Light Readout of Small Scintillators Using SiPM Photosensors

TL;DR

This paper investigates how SiPM sensors can cause non-linear responses when detecting light from small scintillators at high energies.

Contribution

The study identifies how scintillator decay time and SiPM microcell count affect non-linear behavior in radiation detection systems.

Findings

Non-linearity in SiPM response depends on scintillator light decay time and SiPM microcell count.

Saturation effects can be minimized by carefully selecting scintillator and SiPM combinations.

Results help in designing radiation detection systems with better linearity.

Abstract

What are the main findings?

A light readout from scintillators by means of a SiPM can or cannot give rise to non-linearity at a high energy, depending on the light decay time and on the number of SiPM microcells.

What is the implication of the main finding?

A careful choice of scintillator and SiPM can prevent or minimize non-linear behavior at a high deposited energy.

During the last two decades, relevant progress has been achieved in silicon photomultiplier (SiPM) technology, such that in an increasing number of radiation detection applications they are proposed as a viable alternative to traditional photomultiplier tubes (PMTs). Applications where the light from tiny scintillating crystals is detected by a single SiPM raise the question of the possible non-linearity of the response due to the saturation of the number of microcells involved. In other cases, where larger scintillators subtend arrays of SiPMs, the same question could hold. This work tries to disentangle such a question with a realistic numerical approach and a few tests showing that the possible saturation effects depend on the interplay between the features of the scintillator and of the SiPM (array). The quantitative results of this analysis can likely be used to better plan future radiation detection systems and to highlight their linearity boundaries.

Figures

Tables

Table 1. Main features, relevant to this study, of the two selected SiPMs.

	MICROFC−60035−SMTSensL (Now OnSemi) [12]	S14160-6050HSHamamatsu [13]
Number of microcells	18,980	14,331
Microcell recharge time [ns]	100	92

Table 2. Main features, relevant to this study, of the selected five popular scintillators.

	CsI(Tl)	LaBr3(Ce)	CeBr3	BGO	NaI(Tl)
Light yield [photons/keV]	60	70	70	10	45
Decay time [ns]	960	30	20	300	250
Emission spectrum from ref.	[14]	[15]	[16]	[17]	[18]
Refractive index at λ max	1.8	1.9	2.1	2.1	1.8
Weighted PDE SensL [%]	27%	31%	34%	29%	35%
Weighted PDE Hamamatsu [%]	41%	42%	45%	44%	48%

Full text

1. Introduction

The detection of gamma rays is one of the most widespread applications in radiation physics worldwide. Whenever high spectroscopic precision is required, in order to identify the emitting radioisotopes, the gold standard of detectors is the germanium semiconductor [1]. In most other cases, much less expensive and more practical detectors based on scintillators are exploited. A scintillator detector is based on special organic or inorganic materials exhibiting the notable feature of producing tiny flashes of light when hit by radiation. We will not go into detail on how such light is produced at a molecular level, but we will just remind the reader here that the produced visible photons are emitted isotropically according to an exponential time distribution with a decay constant µ characteristic of each material. Organic scintillators are typically faster, with µ values even down to 1–2 ns, whereas inorganic crystals span a wide range from tens of nanoseconds up to milliseconds [2,3].

Over several decades, the light readout of scintillators has mainly been performed by means of photomultiplier tubes (PMTs), until more recently a new kind of solid-state photodetector has been developed, namely the silicon photomultiplier (SiPM). Such a device, suggested many years ago but only recently industrially produced because of relevant technological advances, operates according to a quasi-digital scheme. Indeed, it consists of an array of semiconductor microcells kept in a quiescent state slightly above the breakdown voltage. When a visible photon interacts with one such microcell, it triggers a discharge, limited by a built-in quenching resistor, that gives rise to a signal consisting of the electrical charge stored in the microcell itself. All microcells being identical, each one produces the same signal when hit by a visible photon, and then a light pulse consisting of n photons should produce a signal n times as large as the single photon does.

