Charged-particle detection efficiencies of close-packed CsI arrays
CCharged-particle detection e ffi ciencies of close-packed CsI arrays P. Morfouace , W. G. Lynch , M. B. Tsang National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824,USA
Abstract
Detector e ffi ciency determination is essential to correct the measured yields and extract reliable cross sections of par-ticles emitted in nuclear reactions. We investigate the e ffi ciencies for measuring the full energies of light chargedparticle in arrays of CsI crystals employed in particle detection arrays such as HiRA, LASSA and MUST2. We per-form these simulations with a GEANT4 Monte Carlo transport code implemented in the NPTool framework. BothCoulomb multiple scattering and nuclear reactions within the crystal can significantly reduce the e ffi ciency of de-tecting the full energy of high energy particles. The calculated e ffi ciencies decrease exponentially as a function ofthe range of the particle and are quite similar for both the hydrogen ( p , d , t ) and helium ( He, α ) isotopes. The useof a close-packed array introduces significant position dependent e ffi ciency losses at the interior boundaries betweencrystals that need to be considered in the design of an array and in the e ffi ciency corrections of measured energyspectra. Keywords:
1. Introduction
Silicon strip and pixel detectors are widely used to provide accurate position information regarding charged parti-cles emitted in nuclear and particle physics experiments. To measure energetic light charged particles, such as protonsor alphas with E / A >
18 MeV, which will penetrate through the thickest commercially available Si strip detectors,Si detectors are often backed by scintillators with thickness between 1 to 10 cm. Of the various scintillators suitablefor detecting charged particles, Thallium doped CsI crystals have the virtue of scintillating at wavelengths that can bemeasured with silicon photodiodes. With energy resolutions for charged particle that are typically better than 1%, theyare easily machined and are only mildly hygroscopic in air. Their cost also make them a popular choice to constructhighly-e ffi cient arrays [1, 5, 4]. Fig. 1 shows the maximum measurable energies of p , d , t , He and α particles as afunction of the CsI thickness. As the electronic stopping power decreases inversely with the energy of the detectedparticle, the scintillator thickness required to stop a particle increases rapidly with energy. While longer crystals allowdetection of higher energy particles, the e ffi ciency for the measurement of the full energies of these particles decreaseswith energy due to scattering and reaction losses within the scintillator.This paper focuses on calculating the loss in detection e ffi ciencies of light particles as functions of the particletype and the detector thickness. As the energy of a particle increases, its range in the CsI crystal increases and theprobability that the particle will undergo a nuclear reaction becomes more significant [10]. In addition, there can belosses due to Coulomb multiple-scattering that occurs whenever the scattering deflects the charged particle out of theCsI crystal before it deposits its total energy. Such multiple scattering e ff ects have been neglected in some previouscalculations of reaction losses [10] where it may have been justified by the geometry of the crystal. We show, however,that this can be important whenever the charged particles pass su ffi ciently close to the inner boundaries of the crystalsin close-packed arrays.
