Time and position distributions in large volume spherical scintillation detectors

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Nuclear Instruments and Methods in Physics Research A 574 (2007) 65–82

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Time and position distributions in large volume sphericalscintillation detectors

Paolo Lombardi, Gioacchino Ranucci�

Istituto Nazionale di Fisica Nucleare, Via Celoria 16, 20133 Milano, Italy

Received 12 October 2006; received in revised form 25 January 2007; accepted 25 January 2007

Available online 14 February 2007

Abstract

Large spherical scintillation detectors are playing an increasingly important role in experimental neutrino physics studies. From the

instrumental point of view, the primary signal response of these set-ups is constituted by the time and amplitude of the anode pulses

delivered by each individual phototube following a particle interaction in the scintillator. In this work, under some approximate

assumptions, we derive a number of analytical formulas able to give a fairly accurate description of the most important timing features of

these detectors, intended to complement the more complete Monte Carlo studies normally used for a full modeling approach. The paper

is completed with a mathematical description of the event position distributions, which can be inferred, through some inference

algorithm, starting from the primary time measures of the photomultiplier tubes.

r 2007 Elsevier B.V. All rights reserved.

PACS: 02.50.Cw; 29.40.Mc

Keywords: Spherical scintillation detectors; Photon time of flight; Time and position distributions

1. Introduction

Contemporary large liquid scintillator detectors repre-sent the latest evolution of a mature technology, which hasbeen, since its introduction, a key player in the particledetection arena. The rather easy possibility to accumulatelarge detection mass at not prohibitive costs makes thechoice of massive scintillation detectors particularlyattractive for the field of neutrino physics, where indeed avariety of scintillator set-ups for supernova neutrinodetection [1] and for reactor neutrino oscillations [2–5]have been designed, installed and operated over the past 2decades. Furthermore, in the solar neutrino field themassive spherical detectors Borexino [6] (based upon thefundamental pioneering experience of its pilot prototypeCounting Test Facility [7]) and KamLAND (solar phase)[8] are actively pushing toward an imminent start of datataking in the background challenging sub-MeV region,

e front matter r 2007 Elsevier B.V. All rights reserved.

ma.2007.01.173

ing author. Tel./fax: +39 02 50317362.

esses: [email protected] (P. Lombardi),

[email protected] (G. Ranucci).

while ambitious plans are being shaped to ensure a futureto the SNO apparatus as a multipurpose scintillatorneutrino observatory (SNO+) [9].In the early days of the development of the scintillation

technique a big deal of studies were devoted to thetheoretical and experimental investigation of the timingfeatures of scintillation counters (i.e. assemblies of scintil-lators and phototubes), reaching a fairly accurate descrip-tion of the timing properties of counters of modest size (seefor example Refs. [10–12]).Despite the complexity associated with the size, the

signal characterization in large set-ups in principle canfollow approaches similar to those adopted in smallscintillation equipments, since also large detectors compriseessentially, in this respect, the same 2 major components,i.e. the scintillator in which the interactions occur andthrough which the light propagates, and the phototubesdedicated to the acquisition of the scintillation photons.From a pure instrumental point of view, the detector

signal response to the interactions of interest correspondsto the ensemble of photoelectron signals delivered by eachindividual phototube: specifically, the 2 quantities of

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ARTICLE IN PRESSP. Lombardi, G. Ranucci / Nuclear Instruments and Methods in Physics Research A 574 (2007) 65–8266

relevance characterizing the output pulses are the numberof photoelectrons by which they are formed and the timingevaluated with respect to a common trigger pulse. Fromthese primary measures then the physical quantities ofinterest of each interaction in the detector are computed,essentially the energy of the events and their spatiallocation.

The full understanding of the detector response impliesnormally an integrated 2-fold approach: a thorough MonteCarlo modeling, as accurate as possible, of the lightproduction and transport phenomena from the interactionsite to the detecting photomultipliers, complemented by an‘‘in situ’’ source calibration of the detector. In theparticular case of special interest of spherical geometry, apreliminary, basic understanding of the overall detectorresponse can be, however, gained through some simplegeometric formulas which, although cannot replace theneed for a complete Monte Carlo modeling, can help to getan immediate realistic picture of the performance of thewhole device, especially for what concerns the timedistributions of the phototube pulses. Purpose of this workis to show in details the derivation of these formulas, whichpractically represent a sort of ‘‘zero-order’’ description ofthe performances of a large spherical scintillation counter.

The paper is organized as follows: since it has beenmentioned that the phototube timing in a real set-up isrelated to a common trigger, as introduction to the wholeproblematic in Section 2, the well-known issue of therandom trigger rate from a coincidence arrangement of aplurality of PMT’s is reviewed; then in Section 3, somegeneral considerations on the timing in large scintillationcounters are given, defining also the limit of validity of thederivations presented in the subsequent sections. InSection 4, it is considered the timing response of a singlePMT observing a spherical detector characterized by auniform distribution of events, while in Section 5, the caseof a uniform surface distribution of events is considered. InSection 6, it is illustrated in detail the special case in whichthe detector is characterized by an index of refractiondiscontinuity, as for example when the scintillation vessel issurrounded by a shielding liquid of different nature. InSection 7, it is addressed the complementary issue of thephotoelectron time sequence detected by the phototubelattice as response to an event occurring in a genericlocation inside the detector, being the calculation done inboth cases of single refraction index and 2 refractionindexes. In Section 8, the description of the timing featuresis completed with a digression concerning the expectedfluctuation of the trigger pulse derived from the coin-cidence criterion of NF PMT’s firing in a predefinedwindow.

In the second part of the paper, the question is faced ofthe estimated radial distributions of the event locations,as derived from the primary time distributions. Actually,this problem is addressed without considering in detailsthe procedure to infer the spatial location estimate fromthe time information (for this purpose see for example

Ref. [13]), but it is assumed that the estimation processresults in Gaussian resolution functions affecting equallyeach individual spatial coordinate (x, y, z), and it is thenillustrated how such estimates combine to produce theestimated radial event distributions in various cases ofinterest. Preliminarily, in Section 9, it is demonstrated whatare, individually, the global x, y and z expected distribu-tions in case of a class of events uniformly distributed in thevessel. Then in Section 10, the estimated radial profile ofpoint like events is derived, while in Section 11, the sametype of distribution is obtained for uniformly distributedevents, and in Section 12 for external events. Finally, inSection 13, there are the conclusions.

