Simulation of large photomultipliers for experiments in astroparticle physics
SSimulation of large photomultipliers for experiments in astroparticle physics
Alexandre Creusot a , Darko Veberiˇc ∗ ,a,b a Laboratory for Astroparticle Physics, University of Nova Gorica, Slovenia b Department of Theoretical Physics, J. Stefan Institute, Ljubljana, Slovenia
Abstract
We have developed an accurate simulation model of the large 9 inch photomultiplier tubes (PMT) used in water-Cherenkov detectors of cosmic-ray induced extensive air-showers. This work was carried out as part of the devel-opment of the Offline simulation software for the Pierre Auger Observatory surface array, but our findings may berelevant also for other astrophysics experiments that employ similar large PMTs.The implementation is realistic in terms of geometrical dimensions, optical processes at various surfaces, thin-film treatment of the photocathode, and photon reflections on the inner structure of the PMT. With the quantumefficiency obtained for this advanced model we have calibrated a much simpler and a more rudimentary model ofthe PMT which is more practical for massive simulation productions. We show that the quantum efficiency declaredby manufactures of the PMTs is usually determined under conditions substantially different from those relevant forthe particular experiment and thus requires careful (re)interpretation when applied to the experimental data or whenused in simulations. In principle, the effective quantum efficiency could vary depending on the optical characteristicsof individual events.
Key words: photomultiplier tube, photocathode, Fresnel equations, thin-film, complex index of refraction, simulation
1. Introduction
The Pierre Auger Observatory is the largest detectorbuilt to detect cosmic rays with energies above 10 eV[1]. The surface detector part consists of over 1660 water-Cherenkov detectors distributed over an area of morethan 3000 km . While the longitudinal part of the showerdevelopment is measured with the fluorescence detec-tor, the muonic and electromagnetic lateral componentsare measured at ground level with the surface detector(SD). The individual SD stations consist of 12 t of puri-fied water in a light-tight container with highly reflectiveand diffusive inner surface [1]. The Cherenkov light pro-duced by the penetrating charged particles is observedwith PMTs. The signal levels from the PMTs are con-stantly calibrated against the average response to the at-mospheric background muons that trigger at the lowestthreshold level [2]. Through dedicated experiments [2, 3]the response to atmospheric background muons arrivingfrom all directions [4] has been related to the selectionof atmospheric muons arriving predominantly verticallythrough the center of the SD station. All signals are thusmeasured in relative units of VEM (Vertical EquivalentMuon), i.e. relative to the average signal that would beproduced by the vertical-centered muon. Although this ∗ Corresponding author, tel.: +386 5 3315 255, fax: +386 5 3315 385.
Email addresses: [email protected] (AlexandreCreusot), [email protected] (Darko Veberiˇc) auto-calibration procedure successfully removes most ofthe systematics due to the detector changes, there arenevertheless some applications that require a certain de-gree of knowledge about absolute values. The absolutenumber of detected muons and the size of the electro-magnetic fraction in the signal are important quantities,e.g. in studies of hadronic interactions in air-shower cas-cades [5] or for the purposes of primary particle identi-fication [6, 7], just to name a few. We have developeda detailed simulation of the processes of light detectionin the PMTs to reproduce known calibration data of theSD stations and allow comparison of real and simulatedevents. This study is part of the efforts to produce acomplete SD simulation chain for the Pierre Auger Ob-servatory [8, 9, 10].
2. Photomultiplier tube
The SD station has a cylindrical shape. Its height is1.2 m and its radius is 1.8 m. The Cherenkov radiationemitted in the water of the SD station is captured bythree 9-inch PMTs floating on top of the water container,positioned 1.2 m away from the cylinder axis with 120 ◦ separation in azimuth. The PMTs have been producedby Photonis; the particular PMT model XP1805-PA1 usedby the Pierre Auger Observatory differs from their stockmodel XP1805 [11] only in the additional output from thelast dynode. Together with the usual anode output these Preprint submitted to Nuclear Instruments and Methods in Physics Research A October 22, 2018 a r X i v : . [ phy s i c s . i n s - d e t ] M a r reated byAlexandre Creusot vacuumglassmultipliermirror photocathode neck Figure 1: Schema of the advanced simulation model. The main com-ponents of the simulation set-up are the window glass , photocathode , mirror , effective absorption by the neck , and the multiplier structure. two are used for the monitoring of the dynode–anoderatio and the low and high gain signal acquisition [12].The geometry of the PMT can be well approximatedby an oblate spheroid with two equal semi-principal axesof 108.8 mm length and a third axis 79.4 mm long (outerdimensions). The thickness of the glass window is 2.5 mm.The glass is composed of borosilicate (80% SiO , 13%B O , and 7% Na O) with an index of refraction of 1.47and a strong increase of absorption for wavelengths be-low ∼
300 nm. The photocathode consists of a thin ∼
20 nm layer of bi-alkali metal (KCsSb) and has a wave-length-dependent complex index of refraction [13].Based on this kind of geometry and material speci-fications of the PMT, we developed a simulation modelthat retains the basic geometry properties of the PMTwhile simulating in great detail the physical processesleading to the photoelectron emissions.