2. Simulation
The method described in this paper can be applied to any Si-CsI detection system. To provide concrete examples,we have performed the simulations on individual telescopes of the Large Area Silicon Array (LASSA) [1, 2], the
Preprint submitted to Elsevier December 8, 2016 a r X i v : . [ phy s i c s . i n s - d e t ] D ec sI Range (cm) E ( M e V ) pdtHe a Figure 1: Energy range for p (blue line, closed circle), d (dashed line, open circle), t (red line, closed diamond), He (dashed line, open star) and α (solid line, closed star) as a function of the CsI thickness. High Resolution Array [5] and the MUr `a STrip (MUST2) [4]. The configuration for each telescope used in this workconsists of a Double Sided Silicon Detector (DSSD) backed by a close-packed array of 4, 4 and 16 CsI scintillatorcrystals corresponding to detection in the HiRA, LASSA or MUST2 arrays respectively. These telescopes were chosenbecause the authors are familiar with these devices and because they have geometries that are similar to other existingor planned arrays where we expect the present calculations can be of assistance in estimating the magnitude of suche ff ects. We note however that most measurements with HiRA have utilized shorter 4 cm CsI crystals (HiRA). Therecently completed upgraded HiRA array referred as HiRA10 in this paper has 10 cm crystals. Relevant details of thetelescopes used in the simulations are listed in Table 1.A typical schematic drawing of a close-packed 2 × s × s mm backed with 4 identical crystals.Each crystal is L mm long, with a front width of a mm and a back width of b mm. In order to minimize the multiplescattering e ff ects on the outer edges of the crystal, the front width of two crystals is larger than the active area of theSilicon detector as illustrated in Fig. 2 with 2 e + s = a . The larger edges of the crystals allows the particles to godeeper into the crystal while reducing the e ffi ciency loss near the outer edges of the crystals due to multiple scattering.Some details for the geometry of the HiRA10 and LASSA arrays are listed in Table. 2.For these calculations, we have adopted the NPTool framework [14] that takes the full advantages of both ROOTanalysis framework [15] and GEANT4 simulation toolkits [16]. The NPTool framework utilizes GEANT4 version10.01, a Monte Carlo particle transport model that include the electromagnetic processes or hadronic processes orboth. These processes can occur when the particles travel through the detector materials. In addition, the calculationsfor the MUST2 array utilize C ++ classes that were developed within the NPTool framework [14]. We developedanalogous new classes for the LASSA and HiRA10 detectors so that all the di ff erent simulations and analysis aredone consistently. These calculations take into account the known intrinsic energy resolution of di ff erent elements ofthe telescopes.In this work we will focus on the detection of light charged particles with mass number 1 < A <
4, i.e. protonsto α particles. Similar calculations can be performed for any kind of charged particle. In the following, the e ff ects ofCoulomb multiple scattering are explored separately in Section 2.1 and in conjunction with reaction losses in section2.2. In this subsection, we focus on the multiple-scattering that a particle experiences when going through a givenmaterial. The single and multiple Coulomb scattering influences more strongly lighter charged particles such asprotons that have smaller momenta. To simulate the process we have used the standard electromagnetic package2 b a bLse 3D view Side view
Figure 2: Schematic drawing of a 2 × (deg) c.m. θ ( m b ) Ω / d σ d (deg) c.m. θ
10 20 30 40 50 ( m b ) Ω / d σ d a) b) Figure 3: (a) Angular cross section of the scattered proton for the
Cs target at 30 MeV and (b) for the
Cs target at 80 MeV proton incidentenergy. In both figures the points correspond to the GEANT4 simulations while the solid lines correspond to the DWUCK calculations. µ m thickness (cm) from the targetto the silicon detector (cm)MUST2 [4] 300 4 ≈ ≈ ≈ Table 1: Relevant information for various telescopes.
Detector a (mm) b (mm) s (mm) L (mm) e (mm)LASSA 26.5 33.8 50 60 1.5HiRA10 34.9 44.6 64 100 2.9 Table 2: Details of the geometry of the crystals. See text and Fig. 2 for parameter definition. “option4” in GEANT4. In this package the finite size of the nuclei in the detector material and the penetration of thedetected particle through the Coulomb barrier of such a nucleus is modeled by the Born approximation. The packagesets the nuclear form factor to zero when the scattering angle of the particle from that nucleus is greater than the angle θ max , which is defined to be (chapter 6 of ref. [17]):sin (cid:16) θ max (cid:17) = kR , (1)where k is the wave number of the incident particle and R is the radius of the target nucleus. Within GEANT4, thisform factor modification replaces an optical model calculation of scattering from the Coulomb and nuclear opticalpotential, whose imaginary term models the loss of flux into other channels. It is therefore important to check that thisapproximation does not overestimate the elastic scattering at large angles where the optical potential strongly modifiesthe elastic scattering cross section.To assess the accuracy of this approximation, we simulated the scattering of both 30 MeV and 80 MeV protonparticles in a 5 µ m thick Cs target. We compare the results from our GEANT4 