinion that this procedure leads to a signi"cant increase of the variance of loss-free counting, and that counting statistics would bene"t from the use of #oating-point values for n and hence also the entire spectrum. However, this premise is contested in this work. First consider the case of non-extending deadtime. If the deadtime is shorter than the minimum inspection period length, one needs more than one valid count to increase n by one unit. Assuming that e.g. a"q/IP "0.7, one obtains n"2 for two valid pulses and n"3 for 3 valid pulses, together with a remaining fractional part of, respectively, 0.4 and 0.1. Alternatively, one could apply the true fractional representation of the weighting factors. Then one, two or three valid counts would yield, respectively, n"1.7, 2.4 and 3.1. The average weighting factor is the same in both scenarios. However, the distribution of the #oating-point factors is now calculated from P(n"1#ka)"P(k)
Fig. 16. (bottom) Standard deviation of the LFC weighting factor as a function of the average value 1n2 for a counter with extending deadtime or pileup rejection (q/IP "0.68). The squares represent data from computer simulations and the circles represent approximate values from Eq. (28). (top). The relative error on the standard deviation of loss-free counting as a consequence of using an approximation for p(n).
(30)
in which P(k) is again the Poisson probability for k valid events in one inspection period. In Fig. 17 a typical distribution of n is shown for a counter with non-extending deadtime, for the integer as well as for the #oating-point representation. A "rst remarkable fact is that the distribution of n is still restricted to a discrete number of values; the application of the true fractional representation does not imply a continuous distribution of n. Secondly, the n-distribution does not appear to be signi"cantly narrower than for an integer representation. In fact, numerical tests for a"0.68 and
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Fig. 17. Probability distribution of the LFC weighting factor for a counter with non-extending deadtime in the case of a #oatingpoint representation of n (upper part) (Eq. (30)) and an integer representation (lower part) (Eq. (26)). This example corresponds to an incoming count rate of oq"1.74 and a"0.68. Both representations yield the same average value of 1n2"2.77 and a relative width of p(n)/1n2"39% (#oating point) and 42% (integer).
IP "51.2 ls show that the resulting in#uence on the LFC uncertainty is nearly negligible, i.e. the standard deviation p(N ) is lowered by 2% or less. *$! Then there is also the case of extending deadtime and pulse pileup. Again these loss mechanisms give rise to similar probability distributions of the weighting factor. Fig. 18 shows a nice example, obtained by computer simulation. The graph still shows some discrete character at low n-values, but gradually evolves to a continuous distribution. Yet the width is still comparable with the corresponding integer representation, and consequently also here only minute changes, in the order of 0}2%, are found in the LFC uncertainty.
6. Conclusions The statistical uncertainty of loss-free counting of a stationary Poisson process was scrutinised for di!erent types of count loss, using data from com-
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Fig. 18. Probability distribution of the LFC weighting factor for a counter with extending deadtime in the case of a #oating-point representation of n (upper part) and an integer representation (lower part). (The distributions were subdivided into bins of 0.1.) This example corresponds to an incoming count rate of oq"1.39 and a"0.68. Both representations yield the same average value of 1n2"4.1 and a relative width of p(n)/1n2"64.65% (#oating point) and 65.0% (integer). A similar result would be obtained for a counter with pileup rejection having the same e!ective deadtime.
puter simulations of nuclear counting, experimental tests on a HPGe c-ray detection set-up, and mathematical models. Counting statistics are dependent on the amount and type of non-random loss in the counter. For counters with deadtime only, the relative LFC uncertainty is independent of the size of the considered fraction of the LFC spectrum, even though this is not the case in the corresponding uncorrected spectrum. For counters with pileup rejection, the statistical uncertainty is relatively higher and there is a clear dependency on the spectrum fraction, in anti-correlation with the statistics of the uncorrected spectrum. The mechanism behind the increased count scatter observed on spectrometers with pileup rejection is studied theoretically. Contrary to other loss mechanisms, pileup rejection does not preserve Poisson statistics during the system live-time. The countyield distribution per LFC inspection period is signi"cantly broadened and the standard deviation of
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the count integral of LFC spectra increases proportionally. Such broadening was also observed in "xed live-time spectra taken with a HPGe detector. A new formula (Eq. (2)) was presented to account for the statistical uncertainty of Loss-Free Counting of a Poisson process. It replaces the existing Westphal formula (Eq. (1)), which is valid for pulseloss by extending or non-extending deadtime, but fails in the case of pileup. In addition useful formulas (Eqs. (27) and (28)) are presented to calculate the variance of the LFC-VPG weighting factor, which is an indispensable component of LFC statistics. The combination of Eqs. (2) and (27) or (28) predicts the standard deviation of LFC very well, the error generally being less than 2%. It accounts for the uncertainty of any spectrum fraction, including the full LFC spectrum. The required input parameters can easily be deduced from the measured dual LFC spectra and from basic knowledge concerning the counting device. Furthermore, proof is presented that the LFC uncertainty is hardly in#uenced by the fact that integer values are used for the VPG weighting factors, instead of #oating-point values. On the other hand, it does depend on the ratio between the characteristic deadtime of the counter and the inspection interval of the virtual pulse generator checking the system's availability for counting.
References [1] G. Gilmore, J. Hemingway, Practical Gamma-Ray Spectrometry, Wiley, Chichester, England, 1995. [2] J. Harms, Nucl. Instr. Meth. 53 (1967) 192. [3] G.P. Westphal, Nucl. Instr. and Meth. 163 (1979) 189. [4] G.P. Westphal, J. Radioanal. Chem. 70 (1982) 387. [5] G.P. Westphal, Nucl. Instr. and Meth. B10/11 (1985) 1047}50. [6] K. Heydorn, E. Damsgaard, J. Radioanal. Nucl. Chem. 215 (1997) 157. [7] S. PommeH , J.-P. Alzetta, J. Uyttenhove, B. Denecke, G. Arana, P. Robouch, Nucl. Instr. and Meth. A422 (1999) 388}394. [8] K. Debertin, R. Helmer, Gamma- and X-ray Spectrometry with Semiconductor Detectors, Elsevier Science Publishers BV, Amsterdam, The Netherlands, New York, USA, 1988. [9] S. PommeH , Proceedings of the 2nd International k Users Workshop, Ljubljana, Slovenia, 1996, Jozef Stefan Institute, 1997, p.11}14. [10] S. PommeH , F. Hardeman, P. Robouch, N. Etxebarria, G. Arana, Neutron activation analysis with k -standar disation: general formalism and procedure, Internal Report SCKzCEN, BLG-700, pp. 105. [11] S. PommeH , Appl. Rad. Isotop. 49 (1998) 1213. [12] S. PommeH , B. Denecke, J.-P. Alzetta, Nucl. Instr. and Meth. A 426 (1999) 564. [13] S. PommeH , Is loss-free counting under statistical control?, presented at International Conference on Nuclear Analytical Methods in The Life Sciences, Beijing, China, 1998, Biological Trace Element Research, in press.