Influence of pileup rejection on nuclear counting, viewed from the time-domain perspective

Nuclear Instruments and Methods in Physics Research A 426 (1999) 564}582

In#uence of pileup rejection on nuclear counting, viewed from the time-domain perspective S. PommeH *, B. Denecke
, J.-P. Alzetta SCK ' CEN, Belgian Nuclear Research Centre, Boeretang 200, B-2400 Mol, Belgium
European Commission, Joint Research Centre, Institute for Reference Materials and Measurements, Retieseweg, B-2440 Geel, Belgium Received 6 October 1998; received in revised form in December 1998

Abstract Time-interval density distributions of accepted events in a HPGe c-ray detection set-up are measured with a timeinterval digitiser. In particular, the e!ect of pulse-pileup rejection is investigated. Experimental data are obtained with two types of shaping ampli"ers: a classical ampli"er with semi-Gaussian pulse shaping and a gated-integrator ampli"er. A theoretical model is developed to predict typical time-interval density distributions for stationary Poisson processes passing through a detector with count loss by pulse-pileup rejection. Good agreement is obtained between theoretical, measured and simulated time-interval spectra. It is found that, when counting is a!ected by pileup rejection, the true incoming count rate cannot simply be determined by "tting an exponential to the time-interval distributions. From the Laplace transform of the interval-density distribution, expressions are derived for the expectation value and the variance of the counts. Good agreement is found with experimental counting statistics for di!erent system con"gurations, as well as with data from computer simulations.  1999 Elsevier Science B.V. All rights reserved. PACS: 02.50.Cw; 02.90.#p; 07.85.!m Keywords: Nuclear; Counting; Statistics; c-ray spectrometry; Time-interval; Poisson

1. Introduction Radioactivity measurements are usually based on the counting of individual interactions of elementary particles with the sensitive part of a detector. The counting of events is therefore a basic operation, which certainly deserves the scienti"c

* Corresponding author. Tel.: #32-14-33-27-18; fax: #3214-32-10-56. E-mail address: [email protected] (S. PommeH )

attention it has already received in the past (see e.g. Refs. [1}12]). In particular, the problem of count loss is of vital importance in any type of precision work. To understand counting statistics, i.e. the probability for observing a given number of events, one should look upon the counting process as a sequence of events developing in time, where loss mechanisms directly a!ect the statistical distributions of the intervals between successive events. These time intervals are independent, and form a renewal process [13]. Multiple intervals, i.e. the

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 0 1 6 - 9

S. Pomme& et al. / Nuclear Instruments and Methods in Physics Research A 426 (1999) 564}582

waiting time for k subsequent events, are obtained from a k-fold convolution of the single time-interval densities. The transition from interval densities to counting probabilities can be done by realising that the probability P(N 5k) for counting at least R k events during a time period of length t is equal to the probability P(t 4t) that the waiting time t for I I the kth event does not exceed the counting time. The probability for observing exactly k counts follows directly from P(N "k)"P(t 4t)! R I P(t 4t). I> Poisson processes play a central role in nuclear counting. The corresponding interval densities are characterised by a typical exponential shape. This feature is at the origin of some well-known properties of Poisson statistics, such as the remarkable fact that the variance and the third central moment of the probability distribution are equal to the expectation value k. The standard deviation of nuclear counting is hence often conveniently estimated by taking the square root of the obtained count integral. Any random selection of events from the original Poisson sequence gives rise to a new exponentially shaped interval distribution, hence to another Poisson process with a lower expectation value k. Yet, if the selection is not random in the time domain, the perturbed interval density distribution inevitably leads to new counting statistics. Considering that any detector and electronic circuit has a "nite pulse-pair resolution, the shape of the interval density is always disturbed, certainly in the lowest time-interval regions. Hence, even though an ideal Poisson behaviour is often an excellent approximation for nuclear counting, it is never rigorously realised in practice. Experimental as well as theoretical work was already done on time-interval density distributions for counters with extending or non-extending deadtime and their combinations [1}12]. Surprisingly, the e!ect of pulse-pileup has not received the same attention, even though it is the main source of count loss in contemporary fast spectrometry chains. Moreover, often no clear distinction is made between pileup and extending deadtime. Whereas pileup is indeed of the extending type, it is yet a di!erent loss mechanism [14}18]. In this work, we study the time-interval distributions of events originating from the detection of

565

c-rays from radioactive sources, when processed by a typical HPGe c-ray spectrometer. A theoretical model is developed to describe the time distortion of a Poisson process by a pileup loss mechanism. Experimental time-interval spectra are compared with theoretical expectations for an ideal set-up. A link is made between information in the time domain and counting statistics; e.g. a theoretical expression for the counting uncertainty is deduced from the time-interval density formula. Several aspects of counting are also studied by computer simulation. Special attention is paid to e!ects of pileup and pileup rejection.

2. Experimental method and set-up 2.1. Time-interval digitiser For the experimental determination of time intervals between successive pulses, a dedicated time-interval digitiser of IRMM was applied [19,20]. This device has a pulse-pair resolution of 10 ns and can process three subsequent input pulses within 25 ns and a fourth within 400 ns. A practically loss-free analysis is achieved by switching between two identical channels whenever a pulse arrives: the pulse simultaneously stops the counter in one channel and starts the counting in the other channel. The conversion of time into a binary value is accomplished by counting 100 MHz clock pulses in a 12-bit binary address register. The values are temporarily stored in a fast bu!er memory to prevent loss. The spectrum, having a range of 4096 channels, is read out by a multichannel analyser. The channel width can be selected between 10 ns and 2.56 ls. A more detailed description of the apparatus can be found in Refs. [19,20]. 2.2. Counters with pileup rejection The experimental time-interval spectra were mainly assessed on the c-ray detector set-up `Panoramixa of SCK ' CEN. It consists of a 40% relative e$ciency HPGe crystal (Silena, RGC-P 4018), equipped with a classical RC feedback preampli"er (Silena RFP10), high voltage supply (Silena 7716),