3. Advanced simulation model
The geometry set-up of the advanced simulation mod-el is shown in Fig. 1. The whole shape is approximatedwith a full oblate spheroid with a mirrored back wallreaching 30 mm below the center. The reflectivity of the mirror is set to 97% and only ideal specular reflection isimplemented. As in the case of the real PMTs, the photo-cathode on the inner side of the glass is separated fromthe mirror coating by a transparent gap of 5 mm.
The loss of photons in the extended neck leading tothe PMT base is simulated by the 100% absorbing patchwith a radius of 40 mm placed at the top of the PMT.The multiplier structure (dynode stack) reaches well intothe center of the PMT and is in this particular PMT en-closed in a metallic shield case of cylindrical shape. Inthe photon-tracking experiments we have found that the [nm] l
200 300 400 500 600 700 800 g l ass R e [ n ] , - I m [ n ] , T Figure 2: Real (full line) and negative imaginary (dashed line) parts ofthe complex refraction index of the photocathode as a function of thewavelength. The triangles are experimental data points extracted from[13]. The crosses are extrapolated values. The dotted line representsthe Fermi function used to model the transmission factor of the PMTglass. proper inclusion of this multiplier volume can greatly in-fluence the number of reflected photons (see Fig. 4 for aclear demonstration on how the parallel beam of lightgets focused into the multiplier volume). The multi-plier structure is thus simulated with a cylinder of height60 mm and radius 16 mm. It is made of a copper-likemetal and its effective reflectivity is set to 30% (for sim-plicity specular reflection only).
The material of the glass window limits the spectralsensitivity in the short wavelength region. The borosili-cate glass has a cut-off wavelength of 270 nm (the pointwhere it decreases below 10%). We have modeled thisproperty with a simple transmission factor T glass that isimposed upon the entering photons, T glass = exp [ − d / L ( λ )] = F ( λ ; λ m , ∆ λ ) , (1)where d is the thickness and L ( λ ) is the absorption lengthof the PMT glass. The final transmission factor is mod-eled with a Fermi-function dependence on the wavelength,F ( λ ; λ t , ∆ λ ) = ( exp [( λ t − λ ) / ∆ λ ] + ) with λ t =
285 nmfor the transition wavelength and ∆ λ = The sensitivity of the different types of photocathodesis restricted by the photo-emission threshold for the longwavelengths and can vary with thickness at the shortwavelengths. The bi-alkali component KCsSb is practi-cally the universal choice of photocathode material forCherenkov light applications, and although it has beenknown for almost five decades, the availability of reli-able measurements of index of refraction has been rather2carce. In our simulation of the angular and wavelengthdependencies of the quantum efficiency we have usedthe data points from the experimental compilation of thecomplex index of refraction from [13], where the opticalproperties of this bi-alkali material of similar thicknesshave been reviewed for the purposes of the solar neu-trino detector SNO. Due to the limited number of dif-ferent wavelength measurements we have used a simplelinear interpolation of the complex index of refractionwith wavelength extension along the lines of newer ex-perimental results from [15] (see Fig. 2).