2. Random coincidence rate

The trigger condition in a large self-triggering scintilla-tion detector viewed by NT PMT’s is normally set byrequiring a certain number NF of concurrently firingphototubes within a predefined time window of length T.Such a length is defined to be comparable with the photontime of flight (TOF) across the whole volume of thedetector, while NF is fixed through a trade-off between theminimum desirable energy to be observed and the rate ofthe random triggers due to the fake coincidences originatedby the dark noise pulses emitted by the tubes. The resultingrandom trigger rate may pose a problem when thephototubes work in the single photoelectron regime, sincein that case the noise pulses and the signal pulses areindistinguishable. In order to fix properly NF, it is neededto know the formula which governs the rate of falseoccurring triggers.Since a trigger occurs when NF out of the total NT

devices fire, we have as many trigger configurations as thenumber of the groups of NT PMT’s grouped in classes of

dimension NF, i.e.NT

NF

!. Assuming for simplicity that all

the PMT’s feature the same dark noise rate r, then for eachconfiguration we have, given a PMT firing as first amongthe NF of that configuration, that the probability of theremaining ðNF�1 Þ PMT’s to fire within a time window T

from the first is rNF�1TNF�1, from which by multiplying byr we get the rate, i.e. rNF TNF�1. Actually, this is the ratedue to the considered configuration, but with 1 specificPMT firing as first; the overall rate is thus obtainedconsidering that any of the NF tubes can be the first, whichimplies that the previous rate is multiplied by NF, i.e.NF rNF TNF�1. Finally, taking into account all the possiblePMT’s configurations, one obtains as overall randomcoincidence rate R

R ¼NT

NF

!NF rNF TNF�1. (1)

Essentially the same formula can be found, for example, inRef. [14].

ARTICLE IN PRESS

Fig. 1. Random coincidence trigger rate as function of the number of PMT’s in coincidence and for different dark rates of the phototubes.

P. Lombardi, G. Ranucci / Nuclear Instruments and Methods in Physics Research A 574 (2007) 65–82 67

For a given number of total PMT’s NT, Eq. (1) is asharply varying function both of the number NF ofphototubes in trigger and of the rate r. This is clearlydemonstrated in Fig. 1, where we plot the relation (1) for adetector characterized by NT equal to 2200 and by acoincidence time T equal to 50 ns (for a realistic example,these numbers adhere to the features of the solar neutrinoBorexino detector [6]). The 5 curves in the figure, whichcorrespond to 5 different values of the PMT’s dark rate(chosen in the more likely range of the Borexino PMT’s),i.e. 0.5, 1, 2, 3 and 4 kHz, give the random trigger rateexpressed in events per day for an increasing number ofPMT’s in trigger from 1 to 20. Initially the rate is extremelyhigh, since in the naıve condition of only 1 PMT in triggerthe random rate is obviously equal to the number of PMT’smultiplied by the individual dark rate; the random ratebecomes negligible even for the highest rate of 4 kHz,however, already with only 14–15 PMT’s in trigger.

This result implies that the threshold imposed by therequirement to keep to a minimum the random trigger ratedoes not represent, in practical case, a real limitation on thelow-energy response of a scintillation detector.

3. General timing considerations in large scintillation

counters

As mentioned in the introduction, when referring to thetiming features in scintillation counters one implicitlymeans the characteristics of the time distributions of thephotoelectrons revealed by the observing phototubes asresponse to an interaction occurring inside the scintillator.These features depend upon a multiplicity of factors: theintrinsic scintillation decay time of the scintillator [10], thetime of flight (TOF) of the photons through the detectorprior to arrive to the phototubes, which takes into accountthe geometrical extension of the detector itself, as well asthe attenuation, absorption re-emission and scatteringeffects along the photon path, and finally the timing inthe photomultipliers along the conversion process of thephotons into photoelectrons and the subsequent multi-plication throughout the dynode chain [15].

It should be reminded that the absorption re-emissioneffect stems from the overlap of the emission and absorption

spectra of the scintillation medium, so that a photon canundergo self-absorption followed by a subsequent re-emission [16]. The process is quite complex, and its maineffect is to enhance the tails of the timing distributions. Inprevious works [17] and [18], it was shown a mathematicalprocedure able to deal with the absorption re-emission effectand to quantify its impact on the overall timing properties.On the contrary, in this work the focus is on thedetermination of the TOF geometrical factors in differenthypotheses of intrinsic event distributions, but under thesimplifying assumption of neglecting the absorption re-emission effect (as well as the scattering), and including atmost the simple attenuation effect.Since the overall photoelectron time at the output of

each phototube is the sum of the factors listed above, itsglobal probability density function (PDF) is the convolu-tion [19] of the individual distributions pertaining to eachfactor. For the purpose of the examples in this work, thescintillation decay profile is assumed equal to that of thescintillation mixture (Pseudocumene+PPO at 1.5 g/l)adopted for the Borexino solar neutrino experiment, whilethe PMT time response is modeled as a Gaussiandistribution with sigma equal to 1 ns. In the followingparagraphs we will always show the TOF factor resultingfrom the calculations related to the various cases underconsideration, and, only when useful to illustrate better theobtained results, also the convolution of the derived TOFfactor with the scintillation curve and the PMT response. Itwill be also clear, along the calculations, that the simpleattenuation effect, when considered, will imply essentially amodification of the TOF term.

4. Time distribution detected by a phototube for a uniform

volume distribution of events

Let us consider (Fig. 2) a spherical volume of radius R ofliquid scintillator with events uniformly distributed in it.We want to evaluate the time distribution that such eventsinduce on a PMT observing the volume, neglecting theabsorption re-emission and scattering effects. If weconsider a polar coordinate system centered on the PMT,given a point characterized by coordinates l, y (y is theangular coordinate orthogonal to the plane of the figure

ARTICLE IN PRESS

Fig. 2. Geometrical derivation of the time of flight distribution detected

by a PMT observing a spherical detector characterized by a uniform

internal distribution of scintillation events.

P. Lombardi, G. Ranucci / Nuclear Instruments and Methods in Physics Research A 574 (2007) 65–8268

and this is why it is not reported in the figure itself) and f,we have that the probability that a scintillation eventoccurs in the infinitesimal volume around such a point(shaded area in the figure) is

1

ð4=3ÞpR3l2 sin jdjdydl (2)

and the probability that a photon emitted from this pointhits the phototube is

dS

4pl2cosðp� jÞ. (3)

Hence the joint probability that a photon is generated atthe location l, y and f and is detected by the phototubeunder consideration is

1

ð4=3ÞpR3l2 sin jdjdydl

dS

4pl2cosðp� jÞ. (4)

The integration of relation (4) over y and f gives thus thePDF of the path traveled by the photons arriving at thephototube. Hence

pðlÞ ¼1

ð4=3ÞpR3

Z p

j¼jmin

Z 2p

y¼0l2 sin jdjdydl

dS

4pl2

cosðp� jÞdydj ð5Þ

where jmin is the minimum j that, for each l, correspondsto a real path from the scintillator sphere to the PMT(see Fig. 2).Eq. (5) becomes

pðlÞ ¼1

ð4=3ÞpR3

Z p

j¼jmin

Z 2p

y¼0l2 sin jdjdydl

dS

4pl2

cosðp� jÞdydj ð6Þ

or

pðlÞ ¼2p

ð4=3ÞpR3

Z p

j¼jmin

dS

4pð� sin j cos jÞdj (7)

pðlÞ ¼1

ð8=3ÞpR3

Z p

j¼jmin

�1

2sin 2jdjdS. (8)