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shaping ampli"er (CI-2026) with pileup-rejector (PUR), ultrafast 800 ns "xed deadtime analog-todigital convertor (CI-8715), loss-free counting module with 5 MHz virtual pulse generator (CI-599) and data acquisition board (accuspec-B) for PC. For comparison, time intervals were also taken after replacing the Gaussian shaping ampli"er by a gated-integrator ampli"er (CI-2024). In Fig. 1, a schematic representation of the Panoramix set-up and its connections to the interval digitiser are shown. A LeCroy 688AL level adapter was used to convert the detector signals to fast negative NIM pulses, as required by the interval digitiser. An important aspect of the set-up, in the framework of this paper, is the mechanism of pileup rejection. In order to combine a good energy resolution with a good pulse-pair resolution, the detector signal path is split into a slow amplitude de"ning part and a fast timing channel, where the

time relationship of subsequently arriving pulses is analysed. In the latter, a fast discriminator is used to detect the fast signals and produce short logic signals. The pileup rejector uses the fast channel information to sense whether it should interfere in the slow channel when pileup e!ects may a!ect the signal amplitude. This procedure signi"cantly reduces the amount of clustered events, which otherwise contribute to the spectrum `backgrounda. The pileup rejector circuit will inevitably fail to recognise pairs of pulses which are very closely spaced in time. The minimum spacing is the pulsepair resolution in the fast channel. In the fast-signal channel the signal-to-noise ratio is worse than in the slow-signal channel. A compromise has to be made between fast timing performance and dynamic range. Hence, besides the pulse-pair resolution limit, there is also the limiting factor of the minimum detectable energy below which the pileup

Fig. 1. Schematic representation of the default Panoramix HPGe detection set-up and its connections to the interval digitiser. The `Calvina set-up is almost identical, but the feedback resistor R is then replaced by a reset transistor (shown in "gure with dashed line). 

S. Pomme& et al. / Nuclear Instruments and Methods in Physics Research A 426 (1999) 564}582

rejector cannot respond. Due to the inherent #uctuations of electronic circuits and due to noise, pileup rejectors show an amplitude-dependent pileup rejection e$ciency for small signals. Nevertheless, in the modelling of pulse losses an ideally functioning pileup rejection system will be assumed (see Section 3.1). 2.3. Detector with transistor reset preamplixer

567

calculation of the standard deviations, we used the average value based on the analysis of the previous spectra: i!1 1 1N2 " 1N2 # N G G\ i i

(1)

and 1 i!2 p (N)# (N!1N2 ) p(N)+ G\ G G i!1 i!1

(2)

Some interval spectra were also taken on another spectrometer, `Calvina, a 40% relative e$ciency HPGe crystal (Canberra, GC4018-7500SL-RDC) connected to a transistor reset preampli"er or TRP (CI-2101). Contrary to the RC feedback preamp, the TRP is known to add non-extending deadtime to the system. In the traditional preamp the feedback capacitor, which establishes the detector charge gain (< "Q /C ), is continuously dis   charged with a time constant of e.g. 50 ls. In the case of the transistor reset preampli"er the basic charge sensitive preampli"er circuit is the same, but the feedback resistor is replaced by a special reset circuit. The output of the TRP follows a staircase shape and the reset circuit abruptly discharges the feedback capacitor whenever the output reaches a pre-de"ned threshold level. As a consequence, the ampli"er is thrown into overload by this fast drop of a few volts, and needs a recovery time of roughly two times the normal positive polarity pulses from detector events. The CI-2101 has a 4 Volt dynamic range and a reset time of less than 2 ls typically [21]. Data collection is inhibited during the reset and overload recovery time. 2.4. Counting statistics in c-ray spectra In order to assess deviations from Poisson behaviour in counting statistics due to pulse loss, typically 2000 c-ray spectra of 0.2 s `real timea were taken at di!erent count rates. Due to hard disk limitations, the 16 K spectra were taken in numbers of 50 in succession, analysed automatically, the spectrum "les deleted and then a new acquisition cycle was started. By use of a C-program the number of counts in several regions of interest (ROI) } corresponding to the full spectrum down to a fraction of about 0.02% } was assessed. For the

Fig. 2. Typical pulse shapes and timing parameters involved in spectrometry with a (a) semi-Gaussian pulse-shaping ampli"er (b) gated-integrator ampli"er. ¹ represents the pulse width 5 and PET the pulse evolution time. In this work, the `pileupa pulse loss mechanism only occurs during the PET, while in the following time interval, [PET, ¹ ], `extending deadtimea is 5 assumed.

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in which 1N2 and p (N) are the average value and G G the standard deviation of the number of counts for i spectra. This procedure converges su$ciently fast to the true values, considering the amount of spectra taken. 2.5. Computer simulations In this work, dedicated computer software [14}18] was used to simulate the operation of a counting device. At the input side a stationary Poisson process was simulated by a pseudorandom generator, based on a binomial distribution with low success probability. The corresponding pulses then "nd the hypothetical counter in a status a!ected by previous pulses; `freea if the counter has recovered from the processing of previous pulses or `busya in the case of pileup or if the system is still waiting for the ADC conversion. (The pulse loss mechanism assumed in the program is explained in Section 3.1.) The software allows free choice of the duration of the pulse width (leading and trailing edge) and ADC conversion time. Accepted events are stored in a spectrum. Counting statistics are studied by repeating countings for a "xed acquisition time. Also time-interval distributions are assessed by keeping track of the number of time units between successive valid counts. Both aspects of counting can be followed for the full spectrum as well as for fractions of the spectrum.