Of course, this ideal scheme has several limitations in real sensors:

1. Only a fraction of the photons hitting the SiPM produce signals because of its photon detection efficiency (PDE), which is far from 100%;
2. When two or more photons interact with the same microcell, the produced signal is the same (this is the so-called problem of the multiple hit);
3. Each microcell has a recharging dead time, i.e., a short time interval after being triggered, during which it is inactive because its voltage is being restored to the operating value;
4. As the microcells are kept above breakdown, every now and then they can spontaneously discharge for thermal reasons, giving rise to an overall Poissonian dark noise mainly consisting of single microcell signals plus rarer double, triple and higher order spurious coincidences.

However, the main limitation of SiPMs is their maximum size, which for the currently available devices is 6 mm × 6 mm. Such a small size can also be seen as an advantage should one be interested in setting up a miniature gamma-ray detector. Due to the decreasing cost of these photosensors, the size limitation is currently being overcome by arranging arrays of SiPMs for the readout of larger scintillators. Details on the features and operation of SiPMs can be found in [4,5,6,7] and in the copious list of references therein.

One of the questions arising from the above-mentioned operational limitations is the linearity of the response, in particular, when detecting gamma rays of the order of 1–2 MeV [8,9,10,11]. In this work, we examine the behavior of two different models of SiPM (Table 1) when coupled to five popular scintillators (Table 2) as a function of the energy deposited by gamma rays. In Section 2, we describe the algorithm developed to evaluate the behavior of a SiPM illuminated by scintillation light; in Section 3, the model is applied to a SiPM detecting light from scintillators; and, in Section 4, we show the results obtained with the several examined combinations of scintillator + SiPM, along with the experimental results obtained in two particular real cases. Finally, in Section 5, we discuss the relevant take-home messages arising from our results.

2. Modeling the SiPM Response to Light

2.1. Simulation Procedure

There are two different mechanisms possibly leading to the loss of linearity in a SiPM: (i) the signal saturation due to multiple photons interacting with the same microcell but being detected as one; and (ii) the loss of photons due to microcells being hit during their recharge dead time because of a previous hit. The number of microcells actually fired by a bunch of photons can be smaller than the number of electron avalanches generated in the SiPM, since two or more avalanches generated in the same microcell produce the same signal and thus are seen as one. The most probable number of microcells f(q) actually triggered (i.e., when a photoelectron is produced capable of producing an avalanche) by a bunch of q photons impinging randomly on a SiPM with m microcells and PDE equal to p can be evaluated based on the binomial distribution.

• $[eqn]$ is the probability of a single microcell being triggered by one impinging photon;
• $[eqn]$ is the probability that a single microcell is not triggered by one impinging photon;
• $[eqn]$ is the probability that a single microcell is not triggered by q impinging photons;
• $[eqn]$ is the probability that a single microcell is triggered by at least one of the q impinging photons.

As there are m microcells, the expected number of triggered ones is calculated as follows:

$$[eqn]$$

By exploiting the notable limit shown below

$$[eqn]$$

one obtains the much simpler approximate formula:

$$[eqn]$$

The effect of the possible loss due to the dead time of recharging microcells requires a consideration of the time development of the light pulse. Therefore, we developed a simple algorithm that considers the production of photons exponentially decreasing with time, with the decay constant of the scintillator and using time slots of 5 ns. In each time slot, the number of triggered microcells is calculated by means of Equation (3), and decreased by the number of hits occurring in microcells currently recovering because they were previously triggered and are temporarily unavailable. In reality, during the recovery, there could be a partial signal, whose amplitude grows slowly by a (1 − exp^−t/τ^) function, only significant in the final stage of the recharge, i.e., in the short time it takes the microcell bias to go from the breakdown voltage to overvoltage. Indeed, below the breakdown, no signal is produced. Moreover, the information from the manufacturers only states a stark number for the recovery time, not specifying the relationship with τ. Therefore, we opted for a stepwise yes/no approach, with the recovering microcells switched off during the recovery time, confident that this does not make a relevant difference in the overall behavior, even though it can give rise to some sharper structures in the plots. The algorithm follows the light pulses over a few microseconds, which is much longer than the slowest decay constant of the considered scintillators, and calculates the sum of all triggered microcells, which is proportional to the integral of the output signal.