Eq. (8) after some manipulations becomes finally

pðlÞ ¼3

16pR31�

R2 � l2 � Z2pm

� �24l2Z2

pm

0B@

1CAdS. (9)

In order to obtain the relation (9), it is taken into accountthat, from the triangle formed by R, l, and Zpm (see againFig. 2), the angle jmin is given by

cos jmin ¼R2 � l2 � Z2

pm

2lZpm. (10)

One may wonder which is the meaning of the surfaceelement dS, which appears in Eq. (9). Specifically, it meansthat the integral of Eq. (9) over l gives the probability todetect a photon over the unit surface in correspondence tothe phototube, so that the further integral over the wholedetecting surface (i.e. the sphere of radius Zpm covered bythe observing phototubes) gives 1. Since the detectionprobability for unit surface, for obvious symmetry reasons,is uniform over the entire detecting sphere, such an integralis equivalent to multiply by the area of the sphere of radiusZpm (we do not consider here the obvious fact that in apractical arrangement the detecting sphere is only partiallycovered by the lattice of PMT’s).The above calculation neglects the attenuation. For a

more realistic result also the self-attenuation effect of thescintillator should be considered. Assuming, as it is inmany actual set-ups, that the same attenuation processoccurs inside the vessel and in the path from the vessel tothe phototube, it is enough to consider a multiplicativeexponential factor so that Eq. (9) becomes

pðlÞ ¼3

16pR31�

R2 � l2 � Z2pm

� �24l2Z2

pm

0B@

1CAe�ðl=lattÞ dS (11)

where latt is the attenuation length.Eq. (9) (and similarly Eq. (11)) can be expressed in term

of the variable TOF tf, i.e. remembering that

l ¼c

ntf

ARTICLE IN PRESS

Fig. 3. Time of flight distribution induced on an observing phototube by a uniform distribution of events in the scintillation vessel.


we have

pðtf Þ ¼3

16pR31�

R2 � cn

tf� �2

� Z2pm

� �24 c

ntf

� �2Z2

pm

0B@

1CA

c

ne�ðc=nÞðtf Þ=latt dS ð12Þ

where c/n is the Hessian of the transformation.The above relations are valid for Zpm � RoloZpm þ R

i:e: for nc

Zpm � R� �

otfo nc

Zpm þ R� ��

.

The time distribution for unit surface (Eq. (12) omittingthe factor dS) is plotted in Fig. 3 for R ¼ 4.25 andZpm ¼ 6.52; these geometrical parameters correspond tothose of the solar neutrino experiment Borexino. The 2plots correspond, respectively to either ignoring or includ-ing the attenuation effect, with an attenuation length takenfor the purpose of the example equal to 7m. Since inBorexino the inner containment vessel will be immersed inthe same solvent used as base of the scintillator (Pseudo-cumene) the latter calculation is that more realistic.



distribution of scintillation events only on its surface.


surface distribution of events

The evaluation of the TOF distribution induced by auniform surface distribution of events proceeds similarly tothe procedure illustrated in the previous section. Withreference to Fig. 4 one can note that the eventscharacterized by a same path to arrive to the PMT arethose comprised in the elementary surface subtended by theangle dy. Taking into account the circular symmetry of theproblem, the area of this surface can be written as2pR2 sin ydy. Hence the joint probability that a photonis emitted from the surface subtended by dy and that isdetected by the phototube (corresponding to an elementarydetection area dS) is

pðlÞdldS ¼1

4pR22pR2 sin ydy

dS

4pl2cos j. (13)

The angle y can be expressed as function of l exploiting thecosine theorem for the triangle OP(PMT). i.e.

cos y ¼Z2

pm þ R2 � l2

2RZpm. (14)

By differentiating both members of Eq. (14) we get

sin ydy ¼l

RZpmdl. (15)

ARTICLE IN PRESS

Fig. 5. Time of flight detected by an individual phototube as response to uniformly distributed surface events.



internal distribution of scintillation events and with light refraction at the

boundary.


From the same triangle we can get also cosj

cos j ¼Z2

pm þ l2 � R2

2lZpm. (16)

Introducing the relations (15) and (16) in the Eq. (13)we get

pðlÞdldS ¼1

2

l

RZpmdl

dS

4pl2Z2

pm þ l2 � R2

2lZpm(17)

pðlÞdldS ¼1

16p

Z2pm þ l2 � R2

Rl2Z2pm

dl dS (18)

and so finally

pðlÞ ¼1

16p

Z2pm þ l2 � R2

Rl2Z2pm

dS (19)

where the meaning of the surface element dS is aspreviously explained for Eq. (9).

The relation (19), expressed in term of the TOF, becomes

pðtÞ ¼1

16p

Z2pm þ

cn

tf� �2

� R2

R cn

tf� �2

Z2pm

c

ndS. (20)

As in the previous section, Eq. (20) can also be modified toaccount for the attenuation effect as follows:

pðtÞ ¼1

16p

Z2pm þ

cn

tf� �2

� R2

R cn

tf� �2

Z2pm

c

ne�ðc=nÞ tf=latt dS. (21)

(A similar modification is valid for relation (19), too).The time distribution for unit surface (i.e. Eq. (21) omittingthe factor dS) is plotted in Fig. 5 for the geometricalparameters corresponding to the Borexino detector(R ¼ 4.25 and Zpm ¼ 6.52); as for Fig. 3, the 2 caseswithout or with the attenuation (latt ¼ 7m) are considered.


volume distribution with refraction at the boundary

An unusual situation which requires to modify thecalculation in Section 4 is that in which the surroundingmedium is different from the scintillator solvent, beingtypically water, instead. In this case there is refraction atthe boundary, which greatly complicates the evaluation ofthe TOF at the phototube.As depicted in Fig. 6, the ray emitted from the event

point is bended at the boundary of the 2 mediums. Since

ARTICLE IN PRESSP. Lombardi, G. Ranucci / Nuclear Instruments and Methods in Physics Research A 574 (2007) 65–82 71

the refraction index of the inner medium (scintillator) islower than that of the outer medium (water) the directionof the ray is bended so to increase the angle with the radiusof the sphere drawn through the incidence point. Theevaluation of the time distribution at the phototube can bedone numerically, since a closed form formula cannot bewritten. The numerical evaluation proceeds through 2steps: the first is the determination of the optical path fromthe generic event site to the phototube, the second is theconstruction of the overall distribution from the individualpath lengths.

