Pulse Shape Analysis using CsI(Tl) CrystalsJoel Silva, Enrico Fiori, Johann Isaak, Bastian Loher, Deniz Savran, Matjaz Vencelj, Roland Wirth

Abstract—The decay time of CsI(Tl) scintillating materialconsists of more than a single exponential component. The ratiobetween the intensity of these components varies as a functionof the ionizing power of the absorbed particles, such as γ-raysor protons, and the temperature. This property can therefore beused for particle discrimination and for temperature monitoring,using pulse shape analysis.

An unsupervised method that uses fuzzy clustering algorithmsfor particle identification based on pulse shape analysis is pre-sented. The method is applied to discriminate between photon-and proton-induced signals in CsI(Tl) scintillator detectors.

The first results of a method that uses pulse shape analysis forcorrecting the temperature-dependent gain effect of the detectorare also presented. The method aims at conserving a good energyresolution in a temperature varying environment without the needto measure the temperature of the detector externally.

Keywords—cesium iodide, digital pulse shape analysis, clustering,particle identification, gain temperature dependence, scintillator.

I. INTRODUCTION

FOR many years Tl-doped Csi scintillation crystals havebeen used for detection of all kinds of ionizing radiation.

Its scintillation characteristics were already presented in fewexperimental works (e.g. [1], [2]). In all measurements at leasttwo relevant light components were found with a “short” and“long” decay time, whose intensity ratio were measured be-tween 0.1 and 1.5. Both, the intensity ratio and the decay timesof those components depend on the type and energy of theionizing particle [3]. The CsI(Tl) scintillation characteristicswere found to be also dependent on the ambient temperaturewhen excited by γ-rays [2].

The scintillation light decay times are correlated with thepulse shape. Thus, the use of pulse shape analysis (PSA)applied to CsI(Tl) detector signals represents an adequateapproach to perform tasks such as particle identification andcorrection of gain shift induced by the change of surroundingtemperature. For large detector setups that contain thousandsof CsI(Tl) detectors such as the CALIFA barrel [4] of the R3Bexperiment for FAIR1, this could be a great advantage.

In this paper we present two works that use pulse shapeanalysis applied to CsI(Tl) based detection systems. In sectionII we present a method that uses fuzzy clustering algorithms toachieve particle identification based on PSA [7]. In Section III

This work was supported by the Alliance Program of the HelmholtzAssociation (HA216/EMMI), BMBF (06DA9040I and 05P12RDFN8) andHIC fot Fair.

J. Silva, E. Fiori, B. Loher, D. Savran, R. Wirth are with ExtreMe MatterInstitute and Research Division, GSI Helmholtzzentrum fur Schwerionen-forschung, Planckstr. 1, 64291 Darmstadt, Germany.

M. Vencelj is with Jozef Stefan Institute, Jamova cesta 39, 1000 Ljubljana,Slovenia.

1Facility for Antiproton and Ion Research

we present the first results of a method based on PSA that aimsto correct the gain shift of Csi(Tl) detector due temperaturechanges.

II. PARTICLE IDENTIFICATION USING CLUSTERINGALGORITHMS

A. IntroductionParticle identification (PID) is a fundamental task in many

experimental setups. Because CsI(Tl) scintillation detectors inresponse to different ionizing radiation yield light componentswhose amplitudes depend on particle species [3], the use ofPSA provides an appropriate approach.

A widely adopted method that uses PSA for achieving par-ticle identification is the integration method [5]. This methoduses the ratio of the integral over two distinct regions of thedetector signal to discriminate between particle species. Forbest particle distinction, the performance of the integrationmethod depends on the selected regions. This selection hasto be previously determined for each detector. Factors suchas aging and exposure to radiation might change the detectorresponse over time. Also external parameters (e.g. temperature)might affect the properties of the pulse shape and require aredefinition of the integration regions. Therefore, calibrationprocedures which require human supervision have to be per-formed regularly. That is hardly feasible for large setups withthousands of detectors (e.g. CALIFA barrel [4]) or for detectorused in environments where conditions might vary strongly.For this reason it is highly desirable to develop a procedurethat performs PID calibration of a detector system with theleast requirement of human supervision.