2.2. Cross-Check with an Empirical Formula

In Ref. [19], Grodzicka et al. proposed an empirical formula (Equation (4)) in order to take care of the recharge time of the microcells, the additional hits produced by afterpulses and cross-talk and the light pulse duration. The formula was used by the authors to fit the experimental data produced by means of light pulses into a controlled environment.

$$[eqn]$$

The new variables with respect to Equation (3) are P, which represents the fraction of additional hits due to afterpulses and cross-talk, P_w_, which is the input light pulse duration, and t_d_, which is the microcell recovery time. We remark that the formula implicitly assumes a yes/no approach for the microcell recovery process. We used Equation (4) to calculate the expected number of hits as a function of the number of ideal hits pq (i.e., if no counting loss due to multiple hits was present). The parameters were those of the S10362-33-050C SiPM listed in the article, namely 3600 microcells with a 50 ns recovery time.

We then tuned our model in order to simulate the same SiPM hit by trapezoidal light pulses similar to those employed in Ref. [19], where they were produced by an LED, driven by a pulse generator featuring a 5 ns rise-and-fall slope that excited a very fast plastic scintillator whose light was finally detected by the SiPM. In the model, we used a 1 ns time step to better cope with very short pulses, even though we did not see an appreciable difference when using a 5 ns time step. The authors of Ref. [19] did not specify the p value, so we decided to neglect it both in the formula and in our model, as it is of the order of a couple of percent at a normal operating overvoltage [12,13]. The resulting data, plotted in Figure 1a, show that the predictions of our model are in reasonable agreement with Ref. [19], even though it yields fewer expected hits as compared to the empirical formula. This is likely due to our model finely following the time evolution of the light pulse detection, with hit and recovery of the microcells. We remark that, despite the non-linearity of the SiPM behavior, because of the counting loss deriving from Equation (3), a linear fit in several cases can still reasonably follow the data, depending on the number of photons successfully hitting the SiPM and the pulse duration. However, the price to pay is the need for a compensating offset term, and the linear fit does not pass anymore through the origin, as illustrated in the two examples in Figure 1b.

Confident that our SiPM model can reasonably reproduce the experimental data, we used it to calculate the number of effective hits as a function of the number of ideal ones when our two SiPMs under study are solicited by exponential light pulses, with the decay constants of the scintillators listed in Table 2. In order to further explore the SiPM behavior in terms of the energy deposited into the crystals, the investigated range of ideal hits in the formula was conveniently extended. The results, shown in Figure 2, clearly indicate that, the slower the light pulse, the closer the SiPM response is to the ideal case.

3. Modeling Gamma-Ray Detection with Scintillation Detectors

3.1. Detecting Gamma Rays

The interaction of gamma rays with matter occurs according to three main processes:

• The photoelectric effect, dominating at a low energy, when the gamma disappears, transferring all of its energy to an electron;
• Compton scattering, dominating at an intermediate energy, with the gamma scattering off an electron and imparting it with some kinetic energy;
• e+e− pair creation close to a nucleus, exploiting 1.022 MeV of the incoming gamma and thus being the dominating effect at a very high energy.

For the low-to-medium energy range considered here, namely up to 2.5 MeV, only the photoelectric effect and Compton scattering are relevant. The energetic recoil electron produces ionizations by colliding with other electrons, which are freed and whose total kinetic energy is equal to the energy released by the initial gamma interaction. The scintillation light in scintillators is produced by the interaction of these electrons with “color centers”, i.e., typically dopants in the material capable of reaching an excited level through collision, which deexcites them by emitting visible light. The number of visible photons produced in each gamma interaction is generally proportional to the energy deposited by the gamma ray into the material. The detection and energy measurement of the gamma rays is achieved by measuring the amount of scintillation light produced by means of some photodetector, which converts light into an electric signal. Due to the very small amount of light produced, a physical amplification is quite often required, and the task is accomplished by using a photomultiplier device.