simulations to optical-model cal-culations in Fig. 3. Here, the GEANT4 simulations for 10 incident particles are shown by the points with statisticalerror bars. Corresponding optical model calculations, shown by the solid lines, are performed with the elastic scatter-ing predictions of the DWUCK code [18] using the CH89 global optical potential [19], which is well adapted for theenergy and the mass region we are considering. While the GEANT4 yields do not have di ff raction minima that matchthe minima in the calculated di ff erential cross sections, the average trend with angle is rather well matched suggestingthat the GEANT4 simulation provides a reasonable approximation of the scattering of particles to large angles. In this subsection, we explain how the reaction losses are estimated. We compare the di ff erent parameterizationsavailable in GEANT4 to experimental data, when available, in order to make a suitable choice for each of the particleswe are considering and to estimate the uncertainty of the e ffi ciency calculations. The G4BinaryLightIonReactionModeloption in GEANT4 has been used to evaluate the predicted e ffi ciency losses for di ff erent cross section parameteriza-tions. Two parameterizations are from Shen [20], and Tripathi [21], which provide estimates of the reaction crosssections at energies ranging from few AMeV to few AGeV. Both models apply a correction to reduce the reactioncross sections at low incident energies near the Coulomb barrier. At much higher energies, the cross section is re-duced again, reflecting the energy dependence of the average nucleon-nucleon cross section [23]. To describe protons,we have also used a third parametrization by Grichine [22], which is based on a simplified Glauber approximation.We are mainly interested in the reaction cross section of light charged particles with CsI. However since thereare no suitable experimental reaction cross sections for light charged particles with either Cs or I, we compare theGEANT4 prediction to Sn target, for which some experimental light particle data exist. As
Sn has a mass similarto
Cs, the cross section prediction for Cs is very similar to the one for Sn. However, the di ff erence in Z between Snand Cs increases the predicted cross sections by about 10% relative to Sn. Fig. 4 shows the predicted cross sections4
00 200 300 400 ( b a r n ) R σ ( b a r n ) R σ (MeV) t E
100 200 300 4000.511.52 (MeV) lab E
100 200 300 400 ( b a r n ) R σ (MeV) lab E
100 200 300 4000.511.52
TripathiGrichineShenData
100 200 300 400E lab (MeV) p+ Sn d+
Sn t+ Sn He+ Sn α + Sn Perey“d-fit” a) b) c)d) e)
Figure 4: Reaction cross sections for (a) p , (b) d , (c) t , (d) He and (e) α beams on Sn target as a function of the beam energy using di ff erentparametrization in GEANT4. The black inverted triangles correspond to experimental data from Ref. [25] for protons, Ref. [26] for deuterons andRef. [21] for α particles. The star symbols correspond to the total cross section calculated using Perey and Perey optical parameters from Ref. [29]. (MeV) d E ( b a r n ) R σ (MeV) He E
100 200 300 40000.51 (MeV) α E
100 200 300 40000.51 d+ C He+ C α + Cb) c)a)
Figure 5: Reaction cross sections for (a) d , (b) He and (c) α on C as a function of the particle energy using di ff erent parametrization in GEANT4.The inverted triangles correspond to experimental data coming from Ref. [27]. of these three models (Shen, Tripathi and Grichine) for hydrogen and helium beams incident on a Tin target, as afunction of incident beam energy. The solid inverted triangles show the experimental data for protons [25], deuterons[26], and α particles [21]. Except for proton and He projectiles, cross sections provided by the Shen [20] and Tripathi[21] parameterizations are similar. Unlike the others, the Grichine parameterization [22], based on Glauber model,is constant for d , t , He and α particles. Only the proton cross section displays a significant energy dependence. Thepredicted Grichine cross section for protons is comparable to the proton data, but its predictions are low and unrealisticfor the other charged particles. The Grichine prediction will therefore only be used for the e ffi ciency calculationinvolving protons. We also tested the intra-nuclear cascade (INCL) model using G4HadronPhysicsINCLXX andG4IonINCLXXPhysics [24] and found that the results are similar to that of Shen and Tripathi.Since the experimental total cross sections on tin are limited, we also compare in Fig. 5 the total cross sections for d , He and α beams on C from [27]. We did not find suitable experimental data for tritons on C. Once again thepredicted cross sections for d , He and α particles provided by the Grichine option are unrealistic. Here, the Tripathiparameterization provides a better agreement with the α particle data. The experimental deuteron and He data arebetter described at low energies by Shen parameterization than by the Tripathi one, but at higher energies the Tripathiparameterization provides a somewhat better description of the He and α particle data. At low energies ( <
50 MeV)the Shen cross section underestimates the measured deuteron total cross section by about 20% and the disagreementwith the Tripathi parameterization is even larger. In the next subsection, however, we will show that the contributionfrom the di ff erent reaction cross section parameterizations to the e ffi ciency from this energy range are insignificant(cf Fig. 7).
