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Non-linearity effects on the light-output calibration of light charged particles in CsI(Tl) scintillator crystals

The light output produced by light ions (Z<=4) in CsI(Tl) crystals is studied over a wide range of detected energies (E<=300 MeV). Energy-light calibration data sets are obtained with the 10 cm crystals in the recently upgraded High-Resolution Array (HiRA10). We use proton recoil data from 40,48Ca + CH2 at 28 MeV/u, 56.6 MeV/u, 39 MeV/u and 139.8 MeV/u and data from a dedicated experiment with direct low-energy beams. We also use the punch through points of p, d, and t particles from 40,48Ca + 58,64Ni, 112,124Sn collisions reactions at 139.8 MeV/u. Non-linearities, arising in particular from Tl doping and light collection efficiency in the CsI crystals, are found to significantly affect the light output and therefore the calibration of the detector response for light charged particles, especially the hydrogen isotopes. A new empirical parametrization of the hydrogen light output, L(E,Z=1,A), is proposed to account for the observed effects. Results are found to be consistent for all 48 CsI(Tl) crystals in a cluster of 12 HiRA10 telescopes.


Introduction
Thallium-activated caesium iodide (CsI(Tl)) scintillation crystals with photodiode (PD) readout are widely used in nuclear and particle physics experiments. They provide a popular solution to detect and identify, with good energy resolution (usually on the order of a few percent) and excellent identification performance, gamma-rays, light charged particles and intermediatemass fragments (IMF) produced in nuclear collisions. CsI(Tl) crystals combine several advantageous features including (1) relatively low manufacturing costs, (2) almost non-hygroscopic behaviors, (3) particle-dependent light output that enables particle identification via pulse-shape analysis, and (4) high density (ρ ≈ 4.5 g/cm 3 ). The high-density of the material is a particularly important feature since it allows stopping for highly energetic particles with relatively short detectors, minimizing the sensitivity to surface effects in the light response [1].
These characteristics make CsI(Tl) crystals particularly well-suited for being used as residual energy stages for silicon-based telescope systems, enabling the identification of fragments via the so-called ∆E-E technique [2]. Many large acceptance detection systems such as INDRA [3], CHIMERA [4,5], NIMROD-ISiS [6], GARFIELD [7] or FAZIA [8,9], have been used successfully in detecting and identifying particles in wide energy and mass ranges. Recently, detectors based on modular strip Si-CsI telescopes have been developed to provide high angular resolution in addition to high energy resolution for nuclear structure, particle-particle correlation and nuclear reaction studies. LASSA [10], MUST2 [11], FARCOS [12,13] and HiRA [14] are examples of these modular systems.
Such detectors are typically designed to have an optimized response to the detection of lighter particles and to have a high degree of versatility compared to the previous generation of large-acceptance detection systems. The higher stopping power of charged particles in CsI crystals enables a significantly larger dynamic energy range than that achieved with Si-Si modular telescopes (such as OSCAR [15]). However, the energy calibrations are more complicated as the light response of the CsI crystals is not linear and depends on the charge and mass of the charged particles as well as the length of the detector. The light response of CsI(Tl) scintillators is known to be non-linear for highly ionizing charged particles, due to the quenching phenomenon. Quenching is microscopically caused by an inefficient charge carrier recombination in the presence of high carrier density (eventually enhanced by lattice imperfections due to the interaction of a slowing down fragment) [16], and, therefore, is linked to the stopping power (dE/dx) of the incident radiation. The connection of the scintillation differential-light output to the ion stopping power in inorganic crystals was proposed by Birks in the 1960s [17]. In addition, the production of energetic δ-electrons affects the light output of inorganic crystals for heavy ions at high energy [18] making it impossible to describe the scintillation efficiency of CsI crystals uniquely as a function of the radiation stopping power alone. To account for this, the authors of Refs. [16,19,20] recently developed the "recombination and nuclear quenching model" (RNQM), that was successfully and systematically applied to produce energy calibrations of heavy ions (2 ≤ Z ≤ 60) in CsI(Tl), readout by photomultipliers (PMTs), with an accuracy better than 3% for E/A > 5 MeV and 5 to 15% at lower energies. For light charged particles with Z ≤ 6, the approximation dE/dx ∝ −AZ 2 /E has been used to analytically integrate the differential light output of Ref. [17] into an expression easier to use [21,22] and with an explicit dependence on the charge (Z) and mass (A) of the fragment, with an accuracy usually better