Different scintillation detectors have a different linearity between the deposited energy and scintillation light, especially at a high deposited energy where some saturation of the light yield could be expected. Saturation can also be expected due to the employed photosensor, especially in the case of photoelectric interactions when the full gamma energy is transferred to the material. Starting from our previous experience with detectors based on scintillators and SiPMs [20,21,22,23], we examined the five popular scintillation materials, whose main features relevant to this study are listed in Table 2 [24], and their response when coupled to two models of 6 mm × 6 mm SiPM [12,13], whose features are listed in Table 1.

3.2. Collecting the Scintillation Light

The number of scintillation photons collected on the photosensor depends on the geometric features of the detector and, more importantly, on the type and quality of the outer surface of the scintillator. Indeed, a bare scintillator would lose most of the light through its outer faces, basically collecting only those photons traveling straight from the emission point to the photosensor. This is why scintillators are generally coated with a highly reflective layer (paint, resin, …) to maximize the light collection efficiency, allowing photons to be collected also after several internal reflections. The reflector is not specular, but is white so that, contrary to the case of geometrical reflection, the light path inside the scintillator is quickly randomized. This way, any possible variation in the light collection efficiency with its emission position inside the scintillator is minimized. The typical reflectivity values of the employed reflector materials range from 0.9 to 0.96.

Two detector geometries have been examined, representative of two major possible approaches to the spectroscopic detection of gamma rays with scintillators and SiPMs. The first one concerns applications such as miniature detectors and dosimeters, whereas the second has a crystal size and shape typical of several existing commercial products:

• A compact configuration with a 1 cm × 1 cm × 1 cm scintillator coupled to a single 6 mm × 6 mm SiPM (Figure 3a);
• A bigger one with a cylindrical scintillator of 3.81 cm diameter and 3.81 cm height (1.5″ × 1.5″) coupled to a square array of 4 × 4 SiPMs (Figure 3b).

The results and considerations that we are going to describe in the following can be easily rescaled to similar geometrical configurations with larger or smaller scintillators and a different number of SiPMs.

Due to the compact size of both geometries with respect to the (re)absorption length of several tens of centimeters for all of the scintillators under consideration, in this study, we decided to neglect the possible self-absorption of scintillation photons in the scintillator material. The light collection efficiency, i.e., the fraction of photons reaching the SiPM, was first estimated by means of a simple naive approach:

• No real geometry is considered for the system, and no light propagation is implemented;
• A scintillation photon produced somewhere inside the crystal reaches a point on the inner surface;
• We assume it can hit the SiPM with a probability ε equal to the ratio between the area of the SiPM and the total area of the crystal;
• Otherwise, it can be reflected or absorbed with probability r and (1 − r), respectively, r being the reflectivity of the inner surface;
• We denote with P1 = ε the probability that the photon is collected directly on the first step, with P2 being the probability that the photon is collected after one reflection (i.e., at the second step), and so on;
• After each step, the probability of the photon being still available is (1 − ε)r (i.e., not collected and reflected), whereas the probability of being collected at the following step is still ε;
• The sum of all the probabilities of collection in any number of steps, regardless of the number of reflections, represents the light collection efficiency (Equation (5)).

This calculation was conducted for several values of reflectivity, ranging from 0.9 to 1, with the elementary collection probabilities ε = 0.06 and ε = 0.084 given by the area ratios for the cases of Figure 3a,b.

$$[eqn]$$

In order to support or disprove these results, we performed a set of more sophisticated Monte Carlo simulation runs by means of Geant4 [25]. The inner surfaces of the scintillator were assumed to produce diffuse Lambertian reflection [26], each run with a different value of the reflectivity. Inside the crystal, we generated randomly 10^5^ scintillation photons per run, following each one throughout its path and reflections until being absorbed in a wall or reaching the photosensor. The two geometries of Figure 3 were implemented, and eleven runs per configuration were performed, with the reflectivity of the walls ranging from 0.9 to 1 in steps of 0.1. The runs with a reflectivity equal to one showed that the average times for the detection are 0.62 ns and 1.6 ns, respectively, which, assuming a refractive index around 1.9, correspond to average path lengths of the order of 10 and 25 cm. In the case of reflectivity equal to 0.95, these values roughly halve to about 5.5 and 13 cm, much smaller than the attenuation length values in the considered crystals, thus justifying the choice of neglecting the self-absorption.