The construction in Fig. 6 has the purpose to illustratethe determination of the optical path. Given B the eventpoint, the path traveled by the photon to arrive to the PMTis the sum of the 2 segments l1 and l2 shown in the figure. Inthe adopted reference coordinate system, B has the genericcoordinates x and y and the abscissa of the PMT is Zpm.The direction of the radius passing through B is specifiedby the angle b. The incident angle is denoted as gi, therefraction angle as gr, the angle formed by the photon withrespect to the radius passing through B is denoted as sUThe straight distance from the point B to the PMT isobtained via the cosine theorem applied to the triangleOBP

BP ¼ Z2pm þOB2

� 2OBZpm cos b. (22)

In order to compute the right path l1+l2 bringing thephoton from the event site to the PMT, an iterativeprocedure is adopted: many trial values of the angle s aretried (starting from 0) and for each of them the followingvalues are computed:

l1 ¼R

sin ssinðs� giÞ (23)

from the sine theorem applied to the triangle OBA, and

l2 ¼R sinðb� sþ giÞ

sinðgr � bþ s� giÞ(24)

from the sine theorem applied to the triangle OAP0.The special values of l1, l2 for which they form together

with BP0 a triangle with vertexes B, A and P (i.e. when Pand P0 coincide) are the required correct segments of the

Fig. 7. Time of flight distribution induced on an observing phototube by a unifo

medium (scintillator and water). The example has been computed for the para

optical path, and at this point the iteration is stopped.Again with reference to Fig. 6, it can be demonstrated thatpre-requisite for the refracted ray to hit the phototube isthat the condition a ¼ gr � bþ s� gi40� holds, wheregr � bþ s� gi is the angle formed by the path l2 with thepositive x axis. In the opposite case a ¼ gr � bþ s� gio0�

the ray l2 would not hit the positive x-axis, but on thecontrary it would be the prolongation on the opposite sideof l2 that would hit the negative x-axis (forming theangle—a), thus giving origin to a construction in which thePMT is surely not hit by the light.The light paths from all the points of the sphere can be

properly grouped to form the overall distribution of theTOF to the PMT. Taking into account the sphericalsymmetry of the problem, it is enough to consider a grid ofcalculation points in a semicircle of the sphere itself:assuming a certain binning for the TOF distribution to bederived (for example 0.2 ns) each path corresponding to apoint of the grid is converted into a bin number, and to thecontent of the bin so identified it is added a factor whichrepresents the weight of the calculated path. Since eachpoint of generic coordinates x, y actually stands for anentire circular corona of radius y (in the sense that all thepoints comprised in this corona produce the same path tothe phototube), the first factor in the weight is simply thevolume of such a corona, i.e. 2pyDxDy, being Dx and Dy

the widths of the cells of the grid (Fig. 7).The second factor in the weight gives the probability

that the photon actually impinges the detecting photo-tube, and it is thus expressed as ðdS=4pðl1 þ l2Þ

2Þ

cosðgr � bþ s� giÞ. Therefore the weight is globally givenby

W ¼ 2pyDxDydS

4pðl1 þ l2Þ2cosðgr � bþ s� giÞ. (25)

In practice dS is omitted, as above, and the calculation arereferred to the unit area of the detecting sphere of thephototubes.A detector in which the refraction effect is important is

the Counting Test Facility (CTF) [7], prototype of theabove-mentioned Borexino detector. In CTF the inner

rm distribution of events in the scintillation vessel in the case of 2 different

meters characterizing the CTF detector.

ARTICLE IN PRESS

Fig. 8. Map of the points inside the CTF vessel for which an observing PMT is blind.


vessel (R ¼ 1) containing the scintillator with refractionindex equal to 1.5, is surrounded by water (refraction index1.33). The output of the determination of the TOF factorwith these parameters (and considering that in CTF thephototubes are located at Zpm ¼ 3.3 from the center) isshown in Fig. 7, where for comparison it is reported alsothe TOF distribution that would result if the 2 mediumswere the same (with ns ¼ 1.5) everywhere. The shift of the 2curves is due to the systematically longer optical path in thelatter situation, due to the assumption of having in thewhole detector the medium with the higher refractionindex. Besides this, in the refraction case the curve appearsalso distorted with a sort of ‘‘missing portion.’’ Thispeculiar shape is actually linked to the presence of points inthe vessel which are missed by the phototube, in the sensethat the light rays emerging from them cannot find a pathsuitable to reach the observing phototubes because of therefraction effect, which can also give rise to a light trappingat the border. For completeness, in Fig. 8 the map of theobscured points is reported.

In this respect, it is interesting to point out that, if thedistribution is referred to the whole detecting surfacethrough an integration which is equivalent to a multi-plication by the area of such a surface, the re-scaleddistribution will have an integral, less than 1, which givesthe fraction of the sphere which is not obscured. In thepresent CTF example this fraction amounts to 91.3%.

If required, in the procedure described in this section itcan be included, as well, the attenuation effect, byintroducing in the weight (25) the appropriate exponentialfactors containing l1 and l2, depending upon the actualphysical situation. In the CTF example this correction hasnot been included for the 2-fold reason that the attenuationoutside is negligible, because the medium is water, andinside is negligible also, because of the limited size of thedetector.

Finally, it must be mentioned that as a cross-check, theprocedure adopted in this section has been applied to thecase of same index of refraction everywhere, obtaining adistribution coinciding with the correct curve described byEq. (12) (without the attenuation factor).

7. Time profile of single events

Up to now we have considered how a single phototubedetects the time distribution of the photons coming from auniform (or surface) distribution of events in the scintilla-tor containment vessel. An alternative distribution ofinterest is how the photons from a specific event site aredistributed in time. Essentially, this is the oppositesituation: instead of evaluating the distribution inducedon a single PMT by an ensemble of events, we are nowinterested to the time distribution on all the PMT’s of thedetection times of the photons induced by a singleinteraction site.Again we neglect the absorption and re-emission effect

and limit ourselves to consider the ‘‘zero-order’’ effects,due to the geometry and to the attenuation process. Thecalculation in the next 2 sub-paragraphs takes into account2 different situations: either same scintillation mediumeverywhere or 2 different mediums inside and outside thevessel. In both cases, the first step of the calculation is thedetermination of the ‘‘TOF’’ factor, which in this situationis the distribution of the arrival times to the detectingsphere (i.e. the lattice of phototubes) of the photons as ifreleased in an ideal instantaneous scintillation process; thesecond step is the convolution of the TOF term with theintrinsic scintillation decay curve and the response of thePMT in order to obtain the overall time distribution ofthe detected photoelectrons.

7.1. Same index of refraction

With reference to Fig. 9, where the emission point isdenoted with Zev, the generic arrival point with P, and thepath traveled by the photon with l, if there is no absorptionthe TOF distribution of the photons arriving to thedetecting sphere of radius Rd is determined by puregeometrical arguments.Indeed it is obvious that the distribution of the TOF is

dictated by the distribution of the solid angle O.Specifically, the photons emitted in the infinitesimal solidangle dO will be characterized by the same TOF. Hence we

ARTICLE IN PRESS

Fig. 9. Geometry for the derivation of the time of flight distribution

recorded by a detecting sphere of radius Rd as response to a localized

interaction in the scintillator.


have to transform the probability that a photon is emittedbetween O and O+dO, in the probability that its TOF iscomprised between tf and tf+dtf.