than 5%, resulting in an almost linear trend at low stopping powers. However, significant deviations from the linearity have been observed in Refs. [1,23], and more recently in Ref. [24], in the medium and high energy region of Z ≤ 3 isotopes, suggesting the existence of possible additional efficiency factors that could play a role in the energy-light calibration of CsI scintillators. This clearly demands more extended investiga-tions of the CsI(Tl) light output to light charged particles. In this paper, we study the light output of long (≈ 10 cm) CsI(Tl) crystals incorporated in the upgraded High-Resolution Array (HiRA10) detection system. Our effort focuses particularly on the case of hydrogen isotopes, for which a reliable calibration is particularly needed in view of our recent efforts to explore collective properties of the emission of protons, as well as of other light fragments, produced in heavy ion collisions. Benefiting from the length of the HiRA10 crystals, we exploited high quality data collected in our recent experimental campaign at the National Cyclotron Superconducting Laboratory (NSCL), combined with low energy data from a dedicated experiment at the tandem accelerator of Western Michigan University, to build a consistent CsI calibration data set for hydrogen isotopes (from 1 MeV up to energies of around 200 MeV 1 H and 300 MeV 3 H).
For particles with 2 ≤ Z ≤ 4, the detected energy of the particles is up to 250 MeV. Mainly as a result of the crystal non-uniformities in the Tl doping and in the light collection efficiency, the light output to hydrogen isotopes is found to exhibit a non-linear trend. Based on our observations, we propose a new empirical calibration formula to describe the hydrogen light output with a unique set of parameters, accounting for the observed non-linearity effects. Such effects are found to be negligible within the explored energy range for heavier (Z ≤ 2) isotopes. The paper is organized as follows: Section 2 describes the experimental procedure used to obtain CsI calibration data sets, Section 3 provides a discussion of the effect of crystal non-uniformities on the light output; finally, results reported in Section 3 are used as the starting point to construct the CsI energy calibration described in Section 4.

The experimental setup: HiRA10
Experimental data used to calibrate the light output of the HiRA10 CsI(Tl) crystals are obtained by combining several independent data sets: (1) data extracted by using ∆E-E loci and the Ziegler energy loss tables [25], (2)   to 0.5% energy resolution. The experiments used 58,64 Ni and 112,124 Sn targets.
For calibration purposes, a 10 mg/cm 2 thick CH 2 target was used to obtain the proton recoil as described below. The position of the beam impinging on the target was monitored during beam tuning by means of a camera installed in the vacuum chamber and a luminescent viewer with a calibrated reference scale.
The emitted charged particles from nuclear collisions induced by the Ca beams are detected by 12 HiRA10 telescopes arranged in 3 towers in the 53" vacuum chamber in the S2 vault, as visible in the top view in the left panel of Figure 1.
The HiRA10 array covers angles ranging from around 25 to 75 degrees in the laboratory frame and an azimuthal angular region shown by the right panel of This provides a test of the light output produced by a particle stopping close to the front surface of the crystal. As specified, non-uniformities are found to be less than ±1.0%, testifying the good homogeneity of thallium along sections of the crystal orthogonal to its longitudinal direction. After fabrication of the final crystal, the uniformity of each crystal over the entrance surface was more extensively tested with a 241 Am alpha-source and an 8 × 8 grid configuration.
Results are found to be consistent with those obtained with the prototype. To test the longitudinal distribution of thallium inside the crystal, a collimated 137 Cs gamma source was used to laterally scan the finished crystal across different depths. The light response was found to exhibit a linearly decreasing trend as a function of the crystal depth, with overall non-uniformity values not exceeding 4%-5% between entrance and exit windows of the crystal (see Figure 9). This is a key point of the present paper and will be discussed in more detail in Section 3.

The energy loss method
If the incident particle is completely stopped in the CsI detection stage, the correlation of the energy deposited in the first detection stage (usually indicated as ∆E) and its residual energy measured by the corresponding CsI can be used to identify the mass and charge of the incident particle (the ∆E-E identification technique). An example is shown in Figure 3, which is produced by combining NSCL experiment data for each beam and target combination for one of the  we have considered an equivalent silicon dead layer of 0.6 µm, as obtained from a previous dedicated investigation [27]. DSSSD electronics linearity has been verified by sending a series of calibrated pulses to the preamplifier of each strip.