The resulting values of the light collection efficiency for the two approaches and for the two detector geometries are plotted in Figure 4. Surprisingly, the differences between the simple and the Monte Carlo approaches are quite small, thus suggesting that the simple formula of Equation (5) can realistically be used for future evaluations of similar configurations. The ratio between the simple and Geant4 values is plotted in Figure 5 for the cube and the cylinder geometries, with the statistical error bars calculated from Geant4 hits. The data in Figure 4 and Figure 5 are also listed in Table A1 and Table A2 of Appendix A. For all of the following calculations, we used the light collection efficiency values of p = 0.56 and p = 0.65 resulting from Equation (5) for cube and cylinder geometries, respectively, choosing a reflectivity value of r = 0.95 and the above-mentioned elementary collection probabilities ε = 0.06 and ε = 0.084.

3.3. Detecting the Collected Light

The physical quantities to be considered for the detection with SiPMs are the light yield of the crystal, the light decay time, the emission spectrum, the collection efficiency of the chosen detector geometry, the SiPM’s PDE, its number of microcells and the microcell recovery time (i.e., dead time). If we want to calculate the expected response of the photosensor, we have to take into account all these quantities at the same time. We used the values listed in Table 1 and Table 2, and, in order to show the behavior of the decay time in real operation, we plotted in Figure 6 a signal waveform acquired from a CsI(Tl) scintillator coupled to a SensL SiPM in the configuration of Figure 3a with a digital scope. An exponential fit to the waveform produces a decay constant of µ = 0.96 µs as expected.

As for the PDE, we multiplied the PDE(λ) function of each SiPM [12,13] by the light emission spectra of the five scintillators [14,15,16,17,18] (Figure 7), normalized to the unit area. The integral of such a convolution represents the effective PDE of each SiPM when detecting the light emitted by each scintillator. The resulting values are reported in Figure 8. Obviously, the light collection also depends on the optical coupling between the crystal and the SiPM, which cannot be easily reproduced and controlled numerically. We decided to assume a perfect coupling.

4. Results

The combination of the light decay constant of the scintillator with the microcell recharge time of the SiPM produces an initial decrease in the number of available microcells that could or could not significantly influence the linearity of the SiPM response depending on the deposited energy, on the scintillator type and on the total number of microcells. In Figure 9a,b, we show as an example the number of triggered and of available microcells as a function of time, for the case of 2 MeV deposited in a 1 cm × 1 cm × 1 cm CsI(Tl) crystal read by the SensL and Hamamatsu SiPMs, respectively. Also shown is the ideal number of microcells that would be triggered if no recharge dead time were present. Figure 10, Figure 11, Figure 12 and Figure 13 show the corresponding plots for all the other combinations of SiPMs and scintillators listed in Table 1 and Table 2, respectively. These plots are useful for understanding the behavior of the SiPM during the development of the light pulse. It can be immediately observed that LaBr3(Ce) and CeBr3 give rise to a relevant counting loss due to the large amount of scintillation photons reaching the SiPM in a very short time interval, thus almost blinding it for a while and losing a considerable fraction of the light signal. This effect is more pronounced with the Hamamatsu SiPM due to its smaller number of microcells. Figure 14 summarizes the ratio of the fired-to-ideal number of microcells as a function of time for the five scintillators and the two SiPMs. Such a ratio is mainly determined by the number of available microcells at each instant, as it is evident from the shape of the curves as compared to the green curves in Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13. We remark that the vertical scale starts from zero, thus highlighting how the fast LaBr3(Ce) and CeBr3 scintillators almost completely blind the SiPM at the beginning of the light pulse, more heavily for the Hamamatsu than for the SensL due to the higher PDE and the lower number of microcells.