3. Results
Unless otherwise indicated as in the case of calculations using Charity parameterization, one should assume thatCoulomb multiple scattering e ff ects are included in calculations shown in this section.6 dt He α pt d He α a) b) Figure 6: (a) Particle identification spectrum from the ∆ E − E method with a zoom as an inset. (b) Correlation between the initial simulated energyversus the reconstructed energy. See text for details. ff ects on particle identifications To calculate the e ffi ciency for detection of the full energy of the particle, we simulate the interaction of the variousparticle species ( p , d , t , He or α ) in CsI by assuming an initial flat energy distribution between 0 < E i < E max MeV,where E max is the energy for which the range in CsI equals the thickness of the CsI crystal. Particles are assumed tobe emitted from a target located at the design distances of 35 cm, 20 cm and 17 cm in front of the HiRA, LASSAand MUST2 telescopes, respectively. We reconstruct the total kinetic energy of each particle by adding the calculatedenergy lost in the DSSD Si detector ( ∆ E ) and the energy detected in the CsI crystals ( E ).The various particle species can be distinguished via a Particle Identification Spectrum (PID) constructed byplotting calculated values in a two-dimensional ∆ E vs. E spectrum. Such a spectrum is shown in Fig. 6 for simulationsinvolving the HiRA10 telescope. The inset in the figure depicts an expanded view at low values of ∆ E that allow oneto view the energy loss of energetic particles that stop completely in the CsI crystal. Clearly, one can easily distinguishthe particle species.The blue haze outside the particle lines corresponds to events when a nuclear reaction occurs in the crystal a ff ectingthe measured energy or when the particle scatters out of the side of the crystal. Both e ff ects result in the mis-identification of the particle. This incomplete energy collection is clearly illustrated in the right panel of Fig. 6, wherethe initial energy E i is plotted against the reconstructed energy E det . The 45 ◦ line corresponds to the well reconstructedevents with good PID while all the events below the line correspond to the blue haze in the PID plot where the energyis not reconstructed correctly. In the following we label a particle as fully detected when | E det − E i | < E det = E S i + E CsI . This range is wide enough to prevent particles on tails of resolution functions from being mislabeledas an out-scattered or reaction event. ffi ciency as a function of energy Fig. 7 shows the e ffi ciencies for well detected p , d , t , He and α particles in the HiRA10 crystals up to maximumenergies of 198, 263, 312, 708 and 793 MeV, respectively. These e ffi ciencies are also valid for the original HiRA CsIcrystals up to the maximum energies of 115, 155, 183, 410 and 462 MeV for p , d , t , He and α particles, respectively,corresponding to a range of 4 cm in CsI. Clearly, the calculated detection e ffi ciencies for all particle species decreasewith incident energy reflecting an increased average number of interactions as the particles penetrate further into thecrystals.The open triangles in each panel show the e ffi ciency losses that are solely due to the e ff ects of multiple scatteringout of the crystal. Protons have significantly larger e ffi ciency losses due to multiple scattering than do the otherparticle species. This reflects the smaller momenta of protons for a given kinetic energy, which are more comparableto the probable momentum transfers resulting from Coulomb interactions with the Cs and I atoms in the CsI crystals.7 (MeV) lab E E ff i c i e n cy (MeV) lab E E ff i c i e n cy (MeV) lab E E ff i c i e n cy (MeV) lab E E ff i c i e n cy (MeV) lab E E ff i c i e n cy Multiple ScatteringTripathiCharityGrichineINCLShen proton deuteron triton He α “d-fit” a) b) c)d) e) Figure 7: E ffi ciencies to detect the full energy of the proton (a), deuteron (b), triton (c), He (d) and α particles (e) in the HiRA10 as a function ofthe energy. The open triangle symbols show the e ffi ciency when only the multiple scattering (MSC) is taken into account. The full blue circles, openred squares and black full stars correspond to the e ffi ciency using the Tripathi [21], Shen [20] and INCL parameterization respectively. The blackdashed lines correspond to the determination of the e ffi ciency using the Charity parameterization. For protons, the dotted dashed line corresponds tothe e ffi ciency using the Grishine [22] parameterization. For deuterons, the light grey line corresponds to the result using the “d-fit” parameterizationof the experimental deuteron cross section shown in Fig. 4. ange (cm) E ff i c i e n cy pdtHe He Figure 8: E ffi ciency to detect the full energy of p , d , t , He and α particle in HiRA10 as a function of the range in cm in the CsI crystal using theTripathi parametrization [21]. The other symbols show the lower e ffi ciencies that result from hadronic reaction losses using the Tripathi (solidcircles), Shen (open squares), INCL (solid stars) and Grichine (dotted line) parameterizations for the nuclear crosssection [21, 20, 22]. We note that the results for the Grichine parameterization are only calculated for protons becausethis parameterization severely under-predicts the reaction cross sections of the other species.In Ref [10], Charity et. al calculated e ffi ciency losses due to nuclear reactions using optical model parameteriza-tions of Perey and Perey [29]. We parameterize the nuclear reaction losses of Charity [10] and incorporate them intothe GEANT4 simulation including multiple scattering. These e ffi ciencies are shown as dashed lines in Fig. 7. Detailsof the implementation of the Charity parameterizations in this work are discussed in Appendix A.All the particle e ffi ciencies derived from Shen, Tripahti and INCL agree to within 1%. This is consistent withFig. 4. For protons, the Grichine parameterization provides the lowest e ffi ciency. This can be expected from Fig. 4a),where it over-estimates the measured reaction cross section at low energies. Except in the case of the proton e ffi cien-cies and the e ffi ciencies of other species at low energies, the Charity parametrization provides the lowest e ffi ciencyvalues. This is not surprising because the cross section is assumed to remain high and constant as the energy isincreased, which over estimates the reaction cross section at higher energies (cf Tab. A.3 for Charity cross-sectionvalue).In the case of deuterons, no parameterizations describe the existing reaction cross section data. In order to estimatethe change that could be expected for a cross section that reproduces the reaction cross section at low energies, yetdecreases with energy at higher energies, we have modeled the cross section for deuterons by the grey solid line shownin the deuteron panel in Fig. 4 and labeled as “d-fit”. With this parameterization, we followed the technique used forCharity parameterization detailed in Appendix A. As shown in Fig. 7 for deuterons, the calculated e ffi ciency usingthe “d-fit” cross section gives somewhat lower e ffi ciencies than that of Tripathi and Shen, but the di ff erence is lessthan 2%. This comparison shows the consistency of the calculations using di ff erent parameterizations and suggeststhat the e ffi ciency can be determined to within a few percent at the highest energies.We regard the “d-fit” parameterization as a reasonable best estimate of the e ffi ciency for deuterons. For protons,the best estimate lies midway between the Shen and Grichine parameterizations while for the t , He and α particles,our best estimate for the e ffi ciency lies between the calculations for the Shen and Tripathi parameterizations. Basedon the variance of the e ffi ciency calculations, we estimate the uncertainty in the e ffi ciency to be δ = . EE max .Since the e ffi ciency is reduced by scattering from the Cs and I atoms in the crystal, the e ffi ciency should decreaseexponentially as a function of the range in the CsI. This is shown for the Tripathi parameterization in Fig. 8 forHiRA10 configuration. Interestingly, the e ffi ciency is very similar for all the particles when plotted as a function ofthe range. As expected, the e ffi ciency decreases exponentially with range from a value of 100% at low energies toabout 60% at 10 cm. This general exponential decrease illustrates that the detailed dependence of the reaction crosssection does not strongly influence the energy dependence of the e ffi ciency.9 ange (cm) E ff i c i e n cy p (HiRA10 2x2)p (HiRA10 3x3)p (LASSA)p (MUST2) Figure 9: E ffi ciency to detect the full energy of a proton for the HiRA10 2 × × ff erent reaction processes The incoming light charged particle can undergo an inelastic reaction, which is a part of the total reaction crosssection. When a nucleus is excited below the particle emission threshold, it usually de-excites by γ decays. Theemitted γ ray can be absorbed by the crystal. Its energy, or part of it, may then be included in the measured particleenergy depending on where in the crystal that the inelastic scattering occurs and on the photo-peak e ffi ciency of thecrystal for γ rays at that location. With our cut of ± γ de-excitation process is handled through GEANT4 with the G4DecayPhysics and