We have recently shown that fuzzy clustering algorithmscan be used to perform the calibration in a fully automaticand unsupervised manner for photon-neutron discrimination inliquid scintillators [6]. Due to the success of this method weapply the same prescription in this work to photon-proton dis-crimination using CsI(Tl) scintillation crystals. Furthermore,we derive a means of using the obtained information from theclustering to perform PID. This procedure is independent ofthe particular features of the pulse shape and thus representsa general approach suitable not only for CsI(Tl) detectors, butalso for other detector types.

Detailed explanation on how the fuzzy c-means algorithmis performed can be found in [6].

The work presented in this section has already been pub-lished in [7].

B. Experimental SetupThe data used to demonstrate the present method was taken

during a beam time at the Mayer-Leibnitz laboratory (MLL) inGarching near Munich. The MLL tandem proton accelerator

978-1-4799-1047-2/13/$31.00 ©2013 IEEE

provided a proton beam with an energy of 24 MeV. The protonswere collided with a CD2 foil target and the elastically andinelastically scattered protons, as well as photons arising fromnuclear reactions inside the foil, were detected by a 8×4 arrayof CsI(Tl) detectors (see Refs. [4], [8], [9]) at an angle of 37◦

with respect to the beam axis. The average count rate of asingle crystal was of the order of 200 Hz resulting in a verysmall amount of pile-up signals. The crystals were read outusing Hamamatsu S8664-1010 APDs2. A Mesytec MPRB-16shaping pre-amplifier, which also provides the bias voltage forthe APDs, was used to obtain the detector signal. This signalwas sampled by a 60 MHz 12-bit analog-to-digital converteron a FEBEX2 FPGA3 board. The Multi Branch System (MBS)data acquisition software [10] recorded per-event signal traces,each 2000 samples in length, and additional data generated bythe FPGA board for offline analysis.

C. Signal Preparation

In order to preserve a stable clustering process, the set ofsignals is prepared by performing baseline subtraction, noisereduction, time alignment and removal of distorted signals.These steps are described in the following.

First, the signal baseline voltage is estimated by averagingover the first 100 samples of the trace where no signal ispresent and then subtracted from the signal. Next, the signalsare filtered twice using a FIR4 low-pass filter with a cutofffrequency of 1.8 MHz. The cutoff frequency of the low-passfilter is large with respect to the characteristic time scale ofthe signal: the shortest rise times are larger than 1 µs. Thus,distortions introduced by this step are minimal.

The time alignment of the signal is performed applying adigital CFD5: It returns the sample number kcfd in whichthe signal amplitude is closer, from below, to 20 % of themaximum amplitude.

The energy of the events is estimated by taking the averageof the five samples at the range kmax − 2, .., kmax + 2, withkmax denoting the sample number where the maximum isassumed. Because the clustering performance can be com-promised when the ratio between photon- and proton-inducedsignals are too far from one [6], the energy region between 2MeVee and 10 MeVee, which contains comparable amount ofsignals of each species, is selected (see Fig. 1). Since onlyproton pulse shapes show energy dependence, it would bepossible to extend the clustering to higher energies by firstapplying the fuzzy c-means to the selected region and thenrunning the algorithm in the region E >10 MeVee whilekeeping the photon cluster centroid fixed.

The signals are then cropped so that 50 samples beforeand 450 samples after kcfd are kept. The cropped signals arenormalized such that the signals’ maximum is equal to unity.

Because a contamination of the signal set with distortedsignals (e.g. due to pile-up or clipping) may lead to inferior

2avalanche photo diodes3field-programmable gate array4finite impulse response5constant fraction discriminator

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ox.E

nerg

y[M

eVee

]

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protons

photons

Fig. 1. Energy of events as a function of the Q1/Q2 ratio used by theintegration method (see section 4.1) to discriminate between particle species.The integration method separates both particles well. In the region above 8MeVee only protons are present, whereas photon dominate the region below4 MeVee. Therefore, the shaded region between 2 MeVee and 10 MeVee isselected because it contains equivalent number of signal of each signal class.

clustering results, these signals have to be eliminated (see [6]for details). Therefore, a mean pulse shape

v(k) =1

N

N∑i=1

vi(k) (1)

is calculated, where N is the number of signals still inthe set. The signals with euclidean distance larger than 2.7compared to this mean shape are eliminated from the set.Signals that contain piled-up pulses starting inside the croppedregion are considered as distorted and removed by this process.