As for the cylindrical configuration, we only show the time evolution plots for the worst case, which is the CeBr3 scintillator that features the highest light yield and the fastest decay time, in Figure 15a and b, respectively, for the SensL and Hamamatsu SiPMs. The ratio of the fired-to-ideal number of microcells as a function of time for the five scintillators and the two SiPMs in this configuration is shown in Figure 16. We remark that, in this case, no relevant SiPM blinding occurs (notice that the vertical scale starts from 0.8), thus indicating how a bigger crystal coupled to 16 SiPMs, featuring 16× microcells, strongly reduces the counting losses.

Then, we also calculated the response of each detector configuration as a function of the deposited gamma energy in the typical range up to 2.5 MeV. Figure 17 shows how a single SiPM coupled to small cubes of CsI(Tl), NaI(Tl) and BGO behave almost linearly, whereas when coupled to LaBr3(Ce) and CeBr3 they have a strongly non-linear behavior. The case of the bigger cylindrical scintillator coupled to 16 SiPMs, plotted in Figure 18, shows a much better behavior, apparently perfectly linear for all the combinations. The values reported in the four plots are listed in Table A3, Table A4, Table A5 and Table A6 of Appendix B. In order to numerically evaluate and compare the goodness of the linearity, we made a linear fit for each curve of Figure 17 and Figure 18 and then calculated the respective coefficient of determination (R^2^) values according to Equation (6).

$$[eqn]$$

where y_i_ is the number of hits produced by the model, $[eqn]$ is the corresponding value obtained by the linear fit and $[eqn]$ is the average of the y_i_ values, with 0 ≤ R^2^ ≤ 1 and R^2^ = 1 indicating a perfect linear behavior. Figure 19a,b summarize the found R^2^ values for each detector setup, for cubic and cylindrical configurations, respectively. The considerations that were performed by looking at Figure 17 and Figure 18 find numerical support here.

By inverting the linear fit, we can reconstruct the expected new energy scale and compare it with the true original one. In Figure 20, we plotted the relative difference between the expected energy value from the linear fit and its true value for the SensL and the Hamamatsu SiPMs in the case of the cubic scintillator. The plots represent the relative displacement (distortion) of each energy value reconstructed by a linear fit with respect to the true one. The corresponding plots for the cylindrical configuration are shown in Figure 21. We remark that the heavy distortion at a low energy is caused by the presence of an offset in the linear fit, necessary because of the form of Equation (3) (as already shown in Figure 1).

In Figure 22, we show two sets of spectra, taken with 1 cm × 1 cm × 1 cm CsI(Tl) crystals, read by a SensL and by a Hamamatsu SiPM, respectively. Each spectrum is related to one of four gamma sources (^226^Ra, ^22^Na, ^137^Cs, ^60^Co). The data were available from previous experiments, and unfortunately they were taken at different times using different sets of sources with different activities. This is why each spectrum was normalized to its own integral. The employed readout electronics were different as well; however, all the spectra were built by acquiring the waveforms, subtracting the baseline and numerically integrating the signal area (i.e., proportionally to the light detected by the SiPM) event by event. The spectra with the Hamamatsu SiPM were acquired in a noisier electronic environment and using lower activity radioactive sources, in particular, the ^226^Ra one. This is why two peaks, at 242 and 295 keV, could not be used, while the one at 352 keV was included in the following analysis with some doubts. Nonetheless, despite the very different experimental conditions in the two cases, the position of several energy peaks could be easily determined for the calibration. We selected the peaks up to 1764 keV (see Table 3) and performed the energy calibration by calculating a linear fit between the known energies and the observed peak positions in the spectra. The corresponding calibration plots are shown in Figure 23 and feature good R^2^ coefficient values of 0.9992 and 0.9982 for SensL and Hamamatsu, respectively. The normalized residuals of these two fits are plotted in Figure 24. Table 3 lists the energy values of the peaks used in the calibration for each source. The two highest energy peaks, from ^226^Ra, were not used for the calibration but just for a linearity cross-check. Indeed, the points at these two energies in Figure 23 and Figure 24 hint at a similar worsening linearity for both detectors.