Taking advantage of the high-quality calibrations of the DSSSD, one can extract, for each individual CsI ADC channel E ch CsI , the y-axis centroid (∆E) of the calibrated ∆E distribution obtained by gating on the ∆E-E locus of a certain isotope. The kinetic energy of the isotope impinging on the DSSSD layer (E) can be deduced by a numerical inversion of Zieglers energy loss tables [25] using ∆E. The energy impinging on the CsI crystal (E M eV CsI ) is then calculated by subtracting the energy loss by the ion in the silicon, the dead layer located at its exit face and the Mylar foil used to wrap the CsI entrance face:

Particle punch-through calibration points
The maximum energy that a certain type of incident charged particle can deposit in the crystal is the one for which the particle is completely stopped in the crystal just before punching through. This is known as the punch-through energy of the particular ion and represents the transition point of the ∆E-E line from the usual shape to the reversed band. The additional reversed band is caused by a particle striking a crystal with a high energy, punching through the entire crystal, and only partially depositing its energy. This is clearly visible Fermi function (f (x) = a0 e ((x−a 1 )/a 2 ) +1 ) and a linear term to account for the background. The observed light output corresponding to the particle punch-through is determined by the a 1 term derived by the fit procedure, while a 2 , which represents the sharpness of the distribution, is used to quantify the associated error. We have carefully verified that our chi-square minimization procedure is negligibly affected by the choice of the interval used to perform the fit. This is true if such interval includes a significant flat region before and after the drop due to the punch-through. The a 2 parameter is affected by a combination of many factors including the light output resolution of the crystal, possible reaction or scattering of the incident particle in the crystal (resulting in incomplete energy collection), and the contribution of different beam angles that results in a distribution of different effective-CsI lengths experienced by the incident particle.   The trend is slightly non-linear across the whole energy range explored, as predicted by the light-energy parametrization of the type given in [21,22]. This trend is quantified by the red (hydrogen) and purple (helium) dashed lines in the Figure, which represent the result of a simultaneous fit of the low energy region of hydrogen and helium obtained by using the standard light-output calibration formula discussed in [21]. For protons and deuterons, (open and full squares), we see a linear trend down to approximately 1 MeV, indicating a negligible quenching effect. This is probably due to the high concentration of the activator element and, results in a negligible separation between 1 H and 2 H.

Non-linearity effects on the light output
We briefly summarize the features observed in Figure 7, which serve as the starting point for the following discussion: (1) the measured light output for Z = 2 isotopes is consistently lower with respect to the one measured for Z = 1 isotopes across the entire energy range; (2) the helium light output is slightly non-linear; (3) the hydrogen-isotope light output does not exhibit significant non-linearities and a negligible separation between Z = 1 isotopes is observed, with a slightly larger light response for 1 H with respect to 2 H. These facts are clearly in unquestionable agreement with the predictions of conventional energylight parametrizations for light ions [21,22]. To better understand the meaning of this statement, let us consider the following equation derived by the Birks formalism for inorganic scintillators: Here S is the scintillation efficiency, which corresponds to the amount of light produced per unit of energy released in the crystal in the absence of quenching, dE/dx is the particle stopping power and KB is the so-called quenching factor.
This equation gives the differential scintillation efficiency per unit of energy. absorptions or leaks, the measured light output is given by the simplified formula This formula is often used in the literature to calibrate the light response of hydrogen isotopes assuming negligible quenching effects [29]. However, the quantity S is usually not a constant since it contains a spatial dependence of the type S(x, y, z)= (x, y, z)ρ(x, y, z), where (x, y, z) is the light collection efficiency, i.e.