D. Particle IdentificationA method that uses fuzzy c-means clustering to determine

a principal pulse for each particle species combined with theuse of discrimination functions is presented. As a referenceto compare the performance of this method, the well knownintegration method is used.

The present results were obtained using signals of twoneighboring detector crystals near the center of the 8×4 array.We show a PID histogram for each of the investigated methodsand each detector. To measure how well each respectivemethod separates the two particle species a Figure-of-Merit(FOM) is given. The FOM is defined as

FOM =|µ1 − µ2|σ1 + σ2

, (2)

where µ1, µ2 and σ1, σ2 are the means and standarddeviations of gaussian functions fitted to the two peaks of eachhistogram.

1) Integration Method: As different particle species havedifferent pulse shapes due to distinct processes induced in thescintillator, also the integral over part of the signal differs fordifferent species. The ratio of two integrals

Q1

Q2=

∑b1k=a1

v(k)∑b2k=a2

v(k)(3)

Q1 Q2

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itud

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Fig. 2. Typical shape of a single normalized signal for photons and protonsyield by a CsI(Tl) crystal with a APD readout and shaped by an amplifier.The shaded areas mark integration limits used by the integration method.

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ber

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protons

photons

Detector 1 (FOM = 4.9)Detector 2 (FOM = 5.9)

Fig. 3. Histogram of Q1/Q2 ratios for two different detectors. The peaksare well separated, but due to different characteristics of the pulse shape forthe two detectors their positions differ slightly in both histograms.

over two different regions [a1, b1] and [a2, b2] is takento make the value independent of the total integral and thussuitable for particle identification. When producing a histogramof Q1/Q2 ratios each particle species produces a peak at acharacteristic Q1/Q2 ratio and cuts can be applied for particleidentification. In general, for getting good identification resultsthe integration limits have to be adjusted to the detector signalcharacteristics.

The integration limits adopted for our signal set are illus-trated in Fig. 2 together with the typical pulse shapes of photonand proton induced signals. The Q1/Q2 ratios histogram usingthese limits are shown in Fig. 3 for two different detectors.The peaks in the two histograms have different positions whencomparing the same particle species for the two detectors.These positions cannot be predicted beforehand and have tobe determined by fitting procedure or by manual intervention.The FOM values for the two detectors are also given in Fig.3.

The energy as a function of the Q1/Q2 ratio is depictedin Fig. 1. The photon signal shape is largely independent ofenergy. However, the same is not true for protons, causing anenergy depending change in the Q1/Q2 ratio.

-0.8

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w(k

)

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k

Detector 1Detector 2

Fig. 4. Weighting functions obtained from Eq. 5 for two detectors, used forcalculation of the discrimination values in the histograms depicted in Fig. 5.

2) Fuzzy Clustering Method: In this method the particleidentification is achieved by comparing the detector signalswith prototype signals for each particle species. These pro-totype signals can be automatically derived from a mixed setcontaining a large number of different species. Thus, the fuzzyclustering method has a two-step procedure with the derivationof the prototype as first step and the assignment of individualsignals to particle species as second step.

As first step, the prototype signal cj and the matrix uijdenoting the degree of membership in each cluster for eachsignal is obtained by applying the fuzzy c-means algorithm tothe signal set, for details see [6], [7]. The intra-cluster varianceper sample k

∆2j (k) =

1

Uj

N∑i=1

uij(vi(k)− cj(k))2, (4)

with the total clustering size Uj =∑

i uij ,and other statis-tical quantities can also be calculated from these results.

The information obtained from the clustering method fa-cilitate the use of a improved integration method. Instead ofsetting the integration region manually, the prototype signalscan be used to determine a set of weighting functions

wij(k) =ci(k)− cj(k)

∆2i (k) + ∆2

j (k). (5)

By relating the difference between the prototype signalsto the sum of the intra-cluster variances of both clusters theweights are larger where the prototypes signals differ the mostwith respect to their variances. The obtained weights from thepresent data set are shown in Fig.4. Note that they are slightlydifferent for the used detectors.