When the photon reaches the thin layer of the photo-cathode it can either get reflected with probability R , itcan get transmitted across the layer with effective proba-bility T , or it can get absorbed by the photocathode mate-rial with probability A = − R − T . If the photon energyin the latter case is larger than the needed exit work, theelectron can leave the photocathode. The correspondingquantum efficiency q pc of the photocathode surface canthus be written as q pc ( λ , θ ) = A ( λ , θ ) p conv , (2)where A ( λ , θ ) , the absorption coefficient, depends on in-cidence angle θ and wavelength λ , and p conv is the con-version factor. The conversion factor p conv does not de-pend on the angle of incidence [16, 17] but in principle itcan still be wavelength dependent [18]. Due to the rela-tively narrow window of relevant wavelengths imposedby the water absorption (see Fig. 3) we approximate theconversion factor p conv for the purposes of our simula-tions with a constant. In section 3.7, we determine its ef-fective value from the separate simulation of the quantumefficiency experiment , along the lines of the experimentsusually performed by the PMT manufacturers. In thenext section we will derive the second missing parame-ter A ( λ , θ ) .At this point it is worth to mention that the quantumefficiency q pc from Eq. (2) is the quantum efficiency of the photocathode surface and is only indirectly related to theoverall quantum efficiency of the PMT, as found in thespecifications of the manufacturers. The main differencecomes from the fact that PMTs are designed to increaselight collection through reflections on the mirror backwall. This increase is to some degree contained in thePMT specification. Nevertheless, its magnitude dependssubstantially on the particular way the quoted efficiencyis obtained experimentally. Therefore, the quoted effi-ciency specifications of the PMT always require properinterpretation when used for a photocathode simulation.This fact is in general relevant also for other physics ex-periments involving PMTs. Studies on PMTs usually follow [19] to derive the an-gular and wavelength dependence of the optical proper- ties. Here, we follow the more concise derivation from[20] that is specifically dedicated to thin film treatment.Furthermore, we adopt from [20] the sign convention forthe imaginary part of the index of refraction (see Fig. 2).Although our PMT has only one layer of thin-filmphotocathode, we will for the sake of clarity derive Fres-nel equations for a stack of m thin-film surfaces with cor-responding thicknesses d j and indexes of reflection n j .The indexes n j vary with different wavelengths but inthe expressions below we do not explicitly notate the de-pendence on λ for the sake of clarity. This stack of thinfilms is on the in-going and out-going side surroundedby two semi-infinite substrates with indexes n in and n out ,respectively.The angles of refraction in the consequent layers canall be obtained from the incidence angle θ in from Snell’slaw n j cos θ j = (cid:113) n j − n sin θ in . (3)This expression should be used for all m layers j = m even when it produces a complex cosine both in caseof a complex index n j or a total internal reflection n j < n in sin θ in . Phase change in the layer j is denoted by δ j = π d j λ n j cos θ j . (4)In the case of thin layers, the transmitted and reflectedlight combine coherently and the vector of normalizedemergent fields f out can be expressed as in [20] with char-acteristic matrices of m stacked layers as f out = (cid:20) f E f H (cid:21) = (cid:32) m ∏ j (cid:20) cos δ j i sin δ j / η j i η j sin δ j cos δ j (cid:21)(cid:33) (cid:20) η out (cid:21) , (5)where f E and f H are normalized electrical and magneticfields, respectively, and η j , η out are the tilted optical ad-mittances. Using short-hand notation for scalar products t = f out · (cid:20) η in (cid:21) and r = f out · (cid:20) η in − (cid:21) / t , (6)the reflectance is expressed as R = r r ∗ = | r | = | f E | η − [ f E f ∗ H ] η in + | f H | | t | (7)and the transmittance as T = η in Re [ η out ] | t | . (8)Absorptance follows from A = η in Re [ f E f ∗ H − η out ] | t | . (9)The three quantities obey the relation R + T + A = η in is always real, η j can be complex due3 igure 3: Wavelength dependency of the water absorption length L andthe reflectivity R of the container walls. Data taken from [21] and [22],respectively. to absorption, and η j , η out can become complex whenthe incidence angle θ in increases above the angle of totalinternal reflection θ tot j defined by sin θ tot j = n j / n in .The equations above have to be considered for twopolarization cases. Using the convention where p-polariz-ation implies a magnetic field component parallel to theinterface boundary (TM wave) and s-polarization impliesa parallel electric component (TE wave), two variants ofthe admittance emerge, η p j = n j / cos θ j and η s j = n j cos θ j , (10)where cos θ j is again obtained from Eq. (3). The samepolarization cases should also be used on η in and η out .The upper expressions in (7), (8), and (9) thus have to beseparately evaluated for the p- and s-polarization; hencethe obtained reflectance, transmittance, and absorptancechange accordingly into R p,s , T p,s , and A p,s . For un-polarized light we can define average quantities R = ( R p + R s ) , T = ( T p + T s ) , and A = ( A p + A s ) which,as before, satisfy R + T + A = ∼
30 ns completely randomized, enable us touse the above polarization-averaged expressions for re-flectance, transmittance, and absorptance.