G4RadioactiveDecayPhysics classes.For an inelastic collision which emits particles, the particle ID for the incident particle will be incorrect and will notbe accounted for in our e ffi ciency determination. However, the typical cross section for inelastic scattering leaving thetarget nucleus in low-lying states is about few millibarns [30], whereas the total reaction cross section is of the orderof a barn. The neglect of a detailed accounting for inelastic processes can be shown to be a small e ff ect by making acomparison to the larger uncertainties in the correction due to the reaction cross section. In the case of the deuteroninduced reactions discussed in section 3.2, the total cross section changes by 25% with the “d-fit” parameterizationcompared to Shen parameterization. However, the e ffi ciency determination changes by less than 2% at high kineticenergies. Keeping this in mind, and considering the small contribution ( ≈ ffi ciency determination. ffi ciencies There are many considerations in designing the scintillation array behind the Si strip detector. In heavy ioncollision experiments and multi-particle resonance decay spectroscopy where charged particle multiplicities are high,it is desirable to construct an array with high granularity to minimize multiple hits in one crystal. In principle, suchmultiple hits render the data in a crystal invalid, and represent an addition source of e ffi ciency loss beyond thatdiscussed above. In all the telescopes we discussed here, the CsI crystals placed behind the Si detectors form a closedpacked geometry; 2 × × × ffi ciency on the actual detectorgeometry, we plotted in Fig. 9 the e ffi ciency of protons as a function of the range using Tripathi parameterizationfor HiRA10 2 × × ff erences in their e ffi ciencies, that are most evident for high energy protons.This reflects the size of the silicon detector and, in turn, the size of the crystals. Four individual LASSA crystals sitbehind a single 50 mm ×
50 mm silicon detector. Consequently, their front faces are smaller at 26.5 × thanare the HiRA10 crystals, which measure 34.9 × on the front surface behind the active area of a 64 mm ×
64 mm silicon detector. We have modeled the e ffi ciency of a hypothetical HiRA10 3 × ×
22 mm front surfaces. Thesesmaller crystals detect energetic protons with a significantly lower detection e ffi ciency than the HiRA10 2 × ffi ciency with regard to budgetary and size constraints. On the other hand, one would like tohave crystals with smaller individual solid angles to minimize coincidence summing, ie. the probability that multipleparticles hit the same crystal. This can lead to e ffi ciency problems when particles scatter out of the crystals.To illustrate this tradeo ff , we simulate the e ffi ciency for detection of mono-energetic protons at 40, 80, 120 and160 MeV with one HiRA10 telescope using a 2 × × ffi ciency as a function of the position where the protonhits the 6.4 cm × ffi ciency is uniform at nearly 100% for most of the surface of the telescope,but it decreases to about 90% near the inner boundaries between crystals reflecting multiple scattering out of the innercrystal boundaries. A comparable reduction of e ffi ciency is not observed at the outer boundaries of the crystals becauseouter edge of the crystal is more than 2 mm outside of the active area defined by the passage of the charged particlesthrough the silicon detector as explained in Fig. 2 and the extra size grows to 12.6 mm at the far end of crystal. Withincreasing proton energies, the e ffi ciency decreases. The decrease is more significant at the inner boundaries betweencrystals with the region of reduced e ffi ciency grows in width, becoming about 5 mm wide and the e ffi ciency decreasesto about 50% at 160 MeVTo minimize the rejection of the events with multiple hits, one can increase the granularity of the array. Thismeans using more crystals to subdivide the active Si detector area more finely, minimizing possible multiple hitsby detecting the particles emitted in close proximity in separate detectors. To illustrate such e ff ects of increasedgranularity on e ffi ciency, we consider a 3 × ffi ciency of such a 3 × ffi ciency near the inner boundaries of the 3 × ffi ciency near theedges of the crystal that is rather similar in width as for the 2 × ffi ciency with energy much worse than forthe 2 × ffi ciency due to multiple scattering depends strongly on the ratio of thedepth that the detected particle penetrates into the crystal divided by the width of the crystals. One way to improvegranularity without worsening the out-scattering ine ffi ciency is to build larger individual detectors and move the arrayfarther away from the target. This however increases the cors of the array.