Using the weights wij(k), a discrimination value for eachpair of particle species

Dij [v(k)] =

L∑k=0

wij(k)v(k), (6)

with L being the length of the signal in number of samples,can be assigned to each signal from the set. This operation

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900nu

mbe

r

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

protons

Detector 2 (FOM = 7.1)photons

Detector 1 (FOM = 6.5)

Fig. 5. Histogram of the normalized discrimination values for two detectorscalculated from the same signal sets as the Q1/Q2 ratios from Fig. 3. Thephoton and proton peaks situated around the discrimination values of 0 and 1and form a completely separated distribution.

can be expressed as a scalar product of two vectors making itsuitable for implementation in fast digital electronics.

Discrimination values can of course also be calculated forthe prototype signals. This is useful because the discriminationvalues can be normalized so that the values of zero and unitycan be assigned to both prototype signals respectively:

Dij [v(k)] =Dij [v(k)]−Dij [cj(k)]

Dij [ci(k)]−Dij [cj(k)]. (7)

The peak positions in the discrimination value histogram forthe two clusters are now Dij [ci(k)] = 1 and Dij [cj(k)] = 0by definition.

Figure 5 shows the histogram of the normalized discrimina-tion values for the same signal set in Fig. 3. As expected, thepeaks corresponding to protons and photons are situated nearzero and one. The peak positions for both detectors are alsoidentical even though the pulse shapes for the two detectors areslightly different, leading to different distribution in the Q1/Q2

ratio histograms (see Fig. 3). Compared to the integrationmethod also the FOM is a slightly improved (FOM valuesgiven in the respective figures). This shows that our approachof defining a weighting function without prior knowledge ofthe signal shapes properties give reasonable results.

E. ConclusionWe have presented a robust and reliable method to perform

PID calibration for CsI(Tl) detector in an unsupervised manner.The method uses fuzzy clustering algorithms to obtain princi-pal pulse shapes for each particle species in a mixed data setwith similar population of proton- and photon- induced signals.These pulse shapes are used to obtain a discrimination valuefor each signal.

After determining the weighting functions for a given de-tector the discrimination values are calculated in a computa-tionally inexpensive way. Thus, they can be used to calculatethe discrimination values for further data online using fastdigital electronics. Recording small amounts of full traces fora detector and regularly re-performing the clustering in this

set, allows to detect changes in the prototype pulse shape canbe detected (e.g. caused by a change of conditions) and actupon by updating the discrimination amplitude.

III. TEMPERATURE DEPENDENT GAIN STABILIZATION OFCSI(TL) DETECTORS

A. IntroductionAs previously mentioned in the main introduction (section

I), a temperature sensitivity of the CsI(Tl) pulse shape charac-teristics when excited by γ-rays has been observed [2].

Also the light yield per MeV energy deposited in theCsI(Tl) crystal is known to be a function of the temperature[11]. Additionally, photo sensors such as photomultiplier-tubes(PMTs) and avalanche photo diodes (APDs) are essentiallytemperature dependent devices: Any temperature change inthe detector may cause a gain shift. Therefore, active gainstabilization is needed in order to maintain energy resolutionand energy calibration.

Here, since the change of the pulse shape with temperatureis a observable independent of the scintillation yield, it mayprovide a solution to adjust the gain (energy calibration) of thedetection system. A similar approach has been demonstratedfor NaI(Tl) detectors before [12].

We present the first steps on studying an adequate methodthat allows to correct temperature gain dependence of γ-raydetection systems based on CsI(Tl) using pulse analysis.

B. Experimental SetupExperimental data has been taken using a 15×15×30 mm

CsI(Tl) crystal coupled to R1464 Hamamatsu PMT with adiameter of about 18.6 mm. A high voltage of 1200 V wasapplied to the PMT in order to yield the highest possibleamplification. The signals have been sampled using an AgilentAcqiris DC282 high speed digitizer (U1065A) at a frequencyof 200 MHz, resolution of 10 bit, an input range of 100 mVand a trigger threshold of 15 mV. No additional electronicswere used.

A 22Na radioactive source was placed about 2 cm from thecrystal’s face on the opposite side of the PMT. Detector andsource were placed at the bottom of a thermally well isolatedbox and covered with ice bricks. With this system it waspossible to slowly reach temperatures from 1 ◦C to 20 ◦C. Thetemperatures of the crystal, PMT and environment surroundingthe box were measured using pt100 resistance thermometerdetectors (RTD) and recorded every minute by the moduleNI6217 from National Instruments. The detector and RTDswere wrapped with a 5 mm layer of plastic foam in order toavoid the direct contact with the ice bricks.