To do this we have to use the relationship between O andy, the vertex angle of the cone that subtends O

O ¼ 2pð1� cos yÞ (26)

from which we get

dO ¼ 2p sin ydy. (27)

Since the probability that a photon is emitted between Oand O+dO, p(O)dO, is equal to ð1=4pÞdO (the emission isisotropic), by multiplying both terms of Eq. (27) by 1/(4p),we get

1

4pdO ¼

1

2sin ydy. (28)

By applying the cosine theorem to the triangle OZevP weobtain

cos y ¼R2

d � Z2ev � l2

2Zevl. (29)

By differentiating the first member of Eq. (29) with respectto y, and the second to l, we get

� sin ydy ¼�2l � 2Zevl � 2R2

dZev þ 2Z3ev þ 2Zevl

2

4Z2evl

2dl

(30)

� sin ydy ¼�2l2Zev � 2R2

dZev þ 2Z3ev

4Z2evl

2dl (31)

sin ydy ¼1

2Zev1þ

R2d�Z2

ev

l2

� �dl (32)

or dividing both members by 2

1

2sin ydy ¼

1

4Zev1þ

R2d�Z2

ev

l2

� �dl. (33)

Through the comparison of Eq. (28) with Eq. (33) werealize that the probability density function of the flightpath of the not absorbed photons is

pðlÞ ¼1

4Zev1þ

R2d�Z2

ev

l2

� �. (34)

We can transform the last expression (34) in probabilitydensity function of the TOF by the usual transformationl ¼ (c/n)tf, thus obtaining

pðtf Þ ¼1

4Zev1þ

R2d�Z2

ev

cn

tf� �2

!c

n. (35)

Eq. (35) is valid on a limited interval of values of thevariable tf; indeed the minimum value of the TOFcorresponds to the path (Rd-Zev) and the maximum to(Rd+Zev). The Eq. (35) hence is valid for

ðRd � ZevÞn

cptfpðRd þ ZevÞ

n

c.

It can be easily checked that, as expected, the integral ofEq. (34) over l (or equivalently the integral of relation (35)over tf) is equal to 1.To account for the attenuation effect, the distribution

(35) must be corrected, as done for the distributions inSections 4 and 5, introducing an exponential factor,therefore obtaining

pðtf Þ ¼1

4Zev1þ

R2d�Z2

ev

cn

tf� �2

!c

ne�ðc=nÞ=tf=latt . (36)

The medium outside the vessel is equal to the mediuminside the vessel itself, leading to the same attenuationeffect everywhere.The TOF factors (35) and (36), with reference to the

geometrical parameters of the Borexino detector, areplotted in Fig. 10 for 3 different positions in the vessel(1, 2.5, 4.25m): it is evident that the broadening effect dueto the geometry, obviously more pronounced for locationsof the events closer to the boundary, is accompanied by apeculiar peaking of the time distributions at shorter values.The curves including the attenuation effect, evaluatedassuming an attenuation length of 7m, feature a verysimilar shapes to those unaffected by the attenuation; theirintegral, which is less than 1, is physically meaningful andgives the probability of a photon to reach the detectingsurface without being absorbed.

7.2. Case of index of refraction discontinuity

With reference to Fig. 11, in case of 2 different mediumsthe overall time distribution of the photons impinging

ARTICLE IN PRESS

Fig. 10. Time of flight distributions of the photons released by events located respectively at 1, 2.5 and 4.25m from the center of the Borexino detector,

with or without the inclusion of the attenuation effect.

Fig. 11. Geometry for the derivation of the time of flight distribution

recorded by a detecting sphere of radius Rd as response to a localized

interaction in the scintillator, with light refraction at the border of the

vessel.


upon the detecting sphere (i.e. the sphere of the photo-tubes) is obtained numerically following a proceduresimilar to that used in Section 6. Preliminarily it isevaluated, for each angle y, the light path l1+l2. Practically,the evaluation is carried out for a fine grid of y values, witha step Dy which can be done as small as desired. The flighttime for each y is then ðns=cÞl1 þ ðnw=cÞl2; upon defining asuitable binning for the desired distribution, for example0.2 ns, the flight time is used to identify the correspondingbin. To that bin it is thus added the proper weightcorresponding to the computed TOF, which is equal to

W ¼ e�ðl1ðyÞ=lattÞ1

2sin yDy (37)

where the exponential term represents the probability ofthe photon of being not absorbed within the vessel(assuming that outside there is water, whose attenuationeffect is hence negligible).According to the Fig. 11, the relevant formulas are

gi ¼ arcsinZev

rvsin y

� �(38)

gr ¼ arcsinns

nwsin gi

� �(39)

l1 ¼ �Zev cos yþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2v � Zev sin yð Þ

2q

(40)

(obtained from the triangles OO0P0 and OO0Zev)

l2 ¼ �rv cos gr þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2

d � rv sin gr� �2q

(41)

(obtained from the cosine theorem applied to the triangleOP0P).The TOF distribution, evaluated according the prescrip-

tion just illustrated, is plotted in Fig. 12 for 5 differentradial positions in the CTF vessel (due to the limited size ofthe detector, the attenuation factor in this case has beendropped from the weight). It is interesting to note the effectdue to the total internal reflection for the points close to theborder: essentially there is a central gap in the timedistribution due to the missing trapped photons. Fordisplay purpose the curves are all normalized to unit area.While this normalization is the right one for the curves notaffected by the light trapping, this is not true for the last 2curves exhibiting the trapping effect. It must be pointed outthat the calculation as carried out provides in principle forthem the proper normalization, thus leading to an integralthat gives correctly the fraction of not trapped photons.Another peculiarity of the trapped curves is the sharp

peak that they feature at the beginning of their second part.While this characteristic stems from details of themathematical calculation that are difficult to predict ‘‘apriori’’, its presence can be qualitatively understood as theeffect of the abrupt transition from the ‘‘dark’’ to the ‘‘non-dark’’ region. To shed more light on this intriguing feature,

ARTICLE IN PRESS

Fig. 12. Time of flight distributions of the photons released by events located, respectively, at 0.2, 0.4m, 0.6, 0.9 and 1m from the center of the CTF

detector; it should be noted the gap in the distributions affected by the light trapping, as well as the very peaked characteristic of the second part of these

distributions.