3. Theoretical derivation of the interval density distribution 3.1. Modelling pulse-pileup as event loss mechanism Pulse-pileup occurs when a new pulse from the preampli"er is fed through the shaping ampli"er before the ADC has had the chance to properly "nish processing the previous pulse. In such cases, an ampli"er-ADC combination with pileup rejection has the ability to inhibit the ADC from processing the composit pulses. In this paper, as well as in previous publications [14}18], a clear distinction is made between count loss of the type `pileupa and `extending deadtime.a

In Fig. 2 two typical pulse shapes and relevant parameters are presented. Pileup, with loss of at least two signals, occurs when a pulse falls within the `leading edgea of the previous pulse. This is referred to as the `Pulse Evolution Timea or PET; i.e. from the time at which the pulse rises above the noise level up to the peak height detection by the ADC. If the pulse falls within the &trailing edge' portion, the "rst pulse is successfully analysed and only the second pulse is lost. So pileup in the trailing edge is fully equivalent with `extending deadtimea of duration ¹ -PET. ADC non-extend5 ing deadtime comes into play if the ADC pulse processing time extends until after the end of the pulse. The model 8715 "xed conversion time ADC is su$ciently fast to add no deadtime to the spectrometer using the traditional Gaussian-shaping ampli"er, and less than 800 ns using the gated integrator ampli"er [21]. The Panoramix set-up with gated-integrator ampli"er is obviously a close approximation of an ideal counter with pulse pileup as the only event loss mechanism. The mentioned concept of pulse-pileup can also be useful in absence of a pileup rejector. In spectrometry usually small fractions of the pulse-height spectrum are considered. The composite pulse of coincident events will (most often) fall outside the region of interest and hence be lost in a comparable manner as with pileup rejection. 3.2. Method to generate the interval density from Laplace transforms The time interval between the arrival times of two consecutive events of a renewal process is described by a probability density f (t). The incoming event train at the entrance of the nuclear counter is assumed to be of Poisson origin, which is fully described by its typical exponential interval density [1}3]: G

f (t)";(t)o e\M R

with the unit step function



;(t),

0 for t(0 1 for t'0.

(3)

S. Pomme& et al. / Nuclear Instruments and Methods in Physics Research A 426 (1999) 564}582

and



o\"

 t f (t) dt. 

Since o\ is the average time interval between pulses, o corresponds to the mean count rate of the process (for a su$ciently long measuring time). The concept of interval density between consecutive events can easily be extended to multiple intervals of order k. Since the renewal process is independent of its `historya, the interval time between an arbitrary event n and event n#k is simply obtained from a k-fold self-convolution of the single time-interval density: f (t)"+f (t),HI. I

(4)

The total density is de"ned as > D(t)" f (t). (5) H H The evaluations are simpli"ed by using Laplace transforms: fI (s),L+f (t),

(6)

since convolutions of f (t) can be replaced by simple multiplications of the transform. For the Poisson process one gets [1,22] o (ot)I\ e\MR f (t)";(t) G I (k!1)!

(7)

D(t)";(t)o. G

(8)

One of the particular features of a Poisson process is the complete lack of `memory e!ectsa; the expected time interval is independent from the time elapsed since the last event. As a result the time origin in a Poisson process can be chosen at will. This property, however, is unique for the Poisson process and therefore not applicable to the event train passing the device. Throughout this paper, we shall assume that the time origin is synchronised with a pulse, so that the interval density f (t) of the  "rst event is equal to that of the following ones. It is our aim to determine the `outputa interval density f (t) of accepted events originating from 

569

a Poisson process in a counter with pulse-pileup as event loss mechanism. In the case of non-extending deadtime one "nds f (t) directly by considering  that the original distribution is shifted over a "xed amount of time, i.e.: f (t)";(t!q)oe\MR\O. (9)  Finding an expression for the case of pileup is less straightforward. We follow a procedure similar to the one used by MuK ller [22] for counters with extending deadtime. In the latter case, the occurrence of an output pulse at time t (t'q) is expected on condition that there is a pulse in the incoming event train and that no event occurred within the preceding interval t!q. This leads to the following relationship between the `incominga and `outgoinga total densities: D(t)";(t!q) D(t) = ([t!q, t]) (10)  G G  with = ([t!q, t]) the probability for having no G  incoming event between t!q and t, when an event has occurred at t"0. Exceptional simplicity is obtained in the case of a Poisson process, for which the probability = ([t!q, t])"e\MO is constant, i.e. independent G  of t (t'q). This leads to the conclusion that the total output density is constant for all intervals t exceeding the length of the extending deadtime q: D(t)";(t!q)oe\MO  or, equivalently o DI (s)" e\M>QO.  s

(11)

(12)

Finally, to arrive at the density distributions for single intervals, one can use a simple relationship which holds for the transformed single and total interval densities [22], namely > DI (s)" + fI (s),H   H fI (s) " M 1! fI (s) M and hence 

DI (s) fI (s)"  . 1# DI (s) 

(13)

(14)

S. Pomme& et al. / Nuclear Instruments and Methods in Physics Research A 426 (1999) 564}582

570

and D(t)";(t!¹ !PET)oe\M25e\M.#2  5 Fig. 3. To derive the formal expression for the single interval density f (t) for a Poisson pulse train passing through a device  with pulse-pileup rejection, use is made of a simple relationship of its Laplace transform with that of the total density D(t),  which is derived from the incoming density D(t). G

In this way it can be shown that the counting of a Poisson process with extending deadtime is characterised by the following time interval density distribution [1,22]:



f (t)"L\



o o#seQ>MO

 [!o(t!j q)]H\ "o ;(t!j q) e\HMO. ( j!1)! H

(15)

3.3. Particular solution for pileup A similar procedure can now be adopted for the case of pileup. Also here the total density D (t) will  be used to calculate the corresponding single interval density f (t) (see Fig. 3).  Two conditions have to be ful"lled when considering the time-interval between two subsequent valid pulses of a Poisson process in a counting device with pileup rejection: no additional event is allowed during the pulse evolution time of the "rst signal nor within a time distance smaller than a pulse width ¹ to the second signal: 5 

D (t)" D (t);(t!¹ ) G 5 ; = ([0, PET]6[t!¹ , t]). G  5

(16)

Like in the case of deadtime, all events falling within the interval [0, ¹ ] have no chance of being 5 accepted; D (t)" f (t)"0 for t(¹ . A di!erent   5 explicit formula for D (t) is obtained in two sub sequent interval regions: 

D (t)";(t!¹ );(¹ #PET!t)oe\MR 5 5

for t3[¹ , ¹ #PET] 5 5

(17)