Data was taken every hour in acquisition runs of roughly3 minutes, while the temperature in the box was rising. Thefirst run was performed 10 hours after the experimental setuphas been assembled in order to reach the thermal equilibrium.Because the maximum temperature change rate measured onCsi(Tl) crystal’s surface was smaller than 17× 10−3 ◦C/min,the temperature can be estimated to be constant during eachrun. For every run 100000 traces each consisting of 4000samples were recorded to a disk.

C. Data AnalysisTo study a method that relates the gain-dependence on

temperature with the pulse-shape of CsI(Tl) crystals, everytrace is primarily prepared by performing baseline subtractionand time alignment. Neither energy nor pile-up discriminationis applied to the signals in the present data set.

First, the baseline is determined by taking an average overthe first 1000 samples (5 µs) in a region where no signal existsand then subtracted from the signal.

The timing and triggering is performed using a leadingedge trigger (LET). The LET returns the sample numberkLET when the signal amplitude crosses above a thresholdof 80. This sample number kLET is used as the pulse timereference. In addition, to avoid recurrent triggering on singlespike signals existent in the traces (see Fig. 6 around k = 750),a calorimetric check is performed. The signal is convolutedwith a 60 samples wide box-filter (see Fig. 6), if the amplitudeof the resulting signal never reaches the selected threshold (80),the LET process continues until new kLET is found.

For calorimetry the signal is convoluted with a 1100 sampleswide (5.5 µs) box filter. The resulting signal is proportional tosignal’s integral over 1100 samples. This calorimetric signalis shown in blue in Fig. 6. The maximum value reachedby the calorimetric signal was then used to obtain pulsearea histograms, shown in the upper part of Fig. 8 for twodifferent temperatures. The integral of the pulse is related tothe energy collected in the detector, though the histograms arenot calibrated in order to show the temperature effect on gain.

In order to visualize and analyze the shape of the signals atdifferent temperatures, mean signals are produced by averaging5000 individual signal after being aligned in time and scaledin amplitude. The time alignment is done by shifting thesignal in the way that pulse time reference is then kLET =1200. The scaling is performed by normalizing the signals tothe maximum amplitude of the calorimetric signal. The Fig.7 shows the such calculated mean signals for two differenttemperatures.

As expected for higher temperatures the signal decay timesdecrease [2]. The mean signals also differ in their ampli-tudes, since the amplitude dependence on the detector gain issuppressed when normalizing the signal to their calorimetricamplitude. The decay of the mean signals follows very nicely asum of exponential curves and in principle the decay constantsmay serve as a pulse shape parameter. However, fitting anexponential function to each single pulse is not very efficient.The difference in the decay constants can be translated intodifference of ratios of integrals over the regions where thechange in pulse shape with the temperature is large. In orderto define a pulse shape parameter P which is independent ofthe signal energy the following expression is used:

Pi =Fi(kLET + 1000)− Fi(kLET + 200)

Fi(kLET + 200)− Fi(kLET − 20)(8)

with Fi being the values of the calorimetric signal at thegiven samples with reference to KLET . The parameter Pdefined in this manner is equivalent to the ratio of integralsover the regions raging kLET + 200 → kLET + 1000 and

−200

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ampl

itud

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baseline subtracted60 samples wide box filter1100 samples wide box filter

Fig. 6. The original (baseline corrected) signal (gray), together with theresulting signal after convoluted with a 60 sample wide box-filter for triggering(red) and after convoluted with a 1100 sample wide box-filter for calorimetriccalculations (blue). The time of each signal is determined using the LETmethod applied to the original signal. The blue signal is used to integrate thesignals regions. Also the energy measurements are done using the maximumof the red signal as full signal integration.

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2.5

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500 1000 1500 2000 2500 3000

ampl

itud

e

k

Temperature = 1.8 °CTemperature = 19.7 °C

Q1 Q2

Fig. 7. Mean pulse shapes determined by averaging 50000 signal for twodifferent temperatures, after being aligned in time and normalized to theircalorimetric amplitude. The faster decay at higher temperatures is clearlyvisible.

kLET − 20 → kLET + 200. These regions are illustrated bythe shaded areas in Fig. 7 with the respective Q2 and Q1 labels.In this way, for each signal within the recorded data set, thepulse shape parameter Pi is determined in addition to the timekLET and the calorimetric signal Fi(t).