Fig. 13. Detected photoelectron distributions induced by events located, respectively, at 0.2, 0.4m, 0.6, 0.9 and 1m from the center of the CTF detector;

the distributions affected by the light-trapping phenomenon exhibit a double-peaked structure reflecting that of the original photon time of flight, being the

second peak more marked than the first.


it may be interesting in this case to carry out explicitlythe calculation to go from the TOF distributions of thephotons to the actual detected distributions of thephotoelectrons, according to the sequence of convolutionoperations outlined in Section 3.

The results of the convolution of the TOF terms inFig. 12 with the intrinsic scintillation light profile and thephototube timing response (we remind that we adopt forthe sake of the example the quantities valid for theBorexino project) are reported in Fig. 13. Also in this casethe curves are all normalized to 1 for the purpose ofcomparison. The interesting result is that for the trapping-related curves the double peak feature of the photon TOFis maintained also in the final photoelectrons distributions,being the second peak more prominent than the first.

This result can be compared, at least qualitatively, withthe data gathered in CTF. In particular, the light-trappingeffect in CTF has been specifically investigated in [20],where it was clearly shown that events originating fromlocations close to the vessel (either intrinsic events or eventsgenerated by a calibration source) exhibit the double-peaked structure retrieved by the present calculation, with

the remarkable confirmation that also in the real data thesecond peak is more pronounced than the first, inagreement with the prediction of the model.For the purpose of completeness the TOF distribution is

reported in Fig. 14 taking into account both the attenua-tion factor and the proper normalization due to the lighttrapping, in the 3 cases of event position at 0.2, 0.6 and 1m.In correspondence of each position the curves relevant tothe 2 different situations of attenuation included or notincluded are plotted. Obviously, for points well inside thevessel the only photon loss effect is that due to theattenuation: for the 0.2m position the fraction of photonslost for the attenuation is 13.1%, and at 0.6m is 11.5%.For points close to the boundary the loss of light due to thetrapping is more important than the loss caused by theattenuation; for example for the position at 0.9m the lighttrapping causes a loss in terms of photons of 17.2%, whilethe attenuation causes and additional loss of only 7%,essentially concentrated in the second part of the curve. Asshown in the figure, the first part is unaffected by theattenuation since it is relevant to photons which fly acrossthe scintillator for a short path.

ARTICLE IN PRESS

Fig. 14. Time of flight distributions of the photons released by events located, respectively, at 0.2, 0.6 and 0.9m from the center of the CTF detector,

with or without the inclusion of the attenuation effect.

Fig. 15. Detected photoelectron distributions induced by events located in contact with the vessel wall at 1m from the center of the CTF detector,

respectively, with and without the inclusion of the effect of the wall thickness. The wall has been chosen on purpose very thick (5 cm) to enhance the

amount of the effect for the purpose of this example.


7.3. Effects of a thick containment wall

The calculations performed in this and in the previousparagraphs do not account explicitly for the effect of thethickness of the wall of the scintillator containment vessel.In practice this effect is modest, especially considering thesmall index of refraction difference between the scintillatorand the materials suitable to be used for the vesselmanufacturing. For example, while for a common scintil-lator mixture the refraction index is of order of 1.50–1.53,that of a typical material like nylon employed for the vesselconstruction [21] is 1.53. The minimal amount of theexpected effect is justified also by the circumstance that inpractical set-ups the nylon wall is extremely thin.

In this respect a slightly different situation will be that ofthe forthcoming SNO+ experiment, mentioned in theintroduction. The containment vessel in this case (still thesame used by the now ended SNO experiment) is made bythick acrylic panels of few centimeter thickness, which willseparate the inner active scintillator volume from theexternal shielding water. Therefore, it can be interesting toextend the calculations of the previous Section 7.2 to thescintillator–acrylic–water interface.

For this purpose, it is enough to generalize the situationdepicted in Fig. 11, by adding to the overall TOF from theevent site to the detecting sphere also the path traveled bythe photon inside the wall (and considering, obviously, therefraction process at the entrance and exit of the wallitself). By denoting with rv and Rv the inner and outerradius of the containment vessel, it can be easily shown bya simple extension of the procedure of the Section 7.2 thatthe 2 paths traveled by the photons, respectively, within thevessel wall and in water (the path in the scintillator isclearly unchanged) are given by

l2 ¼ �rv cos gr þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2

v � rv sin gr� �2q

(42)

and

l3 ¼ �Rv cos g0r þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2

d � Rv sin g0r� �2q

(43)

where

gr ¼ arcsinns

nasin gi

� �(44)

ARTICLE IN PRESSP. Lombardi, G. Ranucci / Nuclear Instruments and Methods in Physics Research A 574 (2007) 65–82 77

is the refraction angle at the scintillator–wall interface, and

g0r ¼ arcsinnsrv

nwRvsin gi

� �(45)

is the refraction angle at the wall–water interface.Assuming a scintillation point in contact with the vessel,

for which thus the optical modification due to finite wallthickness is maximum, in Fig. 15 we show the correspond-ing time distributions both with and without the wall effectfor a detector of the same dimension of CTF, butsurrounded by an acrylic vessel 5 cm thick. The changeinduced to the time distribution in this example is quiteevident, since we choose to enhance the difference anunrealistic too large wall depth. In practice, a thickness ofthe order of several cm is appropriate, from the engineeringpoint of view, for much larger vessels, like the 12mdiameter SNO vessel: with these realistic dimensions it canbe easily verified that the resulting change of the timedistribution is even difficult to appreciate.

In this framework it can be worthwhile to add thatanother factor to account for a more accurate descriptionof the interface effect is the transmission coefficient at theboundary, which would simply play the role of a furthermultiplicative factor in the relation (37).

Finally, we conclude this digression by outlining that asimilar procedure to include the effect of a thick wall can beused to properly modify, if needed, the time distributionsdetected by a single observing phototube described in theprevious sections.

8. Trigger fluctuation

As explained in Section 2, the trigger condition in a largescintillation counter is obtained by requiring the firing of acertain number NF of PMT’s in a suitable time windowchosen on the basis of the detector extension. Obviously,with respect to the ideal 0 time of the actual occurrence ofthe event, the trigger time is affected by a fluctuationdictated by the statistics of the coincidence process. In theevaluation of the time sequence of the photoelectronsproduced in a single event, this amount simply to an overallcommon shift of the time base; instead in the evaluation ofthe global time distribution detected by a single photo-multiplier for a uniform volume or surface distributions ofevents this effect has an overall distortion impact due to thefact that the time measurements which are added together tocreate the global distribution are not referred to a same fixed0 time but are referred to a fluctuating, and hence differenton an event by event basis, trigger time. This implies, inparticular, that the actual measured TOF distribution is theconvolution of the trigger time distribution with the TOFideal factor determined in the previous paragraphs.

It is, thus, of interest to find a way to compute the triggerfluctuation. This can be done, with a good degree ofapproximation, profiting of the known formulas describingthe photon statistics in scintillation counters [22,23].