The simulation of the PMT has been implementedwithin the Auger Offline software framework [23] wherethe water-Cherenkov detector simulation is implementedwith the G eant
Figure 4: Simulation of the Photonis quantum efficiency experiment forestimation of the conversion factor. Absolutely calibrated parallel beamof light is sent to the front of the PMT (photocathode is at bottom of theoval). Due to the reflection on the ellipsoidal mirror at the back of thePMT, the light gets approximately focused at the position of the mul-tiplier (caustics at the sides of the multiplier). The inclusion/exclusionof the multiplier volume can have great effect on the amount of thelight returned to the photocathode surface and thus has to be properlysimulated. In this figure the multiplier (c.f. white patch at top cen-ter) is obscuring most of the returning light in this configuration of theincoming beam. the container walls have been assigned to the G eant eant
4. From this pointon, the photons were tracked by our simulation model,based on the considerations made in the previous sec-tions. Using the index of refraction parametrization fromFig. 2 for n , the individual photons were reflected fromthe photocathode with the probability following from the(polarization-averaged) reflectance in Eq. (7), or transmit-ted according to Eq. (8) into the vacuum ( n out =
1) of thePMT. The simulation proceeded with the photon track-ing inside the PMT which can, upon successful reflec-tions on the inner structure, produce another crossing ofthe photocathode (this time with the exchanged valuesfor the “in” and “out” variables in the upper equations)and reentry into the water. In case of an absorption of thephoton upon the crossing of the photocathode, the asso-ciated photoelectron is produced according to the quan-tum efficiency in Eq. (2).
For the wavelength of 375 nm, Photonis [11] quotesa value of quantum efficiency of the PMT as q pmt = cos 2-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 100.10.20.30.40.50.60.70.80.91 inout Figure 5: Normalized distribution of photons arriving to the photo-cathode as a function of cos 2 θ where θ is the incidence angle, for twopopulations, incoming (i.e. entering the PMT, full line) and outgoing(i.e. exiting the PMT, dashed line) photons. Due to the increased re-flectivity of the water–glass interface a depletion from the isotropic dis-tribution is observed for large incidence angles (cos ( × ◦ ) = − ( × ◦ ) =
1) are suppressed due to the reflection and ab-sorption on the inner structure of the PMT. Note that the distributionfor isotropic arrivals as a function of cos 2 θ is constant and equals to1/2. tron flux Φ pe . Their definition of PMT quantum effi-ciency is then q pmt = Φ pe / Φ where Φ is the incidentphoton flux from the absolutely calibrated beam.We have reproduced this exact experimental setupand the beam geometry in our PMT simulation (see Fig. 4)in order to “reverse engineer” the PMT quantum effi-ciency q pmt into the photocathode quantum efficiency q pc . While using the surface quantum efficiency expres-sion from Eq. (2) we simulated a large number N of the375 nm photons in the incident beam and recorded thenumber of released photoelectrons N pe . To reproducethe quoted PMT quantum efficiency q pmt = N pe / N , the conversion factor inEq. (2) had to be set to p conv = λ =
375 nm slightly more than 40% of the absorbed pho-tons in this geometry get converted to photoelectrons.
Using the obtained conversion factor p conv = A ( λ , θ ) is obtained form the Eq. (9) and the photoelec-tron is released with probability A ( λ , θ ) p conv , as givenby Eq. (2). We can derive the overall, incidence-angle av-eraged quantum efficiency of our PMT simulation modelby dividing the number of released photoelectrons N pe by the number of photons N reaching the photocathode [nm] l
300 350 400 450 500 550 600 qu a n t u m e ff i c i e n cy l q( ) l ( eff q Figure 6: Quantum efficiency q (full line) defined as the probabilityfor a photon to free an electron at the photocathode crossing. Thedashed line corresponds to the effective quantum efficiency q eff , i.e.the probability for an incoming photon to eventually free an electron.The effective quantum efficiency is used in the implementation of thesimplified PMT model. (from any side) for the different wavelengths, q ( λ ) = N pe ( λ ) N ( λ ) . (11)The resulting quantum efficiency is shown in Fig. 6 (fullline). This quantum efficiency, obtained indirectly fromthe refractive index parametrization in Fig. 2, reproduceswell the known features of other experimental data [15]and the specification of the PMT manufacturer [11]. Fur-thermore, it gives the correct estimates of the averagenumber of photoelectrons released by the traversing muonas measured in scintillator-triggered experiments withAuger SD stations [2, 3].