(18)

for t3[¹ #PET, R] 5 Both expressions combined yield D(t)"oe\M25+e\M.#2;(t!q)#e\MR\25  ;[;(t!¹ )!;(t!q)], 5

(19)

in which we de"ne the `characteristic deadtimea q"¹ #PET. 5 Note that in the time region [¹ , 2 ¹ ] the total 5 5 interval density equals the single interval density f (t), since more than one valid event is excluded  for t(2¹ . Therefore, we can already conclude 5 that f (t) is expected to show an exponential be haviour in the time region [¹ , ¹ #PET], corre5 5 sponding to the length of a pulse evolution time, and then to be constant in the time region [¹ #PET, 2¹ ]. In the next time region, 5 5 [2¹ , 3¹ ], the total density is the sum of 5 5 the single and double interval densities; D (t)  " f (t)# f (t). Higher convolutions come into     play when one moves further into time. Convolutions are obtained from the Laplace transform:



DI (s)" 

25>.#2 e\QRo e\MR dt 25



#



e\QRo e\M25e\M.#2 dt

25>.#2



"oe\M25



oe\QOe\M.#2#se\Q25 . s(s#o)

(20)

The transition to the single-interval density is obtained from Eq. (14), written in following shape: 

 fI (s)"! [! DI (s)]H M H

(21)

 H " (!1)H>oHe\HM25 C? e\?M.#2 H H ? ;



 

? 1 H\? 1 1 ! e\QH25>?.#2. s s#o s#o (22)

S. Pomme& et al. / Nuclear Instruments and Methods in Physics Research A 426 (1999) 564}582

The "nal step is to "nd f (t) from the inverse  Laplace transform:  f (t)" (!1)H>L\+ DI (s)H,.   H

(23)

Use is made of following properties of Laplace transforms: L\+ fI (s)e\Q?,";(t!a) f (t!a)

(24)

L\+a fI (s)#bg (s),"a f (t)#b g (t)

(25)




L\



"


 

? 1 H\? 1 1 ! s s#o s#o tH\ e\MR (j!1)!

for a"0

? tL\ 1 (!1)?\L CH\ (26) (n!1)!oH\L ?\L>H\ L H tL\ 1 #(!1)? C?\ for a'0 (n!1)!oH\L H\L>?\ L

with CG "n!/i!(n!i)! and j'0. L The time interval density distribution for counting with pileup is then found to be: 

 f (t)" (!1)H>oHe\HM25;2 H



;(t!j¹ ) 5

(t!j¹ )H\ 5 e\MR\H25#2 (j!1)!

H ;(t!j¹ !aPET) C? e\?M.#2;2 5 H ?




1 CH\ # oH\L ?\L>H\ 2 H (t!j¹ !aPET)L\ 1 5 (!1)? (n!1)! oH\L L



C?\ e\MR\H25\?.#2 H\L>?\

.

Fig. 4 shows some typical time-interval density distributions, for three di!erent values of PET, i.e. PET "0, ¹ /2 and ¹ , and assuming a count 5 5 rate o"1/¹ . The case of PET"0 corresponds 5 to extending deadtime. The typical interval spectrum (Eq. (15)) is indeed found as a special case of the pileup formula 27 (see Fig. 4a). The second case, PET"¹ /2, roughly corresponds to a counter 5 with symmetrical pulse shaping, like e.g. from ampli"ers with Gaussian or triangular shaping characteristics. As expected, an exponential part is found during [¹ , ¹ #PET], followed by a remnant of 5 5 the constant part typically observed with extending deadtime. The slope at higher time-interval regions seems less steep than in the case of deadtime. The most extreme case of pileup, PET"¹ , is in spec5 trometry practice well approached by gated-integrator ampli"ers. The fraction of constant density is gone, and a steeply decreasing density is now expected in the beginning, followed by a rather slowly decreasing density at higher interval times. 3.4. Time intervals for an arbitrary fraction of the events In spectrometry, one is usually interested in a particular part of the pulse-height spectrum and therefore mainly concerned by the statistical uncertainty for the count integral in that region of interest (ROI). The calculation of the time-interval density corresponding to a ROI seems easily feasible for pileup and for extending deadtime. One starts from D (t)" D (t);(t!¹ )  D G D 5 = ([0, PET]6[t!¹ , t]), G  5

? (t!j¹ !aPET)L\ 5 (!1) ?\L (n!1)! L

(27)

571

(28)

with D (t) the total density corresponding to the G  D region of interest. The incoming pulse train, being an arbitrary fraction of a Poisson process, is still a Poisson process with D (t)";(t) fo. The fracG D tion f is a number between 0 and 1. The term = is G  the same as for the full spectrum, since interference (pileup) can be caused also by events not belonging to the region of interest. Eventually one gets a similar formula as expression (27), the only di!erence being the replacement of the factor oH by (fo)H.

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Fig. 4. Time-interval density distributions calculated from formula (27), in the particular cases of a device with (a) extending deadtime ¹ (PET"0), (b) a combination of pileup and extending deadtime (PET" ¹ /2) and (c) pileup (PET"¹ ). The count rate o is 5 5 5 equal to 1/¹ , corresponding to a pulse loss probability of 63%, 78% and 86%, respectively. The data correspond to a 100%, 50% and 5 10% fraction of the spectrum of valid events (see Section 3.4).

In Fig. 4, besides the already discussed timeinterval distributions for the full spectrum of events, also the expected interval distributions for a random fraction of 50% and 10% of the spectrum are shown. The distortions look more or less comparable, though for pileup, there is a more pronounced contrast between the sharply decreasing density in the beginning of the spectrum (exponentially dependent on the total incoming rate o) and the much slower decreasing asymptotic density.

The ordinary moments m (t) of the time variable G t are obtained directly from di!erentiation of the k-fold interval density fI (s) [22], by forming I dGfI (s) m (t)"(!1)G I . (29) G dsG Q Mainly the "rst two moments are of interest to this work. Making use of Eqs. (20) and (21), with the factor f included (see Section 3.4), one "nds the expectation value for the k-fold interval time from

3.5. Link with counting statistics

d fI (s) m (t)"!  I  ds

As stated in the introduction, the time-spacing distribution between successive counts determines the counting statistics. Basic mathematical operations on the Laplace transform of the time interval distribution will be used here to derive an expression for the expectation value and standard deviation of the number of counts obtained during a "xed measuring time.