Specifically, there are 2 applicable formulas in case thatone considers events comprising either a fixed or afluctuating numbers of photoelectrons. In the former case,given the probability density function of the photonproduction in the scintillator p(t), then the probability ofthe time of production of the ith photon out of n photonsexactly originated in a scintillation event is

piðt=nÞ ¼n!

ði � 1Þ!ðn� iÞ!1� F ðtÞ½ �

ðn�iÞ F ðtÞ½ �i�1pðtÞ (46)

where F(t) is defined as

F ðtÞ ¼

Z t

0

pðlÞdl. (47)

In the latter case, considering Poisson fluctuating events,the applicable formula is

piðt; nÞ ¼nie�nFðtÞ F ðtÞ½ �

i�1pðtÞ

ði � 1Þ!. (48)

In the original contest in which they are derived, theseformulas are related to the photon timing in a scintillatorcharacterized by the scintillation light profile p(t). Since inthe present case we are interested to the detection time ofthe photoelectrons as delivered by the phototubes, then thep(t) functions to be considered are the time profiles forsingle events as derived in Section 7: in fact, it is onthe sequence of photoelectrons that obey to those curves(we remind that they amount to the convolution of theintrinsic scintillation light, the PMT response and the TOFfactor, with or without the attenuation term) which isapplied the coincidence criterion for the trigger derivation.It can be instructive to evaluate for some special cases of

number of detected photoelectrons, thus exploiting theformula (46), and for few points inside the scintillationvessel, the resulting trigger fluctuation. For example,considering a detector like Borexino, in which one canexpect of the order of 20 PMT’s in coincidence over a timewindow of 50 ns as trigger condition, it can be assumed astrigger distribution that of the detection time of the 20thphotoelectron, as derived from the formula (46). Clearlythis is an approximation, which is valid if the total numberof photoelectrons is significantly high, since it is equivalentto assume that the photoelectron number 20 occurs withinthe first 50 ns window. In case of a number of photoelec-trons not much higher than the trigger condition, it canhappen that either in a substantial fraction of events thetrigger condition is not met or that it is met, but not in thefirst 50 ns window.In this case, the evaluation of the trigger time could be

done, besides obviously through a Monte Carlo calcula-tion, following a sliding window approach which leads to ascan statistics calculation, which however is beyond thepurpose of the present paper.The trigger distributions in these conditions, for 50

detected photoelectrons and for 5 different radial positionare reported in Fig. 16. As intuitively expected, thedispersion of the trigger time is higher for events located

ARTICLE IN PRESS

Fig. 16. Trigger time distributions in Borexino for events located at 5 different radial positions (0.4, 1.2, 2, 2.8, 3.6m) and for 50 photoelectrons.

The assumed trigger condition is 20 PMT’s concurrently firing in a 50 ns time window.


at increasing radial distances from the center of the detector.Also, it appears that there is a shift on the mean value of thetrigger time distributions depending upon the actual radiallocation of the events, with the events closer to the bordercausing obviously an early triggering with respect to thosemore central. By computing Eq. (46) for increasing numbersof photoelectrons it appears that the trigger fluctuationcurves shrink significantly, and also that their absolute shiftwith respect to the true 0 time decrease, since morephotoelectrons induce obviously an early trigger occurrence.

9. Single coordinate distribution for uniform events

The sequence of the photoelectron-detected timesstudied in the previous sections can be used to infer ‘‘aposteriori’’ the spatial location of the originating event. Athorough maximum likelihood procedure for this purposehas been studied for example in Ref. [13]. In this ‘‘inverseproblem’’ framework the studies of the previous para-graphs can be extended deriving some interesting resultsconcerning the expected spatial coordinates distribution ofthe events in some special cases. The main variable ofinterest is r, the radial coordinate of the location of theevents, but it can be useful to show also the expecteddistribution of the individual x, y and z coordinates in thespecial case of uniform distribution of events. Let usconsider for this purpose that the probability for an eventto be in the elementary volume dxdydz, in a sphericalvolume of radius R, is simply

pðx; y; zÞdxdydz ¼1

ð4=3ÞpR3dxdy dz (49)

from which, by integrating over y and z, it stems theprobability of the variable x to be between x and x+dx, i.e.

pðxÞdx ¼dx

ð4=3ÞpR3

Z ffiffiffiffiffiffiffiffiffiffiffiR2�x2p

�

ffiffiffiffiffiffiffiffiffiffiffiR2�x2p dy

Z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2�x2�y2p

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2�x2�y2p dz (50)

pðxÞ ¼1

ð4=3ÞpR3


�

ffiffiffiffiffiffiffiffiffiffiffiR2�x2p dy2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � x2 � y2

q(51)

pðxÞ ¼3

2pR3


�

ffiffiffiffiffiffiffiffiffiffiffiR2�x2p


qdy (52)

which by symmetry becomes

pðxÞ ¼3

pR3


0


qdy. (53)

By putting y ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � x2p

sin t and hence dy ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � x2p

cos tdt, Eq. (53) becomes

pðxÞ ¼3

pR3Z arcsinðy¼ffiffiffiffiffiffiffiffiffiffiffiR2�x2p

Þ=ðffiffiffiffiffiffiffiffiffiffiffiR2�x2p

Þ

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � x2 � ðR2 � x2Þsin2t

q

� cos tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � x2

pdt ð54Þ

pðxÞ ¼3

pR3

Z arcsin 1

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðR2 � x2Þð1� sin2tÞ

q

cos tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � x2

pdt ð55Þ

pðxÞ ¼3

pR3

Z ðp=2Þ0

ðR2 � x2Þcos2tdt (56)

pðxÞ ¼3ðR2 � x2Þ

pR3

Z ðp=2Þ0

cos2tdt ¼3

4R3ðR2 � x2Þ. (57)

It can be easily checked that the relation (57) is correctlynormalized to 1.If the detection system (i.e. the lattice of the PMT’s) is

spatially uniform similar results are valid for the y and z

coordinates, too.So, if the detector is not affected by biases or systematic

effects, a uniform distribution of events would resultin parabolic distributions of the 3 spatial coordinates.Moreover, if the spread in the reconstruction process of theevent coordinates is characterized (as it is usual) by

ARTICLE IN PRESS

Fig. 17. Distribution of an individual spatial coordinate for events uniformly distributed in a vessel of 1m diameter, with and without the effect of the

finite position resolution.


a Gaussian profile, then the distribution of the recon-structed Cartesian coordinates is simply the convolution ofthe Gaussian resolution function with the parabolicfunction (57).

Eq. (57) and its convolution with the Gaussian spreadare plotted in Fig. 17 for the case of a vessel of 1m radiusand assuming a resolution (s) of 20 cm. Interestingly, theconvolution effect is to change marginally the bulk of theparabolic distribution, while adding 2-side tails whoseextent is determined by the width of the resolutionfunction.