4. Relevance to other astrophysics experiments
There are several important points that can be ex-tracted from the previous sections. For optical inter-faces, the G eant eant
4. The ab-sorption increases up to 50% for incidence angles largerthan the angle of total internal reflection for the glass–vacuum interface. No such increase is observed for thereverse, vacuum–glass transition of the photons. Further-more, the quantum efficiency obtained from the manu-facturer’s calibration setup already contains a nontrivialfraction of the efficiency increase due to the reflectivity5 [nm] l
300 350 400 450 500 550 600 i n c o m i ng pho t on s · Figure 7: Photon spectrum reaching the photocathode for 10 000 in-jected vertical-centered muons. The main features of the curve aredue to the shape of the Cherenkov emission spectrum, the wavelengthdependency of the water absorption length (hump at ∼
420 nm), thewavelength dependence of the reflectivity of the inner walls, and thePMT glass transparency (cut-off at wavelengths <
280 nm). of the inner structure of the PMT. The “intrinsic” quan-tum efficiency of the photocathode layer thus has to bereverse-engineered by a procedure similar to the one de-scribed in Section 3.7. Clearly, both effects are most rel-evant in cases of experiments with a distribution of in-coming photons substantially different from those char-acteristic of manufacturer’s experimental setup.This may be of relevance to experiments employingsimilarly large PMTs in various Cherenkov detectors, likeAmanda, IceCube, and IceTop [25, 26, 27], Kamiokande[28], NESTOR, NEMO, and Antares [29, 30, 31], Mini-BooNE [32], Xenon [33], Baikal and Tunka [34], and north-ern Auger Observatory [35], just to name a few.
5. Simplified simulation model
In a simulation, a vertical-centered muon injected intothe water of the SD station has a 1.2 m tracklength andtypically releases ∼
67 500 Cherenkov photons in the en-ergy range between 1.5 and 5 eV (250 to 830 nm in wave-length; see Fig. 7 for spectrum). Additional ∼
10% ofphotons get produced by the secondary delta rays, in to-tal giving ∼
73 100 photons per muon. All these photonsrequire a complete tracking inside the water container byG eant
4. The photons on their way undergo the absorp-tion in the water medium, get reflected and absorbed bythe container walls. On average, only ∼ [nm] l
300 350 400 450 500 550 600 ou t go i ng pho t on r a t i o Figure 8: Ratio of outgoing vs. incoming photons as counted at thePMT photocathode. the time-consuming simulation of these photons that donot contribute to the PMT signal, a simplified simulationhas been devised. The goal is to estimate the effectiveprobability for an emerging photon to contribute to thesignal and eliminate idle photons from the simulation.The corresponding statistical thinning can thus be appliedalready at the time of their production.
After several reflections on the walls, the swarm ofphotons behaves similarly to a photon gas. As can be al-ready seen from the incidence-angle distribution of pho-tons on the photocathode (see Fig. 5), the distribution ofarrival directions is close to isotropic. Instead of the de-tailed simulation of photon tracks inside the advancedPMT model we can reduce the complexity of the pro-cedure by introducing the simplified model of the PMT.The quantum efficiency in Eq. (11) stemming from Fres-nel equations of the advanced PMT model is replacedwith an effective quantum efficiency q eff that describes theprobability for an incoming photon to produce a photo-electron, q eff ( λ ) = N pe ( λ ) N in ( λ ) . (12)The q eff is obtained from the advanced model by nor-malizing the statistics of the released photoelectrons bythe number of incoming photons. The actual wavelengthdependence of q eff ( λ ) is shown in Fig. 6 (dashed line).In the advanced model only 4.5% of the incomingphotons return to the water medium, i.e. only 0.08% ofall produced photons. In Fig. 8, a wavelength-resolvedfraction of outgoing vs. incoming photons is shown. Thefraction is as low as 2% for short wavelengths and doesnot exceed 22% for long wavelengths.Based on these facts we have developed an imple-mentation of the simplified PMT model that uses only6 reated byAlexandre Creusot inert volume active volume Figure 9: Representation of the simplified model of PMT. [nm] l
300 350 400 450 500 550 600 c o rr ec t i on f ac t o r Figure 10: Correction factor q eff ( λ ) / q ( λ ) as a function of the wave-length. For short wavelengths the correction is of the order of 13% andgoes up to 35% for long wavelengths. a rudimentary geometry as shown in Fig. 9. In this sim-plified model the photons are detected on contact withthe outer surface and are consequentially removed fromthe simulation. No tracking of the photons on the in-ner structure of the PMT is performed. The chance ofproducing a photoelectron on the second crossing of thephotocathode in the advanced model is accounted for inthe simplified model by the usage of the effective quan-tum efficiency from Eq. (12). The relative ratio of thequantum efficiencies from Eqs. (12) and (11) is shown inFig. 10. The ratio is mostly close to ∼
10% increase butthen gradually climbs to ∼
35% for long wavelengths.