. Q "k[ foe\M25>.#2]\.

(30) (31)

Considering that the count rate is the inverse of the average interval time, R"1/1t2, this is mathematical proof that the throughput of a counter with pileup rejection is indeed exponential, like in the case of extending deadtime, but with an `e!ective deadtimea of q"¹ #PET (see e.g. 5

S. Pomme& et al. / Nuclear Instruments and Methods in Physics Research A 426 (1999) 564}582

573

Refs. [1}3,14}16]). Indeed, for k"1, one gets R"foe\M25>.#2.

(32)

The central moment of second order is calculated from d fI (s) k (t)"  I  ds



!m(t)  Q

eMO "k +eMO#2feM.#2!2f (1#oq),. (fo)

(33) (34)

The relative uncertainty on the k-fold interval time follows then from



p(t) 1 " (1#2fe\M25!2f (1#oq)e\MO (35) m (t) k  with q"PET#¹ being the sum of the "xed 5 pulse width and pulse evolution time associated with each incoming event and f the considered fraction of the spectrum. This expression eventually also pertains to the relative uncertainty on the count rate and on the amount of counts in a "xed counting interval (of su$cient length). The asymptotic (t/q1) counting uncertainty in a counter with pileup rejection is therefore predicted to be p(k) "(1#2fe\M25[1!(1#oq)e\M.#2]. (k

Fig. 5. Standard deviation of the count integral N of spectra from a counter with pileup rejection, relative to the square root of 1N2, as a function of the incoming event rate o (multiplied with q"PET#¹ ). The theoretical values (full line) are ob5 tained from Eq. (36) (see text). The symbols represent computer simulation results. (The pulse loss mechanism is explained in Section 3.1).

(36)

If PET"0, the pulse loss is of the extending deadtime type, and expression (36) indeed reduces to the appropriate shape[1,3,14}16]: p(k) "(1!2foqe\MO. (k

(37)

In Fig. 5, the theoretical counting uncertainty is shown as a function of the incoming event rate, for a counter with only extending deadtime (PET"0), with `purea pileup rejection (PET"¹ ) and an 5 intermediate case (PET"¹ /1.9). Each of these 5 situations correspond to actual counter types (see section on experimental results). The count scatter clearly increases by the in#uence of pileup rejection. The validity of expression (36) is demonstrated by the excellent agreement with computer simulation data (Fig. 5). From Fig. 6, one can also verify that for all types of count loss, the counting uncertainty

Fig. 6. Standard deviation of the count integral N of parts of spectra from a counter with pileup rejection, relative to the square root of 1N2, at a "xed incoming event rate (oq"1), and as a function of the considered fraction of the spectrum. The theoretical values (full line) are obtained from Eq. (36) (see text). The symbols represent computer simulation results for di!erent combinations of PET and ¹ (see Section 3.1). 5

gradually approaches the value expected from Poisson statistics when increasingly smaller fractions of the spectrum are considered, exactly as predicted from Eq. (36).

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4. Results and discussion 4.1. Time-interval density distribution of fast ICR signals The ampli"er provides a standard TTL logic signal, `ICRa, triggered by the fast timing signal of the preampli"er. The nominal width of the signal, q "150 ns, is relatively short and pulse loss '!0 should be low. Hence, the most immediate method to obtain the `truea total incoming count rate, o, is by accumulating the ICR pulses directly in a fast scaler, divided by the accumulation time to get the count rate R , and then applying a correction '!0 formula for deadtime loss: R '!0 . o+ 1!R q '!0 '!0

(38)

At the applied count rates (R q 1), this cor'!0 '!0 rection formula is valid for deadtime of the extending as well as the non-extending type. The reliability of this method can be assessed by comparing with the results of an alternative and superior method based on an time-interval assessment of the successive ICR signals. For systems with pulse loss caused (merely) by deadtime } extending or non-extending } it can be shown that the asymptotic exponential slope of the interval densities of output pulses runs parallel with that of the incoming pulse train; i.e. o corresponds with the (negative) exponential slope `at in"nitya [4,20]. In Fig. 7, a typical time interval spectrum is shown of the ICR signals from Cs c-rays being detected in Panoramix. The main part of the spectrum clearly shows an exponential shape, and can be well reproduced by a "t. The spectrum also looks rather clean, free of spurious peaks, which are not uncommon in time-interval spectra of pulses processed by electronic devices. The main deviation from a Poisson process is the loss of counts in the lowest time-interval region. The spectrum distortion looks di!erent than the typical e!ects known for extending and non-extending deadtime. The cut-o! region seems to extend beyond the expected "rst 150 ns. The blow-up in Fig. 7 shows a slight modulation in the very beginning of the spectrum.

Fig. 7. Typical time-interval spectrum of ICR signals from the fast channel of the `Panoramixa HPGe c-ray spectrometer. An exponential function, "tted to the tail of the spectrum, allows a determination of the average count rate of the original Poisson process. The gap in the beginning of the time spectrum is caused by loss due to the limited pulse-pair resolution. The blow-up of the "rst 3 ls shows some unexpected undulations.

As a consequence, the time-interval analysis shows that the pulse losses are apparently higher than expected. In Fig. 8, the ratio between the `measureda and the `reala incoming count rate, represented by R and o respectively, is presented. '!0 The graph shows that expression (38) can still be used to correct for pulse loss, however, on condition that the characteristic deadtime is signi"cantly increased, up to about q "560 ns. '!0 For comparison, similar interval spectra have been taken on HPGe detector `Calvina, equipped with a transistor reset preampli"er (TRP). The basic charge sensitive preampli"er circuit of the TRP is not di!erent from the classical version, but the feedback resistor is replaced by a special reset circuit. As already mentioned in Section 2.3, the system is dead during reset time and, more importantly, during the overload recovery time of the shaping ampli"er. The TRP therefore adds nonextending deadtime to the system. Two typical ICR time-interval spectra at high count rate are shown in Fig. 9; one obtained from a measurement of low-energy c-rays from a Co source (122.1 and 136.5 keV) and another of medium energy c-rays from Cs (661.7 keV).