4. Conclusion
We have simulated the e ffi ciency to detect the full energy of light charged particles in CsI crystals using theHiRA10, LASSA and MUST2 telescopes. In order to correctly determine the e ffi ciency of the experimental setup onehas to carefully determine the e ffi ciency loss due to multiple scattering as well as the nuclear reactions that occur inthe detector. This e ff ect is particularly important when using long CsI crystals ( > ffi ciency drops bymore than 30% (Fig. 8). We evaluate the options available within the GEANT4 environment, find the more accurateoptions and evaluate their accuracy. It is interesting to note that the e ffi ciency decreases almost exponentially as afunction of the range and is rather similar for all the light charged particles.11 a) 40 MeV(c) 120 MeV(e) 40 MeV (f) 160 MeV(b) 80 MeV(d) 160 MeV Figure 10: Front view of one HiRA10 telescopes showing the e ffi ciency to detect the full energy for di ff erent proton energies at 40, 80, 120 and160 MeV for a 2 × ×
12e find that multiple scattering decreases the e ffi ciency significantly, especially near the boundary and for protons.Multiple scattering e ff ects depend strongly on the geometry of a close-packed array. This geometry dependenceindicates that e ffi ciency losses should be an important consideration for the design of a close-packed array of CsIcrystals.The best way to verify the e ffi ciency results presented in the present paper would be to measure them experimen-tally using di ff erent mono-energetic charge particle beams. It would be interesting to perform such measurementsfor long CsI(Tl) crystals such as those for the HiRA10. Such measurements could provide both a direct e ffi ciencymeasurement and a calibration of the energy vs. light output response of the CsI(Tl) crystals to these particles. In thecase of protons, this could be achieved by measuring with protons at beam energies up to the full energy range of thecrystal ( ≈
200 MeV).
Acknowledgments
This work is supported by the NSF under Grant No. PHY-1565546. The authors would like to thank the HiRAgroup at NSCL for the helpful discussion and suggestion.
Appendix A.
The transport of an incident flux of a particular species of charged particles through a CsI scintillator satisfies aBoltzmann equation that is modeled in GEANT4 by Monte Carlo techniques. Coulomb and nuclear elastic scatteringdeflect these particles and degrade their energies, while conserving flux. Nuclear reactions, however, typically changethe charges, masses and energies of particles significantly. This often leads to the mis-identification of the particle,removing the flux of particles whose energies can be measured by stopping them in the detector.We denote P ( X ) as the probability the incident particle passes through a thickness of the detector X withoutreaction: P ( X ) = e − λ ( X ) , (A.1)where λ ( X ) = (cid:90) X ρσ ( E ( x )) dx . (A.2)Here ρ is the density of the material in atoms / cm , x the distance that the particle has penetrated the material and σ ( E )is the energy-dependent reaction cross section for particles species reacting with atoms that compose the scintillatormaterial. In the limit of a constant cross section, P ( X ) becomes P σ ( X ) = e − ρσ X (A.3)and the total e ffi ciency including multiple scattering and reactions (cid:15) tot ( E ) can be written as (cid:15) tot ( E ) = (cid:15) mult ( E ) × P σ ( R ( E )) , (A.4)where R ( E ) is the range of the particle with energy E and (cid:15) mult takes the e ffi ciency loss due to multiple scattering outof the detector into account.Alternatively, one can compute changes in P ( x ) and in the total e ffi ciency as the result of a series of probability losses ∆ P = − ρσ ( E ( x )) P ( x ) ∆ x that occur within a section of the trajectory of length ∆ x using the output of the Geant4 simu-lation. Using this approach, we simulated the di ff erent particles in Geant4 with only the