Similar to the well established method in extensiveair-shower simulations [36, 37], the concept of effectivequantum efficiency enables us to implement efficient sta-tistical thinning of the simulated photons. At the mo-ment of production, out of N pr ( λ ) photons with wave-length λ only q eff ( λ ) N pr ( λ ) photons with weight 1/ q eff ( λ ) undergo further simulation. As a result, such thinnedphotons upon reaching the photocathode have to pro-duce a photoelectron with probability 1. The averagequantum efficiency (in photon energy scale where Che-renkov spectrum is flat) is ¯ q eff = ∼
5. If we take into time [ns]0 50 100 150 200 250 300 350 ave r a g e m uon s i gn a l [ a . u . ] Figure 11: Comparison of the time dependence of the muon signal inan SD station for the advanced PMT model (full line) and the simplifiedversion (dashed line). The main feature is an approximate exponentialtime dependence with a decay time τ ≈
65 ns. The two models agreewell with respect to the small details of the early signal and the latterdecay part. A small discrepancy, limited to less than 5%, is observedonly in the late tail. Note that the oscillating structure for time <
30 nsis due to the nonuniform arrival of photons after the first few reflectionswhen photons can not be treated yet as homogeneous and isotropicphoton gas. account also the losses due to the PMT collection effi-ciency, this number increases to ∼ τ ≈
65 ns. Except in the tailof relatively large times, t (cid:29) τ , where the discrepancyis limited to within 5%, the simplified model reproduceswell the details of the muon signal from the advancedPMT model, especially the overall exponential decay andan oscillatory behavior for t <
20 ns. It is worth mention-ing here that the signal from the PMT in an SD station issampled with a 40 MHz (25 ns) FADC, i.e. all these detailswill lie within one sampling bin.
6. Summary
We have implemented an advanced model for thesimulation of the PMT response in a water-Cherenkovstation of the surface detector. The model is based on thecorrect thin-film treatment of the photocathode opticalprocesses with complex index of refraction. The modelalso includes tracking of the photons on the inner mirrorsurfaces and multiplier structure.To relate the obtained quantum efficiency of the pho-tocathode to quoted efficiency from the PMT specifica-tion we have in simulation reproduced the experimentalset-up of the measurement done by the manufacturer.7his properly accounts for the increase in photon collect-ing due to the reflections on mirrored surfaces and innerstructure of the PMT, and gives the average value of theconversion factor.Since a relatively large number of photons is involvedin the creation of the muon signal, certain details of thesimulation, like the dependence of the absorption on theincidence angles and the exact tracking of the photons onthe inner structure, are washed out and can be treated ina phenomenological way.To reduce the simulation time as well as the complex-ity of the problem we have implemented another, sim-plified model of the PMT. Results for the average pho-toelectron release probabilities for different wavelengthsin the advanced PMT model were in turn used to cali-brate the effective quantum efficiency of the simplifiedmodel. Results from both models agree well with theexperimentally obtained number of produced photoelec-trons per muon injection. Finally, we have shown thatthe time dependence of the muon signal is almost notaffected by the simplification in the simulation strategy.
Acknowledgments
This work was supported by the Slovenian ResearchAgency and in part by the Ad futura Programs of theSlovene Human Resources and Scholarship Fund. Au-thors wish to thank Thomas Paul, Ralph Engel, SlavicaKochovska, and the reviewer for useful suggestions, andthe many colleagues from the Pierre Auger collaborationthat were involved in SD detector calibration and simu-lation.
References