the probability that a photon emitted at (x, y, z) is effectively detected by the photodiode, ( (x, y, z) is typically of the order of few tens per cent) and ρ(x, y, z) is the light production efficiency, which is related to the density of the activator element at the position (x, y, z) of the crystal bulk. Disentangling (x, y, z) and ρ(x, y, z) is extremely complicated, since the first has to be carefully cal- where a is a gain factor and b is the light-output gradient revealed by our gamma-source investigation. It is important to stress that the hypothesis of uniformity of the crystal along surfaces orthogonal to the axis is quantitatively supported by the observed uniformity of the crystals at the entrance surface and at the base (see Section 2). Under this simplification, and by considering, for simplicity, a particle traveling longitudinally, equation 2 can be rewritten into a more general form can be numerically integrated. This results in the approximated analytical ex- where a is a gain factor and γ depends on the nature of the incident ion, being γ p < γ d < γ t < 1. The top right panel of Figure 9 shows the results of a similar calculation for protons, deuterons and tritons, compared to the original lightoutput of equation 2. A saturating behavior analogous to that observed in the experimental data of Figure 8 is clearly produced, justifying our assumption.
The last crucial point of the present considerations relates to the extent to which the described effects affect the light output of ions of different nature. As pointed out by equation 5, a saturating trend like the one observed in Figure 8 is compatible, within the limits of our simplified hypothesis, with our argument.
Anyway, since the saturation parameter γ is affected by the spatial properties of the crystal, it is intimately related to the range of a particle in the crystal.
To better clarify this statement, let us consider the following points. A simple consideration of the dependence of a particles stopping power on its energy indicates that most of the light produced by a particle in its path through the crystal is produced close to the end of the track as a result of the rapid increase of the stopping power of ions at lower energies. This makes these results strongly dependent on the range of the radiation through the crystal. Figure 10  confirming the reduced sensitivity of heavier ions to spatial non-uniformities of the crystal. To be more quantitative, a proton at an energy of 100 MeV will produce light in a CsI crystal along a 3.1 cm-path, while a 4 He will travel only for 2.4 mm before it is completely stopped in the crystal. The two numbers differ by more than a factor of ten. Because of such a macroscopic difference, a crystal will appear spatially uniform to a 4 He in this energy range while it will show non-uniformities to an incident proton of the same energy. Numerically, we obtain γ ≈ 0.95 for protons. However, the b gradient needed to reproduce the trend of the experimental data is larger than the one measured by the gamma-source experiment. This could be caused by the different sensitivity of the light efficiency to a gamma and a charged-particle probe, as observed in Ref. [30], and by the different distribution of optical photons produced for a particle longitudinally or perpendicularly penetrating the crystal. Additionally, the possible presence of ballistic deficit [31] in the processed signals, caused by the relative short shaping time used in the experiment (τ ≈ 3µs), needed for high-rate applications, might contribute to enhance the hydrogen to helium non-linearity effect observed here. Future investigations of the light output in different regions of the crystal with charged-particle probes are clearly needed.
As will be shown in the next section, the non-linearity effects here explored will require a correction term in the light output calibration of hydrogen isotopes. Such a term is found not to play a significant role for heavier elements in the energy range explored here.

Energy calibration
The light output calibration has been obtained by using all the data sets described in Section 2 and by taking into account the considerations of Section 3. We used two separate energy-light calibration formulas, one for hydrogen (that is significantly affected by CsI crystal non-linearity in the relevant energy range) and one for heavier ions (for which the effects described in Section 3 do not play a considerable role). calibration for other isotopes not included in the fit can be obtained as a simple extrapolation to other Z and A values of the used fitting formula. Figures   11 and 12 show the energy-light calibration produced for hydrogen isotopes.
To account for the observed non-linearity effects mainly caused by the crystal non-uniformity, we have used a fitting formula derived from equation 4. The dependence on the ion mass is described by the following empirical parametriza- In the equation, a 0 is a gain factor, A represents the Z = 1 isotope mass number and a 1 , a 2 are empirical non-linearity parameters that satisfy the relation  Figure 11. It is important to stress that the linear hypothesis made in previous hydrogen light output calibrations performed with 4-cm HiRA crystals [29] (where the smaller dynamic range and the limited calibration points did not allow for a detailed study of non-linearity effects) resulted in an incorrect determination of the zero offsets. To demonstrate the validity of our approach for all crystals in the array, Figure 13 shows the same proton-recoil kinematic lines as described by Figure Table   3 He ELoss Table   4 He WMU Exp 4 He ELoss Table   6 Region Constrained by WMU data large detection angles for the lower energy (panel a) is mainly because of the higher angular uncertainty of the HiRA10 pixels due to the larger beam spot on the target achieved for the 28 MeV/u beam during the NSCL experiment. This affects the effective angles at which particles are detected, which has a more pronounced effect for backward detection angles due to the inverse kinematics.