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Fig. 8. Ratio R /o of incoming count rate values, with R determined directly from counting ICR signals with a scaler o and the '!0 '!0 `truea incoming count rate obtained from an exponential "t to the corresponding time-interval spectrum. The full line corresponds to count rates corrected by expression (38), applying q "560 ns. '!0

The most remarkable feature of the interval spectra in Fig. 9 is the second bump beginning at about 65 ls. It corresponds to the counts arriving directly after the reset-overload recovery time, during which the ICR signals are suppressed by an INHIBIT signal from the preamp. The two bumps are clearly separated because of the extremely high count rates used here (243;10 s\ for Co and 191;10 s\ for Cs). The ratio of the number of events between the second and the "rst bump is   and  in the case of Co and Cs, respectively.  (A correction was made for the counts lost in the "rst part of the time spectrum.) This means that 3.6 times as many Co c-ray signals are needed to saturate the preamp. The TRP deadtime of 65 ls corresponds to an average of 0.1 and 0.37 ls per event for Co and Cs, respectively. This is small compared to the e!ective deadtime associated with pileup. Another interesting phenomenon is the appearance of spurious peaks in the time spectrum; i.e. at 74.6 and 15.5 ls. They demonstrate that the system is sometimes triggered by non-events, e.g. the resetting of the preamplier probably causes an extra ICR signal, through interference. Consequently, count rates measured by a scaler using the ICR signals are often an overestimate of the count rate

obtained in the c-ray spectrum with the regular data acquisition system. Such an e!ect has clearly been observed at relatively low counting rates. 4.2. System BUSY signal To perform c-ray spectrometry at optimum energy resolution, a slow, amplitude de"ning signal ampli"cation path is used instead of the fast channel. The latter is nevertheless used by the pileup rejection system (PUR), to sense whether nearby signals may interfere. The time behaviour of the combined deadtime signal from the PUR and the ADC was then observed; it begins when an event from the preamp is detected and ends when the ADC has processed a valid event and the ampli"er output has returned to its baseline. A typical interval spectrum is presented in Fig. 10 (for detection set-up Panoramix). It clearly su!ers from enhanced pulse loss as compared to the fast timing channel and shows a behaviour typical of systems with extending deadtime (cf. Fig. 4). The rise time is not as sharp as ideally expected though. The asymptotic slope corresponds well to the slope of the ICR signal (also indicated in Fig. 10), as expected theoretically for extending deadtime systems [4,20].

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Fig. 10. Time-interval spectrum of deadtime signals in the spectrometer Panoramix, induced by Cs c-rays at an incoming rate of 30;10 s\. Also shown is the interval spectrum for the corresponding fast timing channel signals. Both curves show the same slope `at in"nitya.

Fig. 9. Time-interval spectrum of the ICR signals in HPGe detector set-up `Calvina, equipped with a transistor reset preampli"er: (a) for low energy c-rays of a Co source (b) for medium energy c-rays of a Cs source. The second bump in the interval spectrum, corresponding to the "rst events after a preamp reset, is 3.6 times smaller for Co than for Cs.

A second interval spectrum (Fig. 11), taken at high count rate (24;10 s\), shows an unexpected and alarming bump at about 1750 ls. (At lower count rates, such bump would be hidden by the relatively lengthy time intervals.) This could be interpreted as a symptom of the system occasionally being dead for an unusually long period, which has its implications on the loss correction system. In daily practice, the Panoramix set-up is complemented with a loss-free counting module (LFC) for real-time pulse loss correction, based on the Westphal virtual pulse generator (VPG) method [23]. During a VPG inspection period (IP), the LFC module checks the system availability to process pulses and determines a throughput correction factor for the next inspection period. Loss is then compensated by performing add-n operations to the spectrum instead of add-1 for every accepted

Fig. 11. Time-interval spectrum of deadtime signals measured in the spectrometer Panoramix at 24;10 s\ input rate, showing an unexpected hump at about 1.75 ms. This points to an unwanted source of system deadtime, which can upset the LFC pulse loss correction system.

event. Considering that the VPG frequency is 5 MHz and that a bu!er of 256 valid ticks is required per inspection period, a deadtime of 1.75 ms corresponds to a minimum correction factor of n"35. Any valid event in the next VPG IP would then be registered as at least 35 counts, all falling within the same spectrum channel at once. This explains the occasional appearance of single channel `ghost peaksa in LFC spectra on this set-up.

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4.3. Accepted semi-Gaussian-shaped pulses The time-interval spectrum of the deadtime signal is not representative for the timing of accepted events, since it is also invoked by eventually rejected pileup events. To realise this goal, logic signals inside the ADC were selected from which exactly one could be expected per valid conversion. We choose the signals READYH and BFH (which is equivalent with the linear gate control signal LFGH), accessible on chip pins inside the ADC. The READYH signal indicates that data is available for transfer to the MCA. It is reset after receipt of signal ACEPTH, indicating that the data has been accepted by the MCA. The BFH (or LFGH) is set true at peak detect time. The ADC is run in the overlap mode. The time constant of the Gaussian-shaping ampli"er was set at 4 ls. Such long shaping time is used in practice to avoid ballistic e!ects and to arrive at a uniformous pulse shape, independent of the c-ray energy and the manner in which it is deposited in the sensitive crystal. From an assessment of the detector throughput at di!ering incoming count rates of Co c-rays, an e!ective deadtime of q"PET#¹ "34.8 ls was derived 5 [17]. Combined with a PET value of 12.1 ls (found after "ne-tuning of the LFC module [17]), one can predict a value of ¹ "22.7 ls. 5 Typical time-interval spectra of the READYH signal are shown in Fig. 12, for c-rays of Co (1173.2 and 1332.5 keV), Cs (661.7 keV) and Co (122.1 and 136.5 keV). The graphs are in qualitative agreement with the predicted behaviour in Fig. 4(b); no counts during a time interval ¹ after a valid pulse, an exponential behaviour 5 during a time period of length PET, followed by a constant probability during a ¹ -PET period. 5 Moreover, the empirical values for PET and ¹ seem to agree well with the information from 5 the time domain, at least in the case of Co. The transitions between di!erent phases in the timeinterval spectrum are not as sharp as ideally expected. This is particularly obvious around t"¹ , 5 where the step-function is severely smoothed. The obvious cause is the di!erence in perceived pulse width between low- and high-energy pulses, since the latter remain longer above the noise level. This

