Approximations of coincidence time resolution models of scintillator detectors with leading edge discriminator

doi:10.1016/j.nima.2017.11.009

Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment

Volume 912, 21 December 2018, Pages 149-153

https://doi.org/10.1016/j.nima.2017.11.009 Get rights and content

Highlights

•
Coincidence time resolution (CTR) of scintillator detectors is analytically derived.
•
Dependence of CTR on main parameters is clarified by its closed-form approximations.
•
Scintillator rise time, optical TTS and photodetector jitter equally affect CTR.
•
Approximation results qualitatively and quantitatively agree with well-known ones.

Abstract

Coincidence time resolution (CTR) of scintillator detectors is of high importance in high energy physics, medical imaging, and many other time-of-flight (TOF) application areas. Recent progress in developments of fast silicon photomultipliers (SiPM) and dedicated fast timing electronics resulted in significant improvements in CTR of SiPM-based scintillator detectors. CTR of 10 ps is considered as an ultimate goal of these improvements. Approaching to that goal, the most sophisticated devices (multichannel digital SiPM, 3D SiPM) and methods (TOF estimators based on multi-photon time-stamps, intense computational algorithms) are evaluated.

Despite these cutting-edge approaches, conventional analog SiPMs and vacuum PMTs with a conventional leading-edge discriminator (LED) are expected to be relevant solutions in large-scale experiments and applications for a long time. However, CTR estimation even in these cases requires specific numerical or Monte Carlo simulations because to the best of the author’s knowledge, there are no widely recognized closed-form expressions of CTR.

This study is an attempt to develop simple and reasonably realistic closed-form approximations of the CTR from a filtered marked point process model of the photomultiplier response, and clarify the CTR dependence on the main scintillator detection parameters.

Introduction

To move forward in improvements of time resolution approaching a breakthrough CTR of 10 ps [1], designing new scintillation detectors and associated electronics, to make an optimal selection of SiPMs and PMTs preparing large projects on TOF applications, to conduct unique scientific studies with TOF instrumentation, it would be preferable to be based on analytical models of CTR and their approximations providing reasonable practical predictions of the detector performance and its possible improvements.

These objectives having been pursued for decades. Let us try to glance on how CTR dependence on scintillator and photodetector parameters have been expressed in some well-known studies of CTR since the 1950s. In this review part, we focused on the most relevant studies [[2], [3], [4], [5], [6], [7], [8], [9], [10], [11]] as considered below.

Order statistics of photon detection times have been initially analyzed and applied for the CTR estimation by R. Post and L. Schiff in 1950 [2], and advanced further by others, e.g. [3]. In a convenient form the result [3] based on the order statistics could be rewritten as follows: $σ_{t} \approx \sqrt{\frac{5}{3} \cdot \frac{τ_{d} \cdot τ_{s e r}}{N_{p e}}}$ where $σ_{t}$ is a time resolution (standard deviation), $N_{p e}$ — a number of photoelectrons, $τ_{d}$ – scintillator decay time, $τ_{s e r}$ – photodetector response time (either FWHM or fall time of a single electron response — SER).

Nowadays, order statistics approach is applied with respect to photon counting and photon-number-resolving detectors (SPADs and SiPMs) and ability to process a sequence of timestamps of photon detection events, e.g. [4]. In this case $V a r (T_{i : n}) = {(\frac{τ_{d}}{n})}^{2}$ where $T_{i : n}$ is a sequence of the timestamp values and n is a number of the timestamps ( $\sim N_{p e}$ ).

An alternative approach based on convolutions of light intensity profile and photoresponse shape, mostly associated with a single electron response of PMT has been developed by L.G. Hyman et al. [[5], [6]], Gatti and Svetlo [7] in 1964, and others e.g. [8]. Time resolution was expressed in form $σ_{t} = \sqrt{\frac{τ_{d}^{2} \cdot E N F_{A}}{N_{p e}} \cdot g (\frac{C}{N_{p e}}, \frac{τ_{s e r}}{σ_{s p t r}}, \frac{τ_{d}}{σ_{s p t r}})}$ where ${E N F}_{A}$ is an excess noise factor (ENF) of PMT charge amplification process, C — a LED threshold in photoelectrons, $σ_{s p t r}$ — a PMT jitter or single photon time resolution (SPTR) assumed to be defined by Gaussian profile of the single photon detection times, and g() — some Monte Carlo simulated function without closed-form expression.