The energy-light calibration of heavier isotopes is obtained by using the standard light-output parametrization [21] obtained by an analytical integration of equation 1 under the approximation dE/dx ∝ −AZ 2 /E: where a 0 and a 1 are parameters obtained from a simultaneous fit of data. The expression presents a linear part dominating at high energies which is characterized by a slope given by a 0 and is related to the light efficiency of the crystal. There is also a non-linear quenching term dominating the lower energy region and affected by the fitting term a 1 . The additional AZ 2 dependence accounts for the different quenching because of the different mass and charge of the impinging ions. Figures 13 and 14 show the result of the fit to helium and heavier isotopes, respectively. Additionally, the inset in Figure 14 shows Li ELoss Table   6 Li ELoss Table   7 Li ELoss Table   8 Be ELoss Table   7 Be ELoss Table   9  Results of the energy-light calibration for hydrogen and helium isotopes are summarized in Figure 16, using the same color scheme adopted for the above discussion as a comparison. The low energy region delimited by the black box in the figure is expanded in the inset for clarity. The light output non-linearity observed for lighter isotopes results in a crossover of the Z = 1 and Z = 2 lines mainly caused by the sensitivity of the Z = 1 light output to the spatial nonuniformity of the scintillation efficiency, as already pointed out while discussing the ion to have a macroscopic (of the order of few cm) range in the crystal. The absence of firm energy-light constraints in the energy regions of Z ≥ 2 affected by the crystal non-linearity makes it extremely difficult to perform a systematic correction of the resulting effects. This is a limitation, as also pointed out in [29], for the energy resolution of the crystal in applications involving the detection of high-energy particles. Experiments involving direct beams with different energies to directly strike the crystal at different depths, could help in performing a systematic study of non-uniformity effects experienced by heavier ions.

Conclusions
In this paper, we have studied the light response to light ions of long (≈ 10 cm) CsI(Tl) crystals with a photodiode readout. In real experimental applications, these crystals are assembled in an array of 4 closely packed crystals and used as the second detection stage of the HiRA10 array. High quality data collected in a HiRA10 NSCL experimental campaign, combined with data obtained using direct low-energy beams delivered by the tandem accelerator of WMU, allowed us to effectively constrain the light output in a wide energy range for hydrogen (from 1 MeV up to energies of around 200 MeV of 1 H and 300 MeV of 3 H) and 2 ≤ Z ≤ 4 (for energies up to 250 MeV) isotopes. The spatial non-uniformity of the crystals was probed by using a 241 Am α-particle source to scan several points of the crystal over the entrance and back surface.
A 137 Cs γ-source was used to longitudinally scan the crystal, indicating linearly decreasing, depth-dependent light efficiency. Based on this assumption, we develop a simplified model to study gradient effects on the light output produced by several incident ions and mainly arising from the combination of the spatial non-uniformity of Tl activator concentration and the geometrical light collection efficiency. Because of the longer range experienced by hydrogen isotopes in the CsI, the Z = 1 light-output calibration results are significantly affected by the spatial non-uniformity of the crystal, while the crystal appears almost uniform to heavier (Z > 1) isotopes in the energy domain used to constrain the light output. Accounting for the functional light output derived by our model, we propose a new empirical formula for hydrogen light-output calibration in spatially non-uniform crystals of the type studied here. The formula contains an explicit dependence on the ion mass and allowed us to obtain a simultaneous fit of hydrogen energy-light data. CsI crystal non-linearities are found to play a significant role in the energy-light calibration of light particles. A proper energy-light calibration across a wide dynamic range requires corrections for the deriving gradient effects, which represent the principal limitation to the CsI(Tl) energy resolution in experiments involving a wide range of energetic light parti-cles. Experiments with light beams directly impinging on the crystal at different depths are therefore required to investigate and correct for non-linearity effects affecting the high-energy region of the energy-light calibration, and typically not easily observable within the energy constraints used to calibrate the response of CsI to Z > 1 isotopes.