10. Reconstructed distribution of point-like events

Again in this section we do not consider how the eventlocation is inferred starting from the sequence of the times ofthe detected photoelectrons, but we assume simply that sucha procedure leads to estimates of the x, y and z coordinateswhich are all 3 characterized by a Gaussian uncertainty.What we want to show here is how these estimates combineto give the quantity of interest, i.e. the probability densityfunction of the radial distance r of the event from the centerof the detector. To this purpose we follow closely thederivation illustrated in Ref. [19]. Alternative approaches arealso possible, like for example that reported in Ref. [24].

Let us consider the joint probability that the estimatedcoordinates are located around a generic point ofcoordinates x, y, and z. We can then write

pðx; y; zÞdx; dy;dz ¼1

s1s2s3ffiffiffiffiffiffiffiffiffiffiffi2pð Þ3

q e�

x�x0ð Þ2

2s21

þy�y0ð Þ

2

2s22

þz�z0ð Þ2

2s23

� �dxdydz

(58)

under the assumption that the 3 estimates are independent.Here x0, y0 and z0 are the true, unknown coordinates of theevent.

In polar coordinates we have, as usual x ¼ r cosj cos y,

y ¼ r cosj sin y, z ¼ r sinj and

dxdydz ¼ r2 cos jdrdydj

from which Eq. (58) becomes

pðx; y; zÞdx;dy;dz ¼ 1

s1s2s3ffiffiffiffiffiffiffiffi2pð Þ3p

�e�

r cosj cos y�x0ð Þ2

2s21

þr cosj sin y�y0ð Þ

2

2s22

þr sinj�z0ð Þ2

2s23

� �r2 cosjdrdydj:

(59)

In order to obtain from the relation (59) the desireddistribution in r it is enough to integrate the secondmember over y and j, so

pðrÞ ¼r2


q Z 2p

0

dy

Z ðp=2Þ�ðp=2Þ

e�

r cos j cos y�x0ð Þ2

2s21

þr cosj sin y�y0ð Þ

2

2s22

þr sin j�z0ð Þ2

2s23

� �cos jdj

ð60Þ

The integration can be carried out numerically, but it canbe performed very easily in closed form forx0 ¼ y0 ¼ z0 ¼ 0. In that case we have, assuming forsimplicity s1 ¼ s2 ¼ s3 ¼ s

pðrÞ ¼r2

s3ffiffiffiffiffiffiffiffiffiffiffi2pð Þ3

q Z 2p

0

dy

Z ðp=2Þ�ðp=2Þ

e�

r cos j cos yð Þ2þ r cos j sin yð Þ2þ r sin jð Þ2

2s2 cos jdjð61Þ

or

pðrÞ ¼r2


q Z 2p

0

dyZ ðp=2Þ�ðp=2Þ

e�ðr2Þ=ð2s2Þ cos jdj (62)

pðrÞ ¼2r2e�ðr

2=2s2Þ


q Z 2p

0

dy ¼2r2e�ðr

2=2s2Þ

s3ffiffiffiffiffiffi2pp . (63)

Eq. (63) is the extension to the 3D configuration ofthe well-known Rayleigh distribution, valid for the 2Dcase [19].

ARTICLE IN PRESS

Fig. 18. Estimated radial distributions of point-like events at 3 different true locations from the center of a detector of radius 1m and for s ¼ 0.15m.

Fig. 19. r2 uniform event density and the corresponding estimated radial distribution evaluated for a detector of 1m radius and a resolution of 0.15m.


In Fig. 18, the expected radial distributions of point likeevents located at r ¼ 0, 0.4 and 0.8 in a generic scintillatorvessel of 1m radius are reported.

The sigma for the x, y and z variables assumed in thecalculation is equal to 0.15m. For events located in thecenter the most probable estimated radial position is.0.21m, for events located at 0.4 is 0.45m, and for eventslocated at 0.8 is 0.83. So, the more the events are far fromthe center, the more the peak position of the radialdistribution is close to the true event location.

Eq. (63) is applicable also in the case in which it is neededto perform spatial correlation of different events. Inparticular, if 2 events originate from the same location(as in special radioactive decay correlated sequences) theradial distance of the corresponding estimates obeys torelation (63), with the caveat that the s to be introduced isthat affecting the individual coordinates multiplied by O2.

11. Reconstructed distribution of uniform events

From relation (60) it can be inferred the radialdistribution of events uniformly distributed in the vessel.For this purpose it is enough to multiply Eq. (60) by thedensity of the events as function of r0 (the true radial eventposition, scanning the whole vessel) and then integrate the

result over r0 itself. Furthermore, because of the sphericalsymmetry, it can be assumed without loss of generalityx0 ¼ r0, y0 ¼ 0 and z0 ¼ 0. Hence, since the probabilitydensity function of r0 is

pðr0Þ ¼3

r3vr20 (64)

we have

pðrÞ ¼Z rv

0

dr0pðr0Þr2


q Z 2p

0

dy

�

Z ðp=2Þ�ðp=2Þ

e�

r cos j cos y�r0ð Þ2

2s21

þr cos j sin yð Þ2

2s22

þr sin jð Þ2

2s23

� �cos jdj.

ð65Þ

The expected distribution stemming from Eq. (65) isreported in Fig. 19 for the case of the CTF vessel.In Fig. 19, the r2 geometrical distribution is also

displayed. Interestingly, in the core of the vessel the 2distributions coincide, while it is close to the boundary thatthe effect of the finite resolution manifests with a tail ofevents whose location is estimated outside the physicalregion of the containment vessel.

ARTICLE IN PRESS

Fig. 20. Estimated event densities for uniformly (internal) and exponentially (external) distributed events evaluated for a detector of 1m radius and a

resolution of 0.15m.


12. Reconstructed distribution of external events

A calculation similar to that of Section 11 can lead to theevaluation of the estimated distribution of external events(external in the sense that they are induced inside thescintillator by g-rays originated by radioactive contami-nants in the surrounding medium), in the reasonableassumption that the actual position of such events followan exponential decaying profile inward from the vesselwall. Thus in this case the probability density function ofthese events can be written as

pðr0Þ ¼r20e�ðrv�r0=lÞR rv

0 r20e�ðrv�r0=lÞ

(66)

and the required distribution is obtained from Eq (65) inwhich it is inserted the relation (66) as p(r0).

The expected distribution is reported in Fig. 20 for thecase of the CTF vessel; for comparison it is reported againthe expected distribution for uniformly distributed internalevents.

13. Conclusions

The time and the associated event position distributionsin scintillation detectors can be accurately studied throughdetailed Monte Carlo modeling. In the special case ofspherical geometry, however, useful formulas can bederived to give an approximate analytical description ofthe detector signal responses, which can usefully comple-ment the more precise Monte Carlo outputs. In this paper,we have given a thorough account of their derivation forseveral cases of practical interest.

Acknowledgements

The authors acknowledge the support of INFN to thisresearch.

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Documents

Time and position distributions in large volume spherical scintillation detectors