Fig. 12. Time-interval spectrum from accepted pulses induced by c-rays in HPGe detector Panoramix (cf. ADC READYH and BFH signals), originating from (a) Co (51;10 s\) (b) Cs (53;10 s\, 84% count loss) (c) Co (65;10 s\). The time spectra are typical of a Poisson process su!ering loss by pulsepileup rejection. Di!erent phases in the spectrum can be linked with the duration of the PET and the width ¹ of the Gaus5 sian-shaped pulses. The pulse width is not perfectly constant as a function of energy, leading to some shifting and smoothing of the interval spectra. The time variables PET and ¹ derived 5 from the Co throughput curve and LFC setting are indicated in the left upper corner.

interpretation is corroborated by a quick comparison of the cut-o! position (t"¹ ) in the interval 5 spectra of the di!erent isotopes (Fig. 12), which indeed moves to the left when going from Co over Cs to Co. The low-energy photon interactions of Co already produce a shoulder at t" 11}17 ls. More evidence follows from gating the ADC for certain regions of interest (see Section 4.5).

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Fig. 13. Comparison of the time-interval distribution of the ICR, BFH and system busy signals, all of them corresponding to the same measurement of Cs c-rays in Panoramix (o"23.35;10 s\). The exponential slope at in"nity of deadtime a!ected signals is directly proportional to the incoming count rate. This is no longer true for the BFH signals of ADC accepted signals, due to the in#uence of pileup rejection.

Not shown in the graphs are periodical undulations in the time spectra, due to the synchronisation of the ADC logic signals to the rhythm of the internal clock and the superior time resolution of the interval digitiser. This e!ect has been hidden by the summing of data over a su$cient number of time channels. In Fig. 13, a zoom out is shown of the spectrum of accepted counts and the corresponding ICR signals in the fast channel. Whereas the slope of the latter directly relates to the incoming count rate, this is no longer the case for the pileup a!ected event train. It is clear that the method of the exponential "t would yield here a far to low estimate of the original count rate. This follows also from the theoretical expression (27) and simulation results [16]. 4.4. Accepted events from a gated-integrator amplixer In the Panoramix set-up, the Gaussian shaping ampli"er was then replaced by a gated-integrator ampli"er (see Sections 2.2 and 3.1). Again, a shaping constant of 4 ls was chosen. This corresponds

Fig. 14. Typical time-interval distributions of valid events (BFH signal) from Cs c-rays detected in Panoramix, equipped with a gated-integrator shaping ampli"er, at an incoming count rate of 6, 23, 33 and 115;10 s\ respectively. The time distortions are typical of a counting device with pileup rejection as the only loss mechanism.

to a relatively higher pulse loss, since almost the full pulse width acts as pulse evolution time, which is counted twice in the throughput formula R +e\M.#2. o

(39)

On the scope, the pulses look as expected, with a rather uniformous duration of 34 ls. The trailing edge is extremely short; i.e. less than 1 ls. This is con"rmed by the time-interval spectra taken from the corresponding BFH signals in the ADC. A few examples, corresponding to di!erent incoming count rates, are shown in Fig. 14. The interval

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spectra show a sharp increase starting at ¹ "34.5 ls. Clearly the low-energy signals now 5 have the same length as high-energy pulses, since they are kept well above the noise level during rise time and variations in fall time are of course negligible. Consequently, the interval spectra do not di!er from one c-ray source to another. The interval spectra agree qualitatively with the density distribution predicted in Section 3.3 (see Fig. 4c). The exponential behaviour in the beginning of the spectrum is the most clearly pronounced at high incoming count rates. Again, the asymptotic slope of the interval spectrum does not directly relate to the original incoming count rate. 4.5. Time intervals for an arbitrary fraction of the spectrum The time intervals corresponding to a region of interest (ROI) can be obtained by using the coincidence or anti-coincidence gate of the ADC. In coincidence (anti-coincidence) mode, a READYH and BFH logic signal is only generated in the presence (absence) of a positive gate pulse somewhere during the `linear gate timea. Such gating signals are generated with a single-channel pulse-height analyser with adjustable energy window, using the incoming shaping ampli"er pulses as input. On the theoretical side, the calculation of the interval density has been discussed in Section 3.4. In Figs. 15 and 16, typical experimental timeinterval spectra are shown for the full pulse-height range as well as for selected energy regions, respectively taken with a semi-Gaussian and a gatedintegrator ampli"er. Again, counts are excluded within the "rst time interval [0, ¹ ] following the 5 previous pulse. Then the density function looks again exponential during [¹ , ¹ #PET], with 5 5 the slope corresponding to the total count rate o rather than the expectation value of the region of interest, being fo. The measured time-interval spectra agree well with theoretical density distributions calculated from Eq. (27), as well as with the corresponding simulation data. As an example, theoretical and simulated spectra are shown for PET"¹ and  oq"2.3 in Fig. 17, for direct comparison with experimental spectra in Fig. 16.

Fig. 15. Time-interval spectra for (all or part of) the accepted pulses (ADC BFH signal) from Cs c-rays measured with Panoramix at an incoming rate of 33;10 s\ (oq+1.15), using a semi-Gaussian shaping ampli"er and applying pileup rejection. The di!erent spectra correspond to a di!erent region of interest in the pulse-height spectrum; the fraction which it represents in the full spectrum is indicated in the right upper corner.