5. Discussion

The main indication from our results is that, when planning to set up a spectroscopic gamma-ray detector based on a scintillator and SiPMs, careful consideration should be given both to the choice of components and to the energy range of interest. Indeed, there is a relevant interplay between the light yield of the scintillator, its decay time, the number of microcells featured by the SiPM and the microcell recovery time. The SiPM, as a quasi-digital counter, has a finite number of microcells and this can give rise to a partial saturation of the output signal due to two or more photons interacting with the same microcell (multiple hit), simultaneously or during its recovery. This effect takes place massively with LaBr3 and CeBr3 scintillators, whose light emission produces a large number of photons in a very short time interval. This causes non-linearity when the deposited energy is small, but can almost blind the SiPM for a while in case of a large deposited energy in the small scintillator cube (see Figure 10, Figure 11, Figure 14 and Figure 17). The effect is more pronounced in the SiPM with a smaller number of microcells (Hamamatsu). The result of the above-mentioned interplay is summarized in Figure 19a, where the corresponding values of R^2^ indicate quite a poor linearity. In the bigger cylindrical configuration, there is a relevant advantage, i.e., the photons produced are shared among several SiPMs so that each one sees only a fraction of them. In such a case, the counting losses are strongly reduced, as can be seen in the worst-case plots of Figure 15, Figure 16 and Figure 18.

If one is interested in setting up a small scintillator with a single SiPM, the best candidate seems to be CsI(Tl), either using a SensL or a Hamamatsu SiPM. Indeed, CsI(Tl) represents the best trade-off between linearity, energy resolution (Table 4) and chemical properties:

• Detectors with LaBr3(Ce) and CeBr3, hygroscopic and thus requiring an expensive air-tight case, would be strongly non-linear as they tend to blind the SiPM;
• NaI(Tl) is nearly equivalent to CsI(Tl), but it is hygroscopic;
• BGO has a poor light yield; therefore, the SiPM light readout would be perfectly linear but would provide a poor energy resolution;
• CsI(Tl) is reasonably inexpensive compared to LaBr3(Ce), CeBr3 and NaI(Tl), and does not require any special air-tight case, thus being easy to manipulate. Despite some claims about the possible non-linearity of similar SiPMs when detecting gamma rays above 1 MeV or less when coupled with CsI(Tl) [10], our calculations show no strong evidence of such an effect (Figure 17 and Figure 19a). Indeed, the experimental data in Figure 23 and Figure 24 hint at a slight loss of linearity above 1.7 MeV for both the SensL and the Hamamatsu SiPMs, but this can likely be ascribed to a non-linearity of the crystal itself, as was also suggested in [11].

We remark that the energy resolution values quoted in Table 4 do not consider corrections for the Fano factor, and they were simply calculated as a Poisson uncertainty from the number of triggered microcells (inverse square root multiplied by 2.35). However, they provide realistic indications, in particular for LaBr3(Ce) and CeBr3. In the same table, we also listed the expected position of the 662 keV peak for a ^137^Cs source in the energy scale obtained from the linear fit for the four studied cases. In order to highlight the non-linearity, in the table, we quoted the (1 − R^2^) value for each configuration.

As a further exercise, we used our calculation method to investigate the behavior of bigger cubic geometries, only considering LaBr3(Ce) and CeBr3 scintillators because they are the only ones producing relevant non-linearity with the SiPM readout. Therefore, we examined the response as a function of the energy for the two additional cubic configurations of Figure 25. In Table 5, we summarize the main features of these configurations along with the previously shown cubic and cylindrical ones for comparison, and in Figure 26 we compare the corresponding results, which show an improving linearity trend as the crystal and the SiPM array sizes increase.

6. Conclusions

The results of our numerical analysis can likely be used when planning radiation detection systems based on scintillators and SiPMs, as it highlights their realistic linearity boundaries. The analysis also confirms the interesting properties of the CsI(Tl) scintillator, as it represents a good compromise between cost, performance and ease of use, especially if planning to employ it in miniature detection systems like personal dosimeters. For higher level spectroscopic applications of scintillators, the feasibility of larger cylindrical configurations, coupled to arrays of SiPMs, has been confirmed up to a few MeV. LaBr3(Ce) and CeBr3 are the best options in terms of timing and energy resolution, even though CsI(Tl) remains a notable candidate as well.