Analytical extraction of parametric dependences from Monte Carlo simulations is a new approach recently applied by S.E. Derenzo et al. [9] for the study of fundamental limits of scintillation detectors. As they observed, a photoelectron detection rate (a mean number of photoelectrons $N_{p e}$ per a scintillation decay time $τ_{d}$ , i.e. $N_{p e} ∕ τ_{d}$ ) is found to be a much more influential factor affecting time resolution than an optical transport jitter and SPTR, as well as single electron response (SER) rise and decay times (see Eq. (4) in Box I):

where time resolution TR is given in FWHM (TR $\approx 2.35 σ_{t}$ ), optical transit time spread (OTTS) of photons in scintillator crystal is assumed to be defined by single-exponential process with characteristic decay time $τ_{o t t s}$ , and $τ_{r}$ — scintillator rise time. However, the results and conclusions of the study [9] have inherent limitations in the clarity of observed relationships and its generality because it is based on Monte Carlo simulations as well as many others in this area.

Cramer–Rao lower bound estimations have been recognized as a powerful tool to realize theoretical limitations of the CTR [10]. However, this approach is applied to photon and photoelectron statistics only and does not consider photodetector response. Moreover, closed-form expressions could hardly be obtained by this approach, and in fact, it did not provide them yet.

Rounding up this draft overview, it could be concluded that a common understanding of the CTR dependence on the main scintillator detection parameters is limited to just a couple of proportionalities as summarized, for example, in [11] $σ_{t} \sim \frac{1}{\sqrt{N_{p e}}}$ $σ_{t} | \begin{matrix} τ_{d} ≫ τ_{r} \\ τ_{d} ≫ σ_{s p t r} \end{matrix} . \sim \sqrt{τ_{d}}$ and a tendency that CTR decreases with decreasing of $τ_{r}$ and $σ_{s p t r}$ .

Therefore, the motivation of this study is to derive the CTR dependence on the key timing parameters of the scintillation detection ( $τ_{d}$ , $τ_{r}$ , $σ_{s p t r}$ , $τ_{s e r}$ ) as a closed-form expression and analyze it with respect to some known analytical and experimental results. Development of the closed-form expressions and approximations in this study have been based on a filtered marked point process model [12].

Section snippets

Filtered marked point process approach

The leading-edge discriminator (LED) technique assumes that a light pulse detection event happens when a threshold D is crossed by the output response in a threshold crossing time t = $T_{D}$ , and the time resolution of the LED detection is defined by the probability distribution of random variable $T_{D}$ . Time resolution as a standard deviation $σ_{t}$ of the random variable $T_{D}$ is determined by a derivative of mean $μ_{o u t}$ (t) and standard deviation $σ_{o u t}$ (t) of the photomultiplier output response $V_{o u t}$ (t) with

Simulation results of scintillation detection

This study has been mainly focused on the development of the approximations, therefore its initial verification has been based on comparison with a comprehensively studied scintillation detection of 511 keV gammas using LSO/LYSO with SiPM readout. Studies in this area provide reliable, reproducible, and well-recognized results with a good agreement of simulations and experiments, e.g. [[8], [9], [13]].

Theoretical and experimental analysis [13] of CTR for LSO:Ce codoped 0.4% Ca and MPPC

Conclusion

The time resolution of a scintillation pulse detection using LED technique has been analyzed in a framework of filtered marked point process approach. Closed-form expressions of the CTR have been derived in case of bi-exponential scintillation pulse shape, Gaussian OTTS and SPTR profiles, Heaviside SER pulse shape and, with some limitations, bi-exponential SER due to analytical convolutions of corresponding functions. These results facilitate simulations of the CTR, but, being rather large and

Acknowledgments

This work has been supported by Megagrant 2013 program of Russia, the agreement No. 14.A12.31.0006 from 24.06.2013 and the additional agreement No. 2 from 03.03.2016 and Russian grants # 3.2989.2017/4.6 and 3.8484.2017/9.10.

It has also been supported by the EC in a framework of the SENSE H2020 CSA project 713171 funded under H2020-EU.1.2.1. - FET Open and the FAST COST (European Cooperation in Science and Technology) action TD1401.

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Approximations of coincidence time resolution models of scintillator detectors with leading edge discriminator

Highlights

Abstract

Introduction

Section snippets

Filtered marked point process approach

Simulation results of scintillation detection

Conclusion

Acknowledgments

NIMA

Nucl. Instrum. Methods

NIMA

NIMA

Statistical limitations on the resolving time of a scintillation counter

Phys. Rev.