Finally, one should be aware that the slope at in"nity does not correspond with the true count rate in the region of interest ( f(100%); not only for pileup but also for extending and non-extending deadtime (see Ref. [16]). This means that the method of determining the incoming count rate from the time interval spectra [20] is only applicable when the full spectrum of events is being monitored; it is bound to fail as soon as only a part of the spectrum is observed by the time digitiser. 4.6. Ewect of pileup on counting statistics Besides time-interval spectra, also counting statistics were investigated on di!erent c-ray

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Fig. 17. Theoretical and simulated time-interval spectra for a system with pileup rejection (PET"¹ ), at a count rate of 5 oq+2.3. The graph is comparable with experimental results in Fig. 16.

Fig. 16. Same as Fig. 15, but using a gated-integrator ampli"er instead. The incoming count rate, 33;10 s\, corresponds to oq+2.3.

spectrometers. Fig. 18 shows, as a function of the incoming count rate, the standard deviation on the number of counts N in spectra measured with Panoramix, relative to (N. The counting statistics do not only deviate from Poisson behaviour, corresponding to p(N)/(N"1, but also from the known behaviour of counters with typical types of deadtime [1,5,6,11]. Amongst these, counters with extending deadtime are the closest approximation, but the corresponding count scatter is still signi"cantly lower than with pileup rejection (see Fig. 18). Nevertheless, the theoretical expression (36) developed in this work for the standard deviation seems to reproduce the observed behaviour reasonably well (see Fig. 18), illustrating that the underlying pulse loss mechanism is probably well modelled (see Section 3.1). For the modelling of semi-Gaussian pulse shaping, the ratio between pulse width and pulse evolution time was assumed to be 1.9.

Fig. 18. Uncertainty on the count integral N of c-ray spectra, relative to the square root of 1N2. Each experimental data point is derived from about 2000 c-ray spectra taken on HPGe detection set-up Panoramix. The full line corresponds to the theoretical expression (36), assuming that PET"¹ /1.9. The dashed 5 line shows the behaviour of systems with extending deadtime only (PET"0).

The possible in#uence of non-extending deadtime by a transistor-reset preamp on counting statistics was veri"ed on spectrometer set-up `Calvina, a twin set-up of `Hobbesa. As shown in Fig. 19, the count scatter is almost identical to the case of `Panoramixa, with the classical RC feedback

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Fig. 19. Same as Fig. 18, for Cs c-rays detected on `Hobbesa (twin system of `Calvina), equipped with a transistor reset preampli"er.

Fig. 20. Same as in Fig. 18, except that the semi-Gaussianshaping ampli"er of the standard `Panoramixa set-up was replaced by a gated-integrator ampli"er. Pileup, as de"ned in Section 3.1, was assumed to be the only loss mechanism (PET"¹ ). 5

preamp. The contribution of the TRP reset-recovery time to the total system pulse loss is apparently too modest to induce a signi"cant reduction of the count scatter. Also a pure case of pileup was tested, with the gated-integrator ampli"er in the `Panoramixa setup (see Fig. 20). The count scatter is high, when compared to the results obtained with semi-Gaussian pulse shaping (Fig. 18). The experimental data

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Fig. 21. Counting uncertainty as a function of the considered spectrum fraction, measured with Panoramix with a semi-Gaussian (PET"¹ /1.9) and a gated-integrator ampli"er 5 (PET"¹ ). 5

con"rm the theoretical prediction (Eq. (36)) for PET"¹ , deduced from the interval density dis5 tributions (Fig. 20). The standard deviation obtained with semi-Gaussian shaping is clearly an intermediate case between the extremes of `pileupa and `extending deadtimea. Counting statistics have also been investigated as a function of the size of the considered fraction of the spectrum, an example of which is presented in Fig. 21. It is indeed found that the counting statistics gradually approach Poisson behaviour when increasingly smaller fractions of the spectrum are considered [16]. The experimental results obtained with the gated-integrator ampli"er are slightly lower than theoretically expected, though follow the right trend. The results for semi-Gaussianshaped pulses con"rm theory up to about f"80%. A further increase of the ROI up to (nearly) the full spectrum leads to an unexpected enhancement of the scatter, which explains the slight discrepancies seen in Fig. 18. This is tentatively attributed to some irregularities occurring in the low-energy part of the spectrum.

5. Conclusions Counting statistics depend on the time-interval density distributions of the registered events in the

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counting device. Pulse-pileup in the shaping ampli"er is the main loss mechanism in a typical HPGe c-ray spectrometer with fast ADC. The operation of a pileup rejection system signi"cantly in#uences the time-interval density of accepted events, hence also the statistical uncertainty on the count integral. Most of the prominent experimental features of the experimental interval density distributions were adequately predicted by a theoretical model presented in this work. Also an expression for the counting uncertainty was derived from the timeinterval information. Time-interval distributions between successive events in a few HPGe c-ray detector set-ups with pileup rejection were assessed at di!erent count rates with a time interval digitiser. Some essential parameters could be derived from such time spectra, such as the true incoming count rate, the pulse evolution time (time between arrival and acceptance of event) and the pulse width ¹ . 5 In counters with count loss caused by deadtime alone, the `truea incoming event rate can be obtained directly from an exponential "t to the slope of the interval density `at in"nitya. However, this procedure is not valid for a system with pileup rejection, since the slope strongly deviates from the original incoming pulse train. Nor is it applicable to an arbitrary part of the event spectrum. A distinct departure from Poisson statistics has been observed in series of c-ray spectra, though clearly di!erent from the known e!ects caused by extending and non-extending deadtimes. The observed e!ects were well reproduced by the theory presented in this work, as well as by computer simulation of a spectrometer with pileup rejection. The limiting case of pileup in which the pulse evolution time is zero is equivalent to extending deadtime. It was found that, in general, a study of the timeinterval density distribution of various signals in a counter can yield interesting information on its operation. The time domain perspective is a vital window on the nature of a counting process.

Acknowledgements Thanks are due to J. Uyttenhove (RUG, Belgium) and A. Simonits (KFKI, Hungary) for their

advice on ADC electronics. B. De Sutter (SIEMENS ATEA, Belgium) is acknowledged for useful suggestions concerning the theoretical part of this work.

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