D. Results

In the following we discuss the dependence on the pulseshape parameter P and the detector gain upon temperaturechanges in the range of 1.8 ◦C → 19.7 ◦C.

The temperature values presented in the results were

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Temperature = 1.8 °CTemperature = 19.7 °C

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tive

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on

Temperature [°C]

22Na 511 peak22Na 1275 peak

Fig. 8. Upper part: Energy histogram of the 22Na radioactive source for twodifferent temperatures. In order to show the detector gain shift the histogramshave not been calibrated in energy but are represented in ADC channels.Lower part: Relative peak position with respect to the position at 1.8 ◦C, of22Na photopeaks at 511 MeV (dots) and 1275 MeV (crosses) as function oftemperature measured at the crystal surface.

recorded from the sensor placed in contact with the crystalside face.

In order to study the dependence of detector’s gain withtemperature the spectrum of a 22Na source is used. The averageenergy detected for the 511 keV and 1275 keV γ-rays are,thus, determined by the position of the respective photopeakin the spectrum. The lower part Fig. 8 shows the position ofthe photopeaks as a function of the temperature. The peakpositions have been normalized to its value at 1.8 ◦C

The upper part of the Fig. 9 shows histograms of the pulseshape parameter P for three different temperatures of thecrystal. Each histogram contains the parameter P from 5000signals of a single run. The spread of the distribution foreach temperature is large compared to the differences betweendifferent temperatures, i.e. for individual pulses the pulse shapeis not well defined. However, since the temperature is a slowlychanging parameter compared to the rate of incoming eventsit is not necessary to determine the pulse shape for single

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Temperature = 1.8 °CTemperature = 9.8 °CTemperature = 19.7 °C

0.89

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µP

Temperature [°C]

Fig. 9. Upper part: Histogram of the defined parameter P calculatedfor the same amount of individual pulses at different temperatures. Forlower temperatures the mean of P shifts to higher values. Even thought thedistribution in P is larger than the observed shift, the mean value µP mightserve as a suitable temperature parameter. Lower part: Determined meanvalue µP as function of temperature. A monotonic dependence of µP ontemperature is observed.

signals. Therefore, the mean µP of a set of pulses may beused to determine the current pulse shape.

The result of µP as a function of the temperature is shownin the lower part of Fig. 9. A rather smooth monotonic depen-dence is observed. This proofs, that a pulse shape parameterdefined in the present way may indeed serve to determine thecurrent pulse shape. And thus, to be used as a “thermometer”to correct the gain shifts due to temperature variations.

Finally, also the parameter mean value µP is correlated withthe relative peak positions of the 22Na source. This correlation,illustrated in Fig. 10, shows a rather smooth and linear profilewithin the considered temperature range. Since µP dependslinearly on temperature the profile of µP correlation with thepeak position is mainly dictated by the temperature dependenceof the peak position. However, in the lower plot of Fig. 8and consequently in Fig. 10 a apparent peak stands out fromthe trend. When plotting peak position as function of the run

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0.89 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1

µP

relative peak position

22Na 511 peak

22Na 1275 peak

Fig. 10. Correlation of 22Na photopeaks at 511 MeV (dots) and 1275 MeV(crosses) with the parameter mean value µP . Since µP shows linear depen-dence on temperature, the profile of this correlation is ruled by temperaturedependence of the relative peak position.

time this same data points coincide with the moment when thetemperature measure outside the box abruptly drops 3 ◦C froma rather stable behavior for about 1.5 hours. This temperaturedrop outside box has not affected the temperature measuredin the surface of the crystal. Nevertheless, the agreement ofthe peak position with the external temperature suggests adependence of the gain shifts on external electronics (digitizerand HV supply) due to environment temperature changes.

E. Outlook

The results of the present analysis revealed that the pulseshape of a CsI(Tl) scintillator is a well defined observable,which depends monotonically on the temperature of the crystal(and PMT). A pulse shape parameter which is easy to calculatehas been proposed. It allows a unique identification of thecurrent pulse shape and may serve as a parameter to adjustthe temperature dependent shift in the energy spectrum.

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