Coulomb multiple scatteringthrough the “option4” as explained in section 2.1. Since the reaction cross section does not change trajectories of non-reacted particles in the simulation, the reaction loss ∆ P is then obtained from an analysis of the multiple scatteringsimulation. After the i th step in the analysis of an event in the simulation we get the energy deposited by the particlein the CsI crystal ∆ E ( E ( x ) , ∆ x ) loss , i and the mean energy E ( x ) of the particle during the step. We then calculate thevalue of the fractional probability ∆ P using the reaction cross section at that energy and survival of the particle in thisevent from the following three steps • We calculate a random number p between 0 and 1, 13 if p > | ∆ P | , we assume there is no nuclear reaction and the event is kept, • if p < | ∆ P | , we assume that a nuclear reaction occurs and the event is terminated.If no nuclear reaction has occurred in the thickness ∆ x traversed during the step, we continue the process using thenew energy of the particle E i + = E i − ∆ E ( E ( x ) , ∆ x ) loss , i until it stopes in the scintillator or scatters out of the crystal.If a reaction occurs in the i th step, then this reduces the number of properly detected particles by one event. In ourwork we used this Monte Carlo approach to calculate the e ffi ciency for deuterons using the “d-fit” parameterizationemploying a step size corresponding to ∆ x = µ m.In Ref. [10], Charity et. al calculated the “fractional loss” for various particles in CsI from nuclear reactionsdefined by FL ( E ) = − P ( R ( E )). Their fractional losses are displayed by the points in Fig. A.11a) as a function ofthe range R . To obtain the lines in Fig. A.11a), we fitted their values of FL ( E ) by assuming constant reaction crosssections obtaining values for these cross sections, which are given in Tab. A.3; these values are comparable to thosegiven in Perey and Perey [29] for target nuclei in Cs or I mass region. The fractional loss in Fig. A.11a) correspondsto the e ffi ciency displayed in Fig. A.11b); where the e ffi ciency losses due to multiple scattering are neglected as in thecalculations of Ref. [10]. The e ffi ciencies, labelled as “Charity” in Fig. 7, however include the e ffi ciency losses due tomultiple scattering in addition to those cause by reactions. Range (cm) FL ( x ) − − − − protondeutontriton3He Range (cm) P ( X ) protondeutontriton3He Figure A.11: (a) Fraction loss in a CsI for proton (red), deuteron (yellow) and triton (blue) particule as a function of the range in the material [10].The lines correspond to the fit of the points assuming a constant cross section. (b) Associated probability P ( X ) using this parameterization. Particle σ (barn) p d t He 2.2 α Table A.3: Fit of the fractional loss (Fig. A.11) due to nuclear reaction in CsI material [10]. The nuclear cross section for a given particle is alsoreported eferences [1] A. Wagner et al. Nucl. Instr. and Meth. A et al. Nucl. Instr. and Meth. A et al. Nucl. Instr. and Meth. A et al. Eur. Phys. J. A. et al. Nucl. Instr. and Meth. A et al. Phys. Rev. C et al. Phys. Rev. Lett. et al. Phys. Rev. C et al. Phys. Lett. B et al. Phys. Rev. C et al. Nucl. Instr. and Meth. A et al. Nucl. Instr. and Meth. A et al. Journal. Phys. G: Nucl. Part. Phys Nucl. Instrum. Methods Phys. Res. A et al. Nucl. Instrum. Methods Phys. Res. A // geant4.web.cern.ch / geant4 / UserDocumentation / UsersGuides / PhysicsReferenceManual / fo / PhysicsReferenceManual.pdf[18] P. D. Kunz, computer code DWUCK4, University of Colorado (unpublised)[19] R. L. Varner et al. Physics Report et al. Nuclear Physics
A491 et al. NASA Technical Paper
TP-1999-209726 (1999)[22] V. M. Grichine
Eur. Phys. J. C Physical Review C et al. Progress in NUCLEAR SCIENCE and TECHNOLOGY Atomic Data And Nuclear Data Tables Nuclear Physics A
Physics Letters
Phys. Rev. C At. Data Nucl. Data Tables Nuclear Physics A65-80 (1973)