A feasibility study of orthopositronium decays measurement with the JPET scanner based on plastic scintillators
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Abstract
We present a study of the application of the Jagiellonian positron emission tomograph (JPET) for the registration of gamma quanta from decays of orthopositronium (oPs). The JPET is the first positron emission tomography scanner based on organic scintillators in contrast to all current PET scanners based on inorganic crystals. Monte Carlo simulations show that the JPET as an axially symmetric and high acceptance scanner can be used as a multipurpose detector well suited to pursue research including e.g. tests of discrete symmetries in decays of orthopositronium in addition to the medical imaging. The gamma quanta originating from oPs decay interact in the plastic scintillators predominantly via the Compton effect, making the direct measurement of their energy impossible. Nevertheless, it is shown in this paper that the JPET scanner will enable studies of the \(\text{ oPs }\rightarrow 3\gamma \) decays with angular and energy resolution equal to \(\sigma (\theta ) \approx {0.4^{\circ }}\) and \(\sigma (E) \approx 4.1\,{\mathrm{keV}}\), respectively. An order of magnitude shorter decay time of signals from plastic scintillators with respect to the inorganic crystals results not only in better timing properties crucial for the reduction of physical and instrumental background, but also suppresses significantly the pileups, thus enabling compensation of the lower efficiency of the plastic scintillators by performing measurements with higher positron source activities.
Keywords
Discrete Symmetry Plastic Scintillator Gamma Quantum Compton Effect Secondary Scattering1 Introduction
The positron emission tomography (PET) is based on registration of two gamma quanta originating from a positron annihilation in matter. However, the \(e^+ e^ \rightarrow 2 \gamma \) process is not the only possible route of positron annihilation. Electron and positron may annihilate also to a larger number of gamma quanta with lower probability, or form a bound state called positronium. In the ground state with angular momentum equal to zero positronium may be formed in the triplet state (with spin S = 1) referred to as orthopositronium (oPs), or singlet state (S = 0) referred to as parapositronium (pPs). Positronium, being a boundstate built from electron and antielectron bound by the central potential, is an eigenstate of both charge (C) and spatial parity (P) operators, as well as of their combination (CP). Therefore, it is well suited for the studies of these discrete symmetries in the leptonic sector. These symmetries may be studied by the measurement of the expectation values of various operators (odd with respect to the studied symmetry) constructed from the momenta of photons and the spin of the orthopositronium [1]. Such studies are limited by the photon–photon interaction, however it was estimated that the vacuum polarisation effects may mimic the CP and CPT symmetries violation only at the level of 10\(^{9}\) [2], which is still by six orders of magnitude less than the presently best known experimental limits for CP and CPT violations in the positronium decays which are at the level of 0.3 % [3, 4]. Orthopositronium is symmetric in space and spin and, therefore, as a system built from fermions it must be charge symmetry odd. Parapositronium, in turn, as antisymmetric in spin and symmetric in space, must be charge symmetry even. C symmetry conservation implies that the orthopositronium annihilate into odd number of gamma quanta, \(3\gamma \) being the most probable, with lifetime 142 ns and parapositronium decays into even number of gamma quanta with lifetime 125 ps [5, 6, 7, 8]. Such a huge difference in the lifetimes enables an efficient experimental disentangling of oPs from pPs decays.
Summary of major physical characteristics of betaplus isotopes useful for PET imaging and positron annihilation lifetime spectroscopy (PALS) investigations. For isotopes that decay into excited states the properties of emitted gamma quanta are denoted. Data were adapted from [27]
Isotope  Halflife  \(\beta ^+\) decay  \(E_{\gamma }\) (MeV)  \(E_{e^+}^{max}\) (MeV)  Excited nuclei lifetime 

Isotopes for PALS and PET imaging  
\(^{22}\)Na  2.6 (years)  \(^{22} \text{ Na } \rightarrow ^{22}\text{ Ne } + e^+ + \nu _e +\gamma \)  1.27  0.546  3.63 (ps) 
\(^{68}\)Ga  67.8 (min)  \(^{68}\text{ Ga } \rightarrow ^{68}\text{ Zn } + e^+ + \nu _e +\gamma \)  1.08  0.822  1.57 (ps) 
\(^{44}\)Sc  4.0 (h)  \(^{44}\text{ Sc } \rightarrow ^{44}\text{ Ca } + e^+ + \nu _e +\gamma \)  1.16  1.474  2.61 (ps) 
Isotopes for PET imaging  
\(^{68}\)Ga  67.8 (min)  \(^{68}\text{ Ga } \rightarrow ^{68}\text{ Zn } + e^+ + \nu _e\)  –  1.899  – 
\(^{11}\)C  20.4 (min)  \(^{11}\text{ C } \rightarrow ^{11}\text{ B }+ e^+ + \nu _e \)  –  0.961  – 
\(^{13}\)N  10.0 (min)  \(^{13}\text{ N } \rightarrow ^{13}\text{ C }+ e^+ + \nu _e\)  –  1.198  – 
\(^{15}\)O  2.0 (min)  \(^{15}\text{ O } \rightarrow ^{15}\text{ N }+ e^+ + \nu _e\)  –  1.735  – 
\(^{18}\)F  1.8 (h)  \(^{18}\text{ F } \rightarrow ^{18}\text{ O } + e^+ + \nu _e\)  –  0.634  – 
 (i)
positron emission and thermalisation in the target material,
 (ii)
angular and energy distributions of gamma quanta originating from orthopositronium annihilation,
 (iii)
Compton interactions of emitted gamma quanta in the detector built from plastic scintillators,
 (iv)
determination of gamma quanta hitposition and hittime in the detector with experimentally determined resolutions,
 (v)
multiple scattering and accidental coincidences,
 (vi)
reconstruction of registered gamma quanta fourmomenta,
Section 2 gives a general introduction of positron emission and interaction with matter together with the formation of positronium and the description of orthopositronium annihilation into three gamma quanta. Possible detector geometries are summarized in Sect. 3. Properties of JPET detector, comparison between simulated and experimental spectra and the method of background rejection are presented in Sect. 4. Section 5 contains the detector efficiency estimation as well as the energy and angular resolutions.
2 Performance assessment: Monte Carlo simulations
The following paragraphs contain the description of Monte Carlo simulations of positrons emitted from \(\beta ^+\) source (\(^{22}\)Na) that bind with electron and form positronium. Simulation takes into account the effects of finite positronium range and nonzero residual momentum of the annihilation positronelectron pair. Special emphasis is put on a proper description of available phasespace of photons from orthopositronium annihilation and their further detection in the JPET detector that consists of plastic scintillators.
2.1 Positron source and positronium formation
Some of the \(\beta ^+\) emitters, e.g. \(^{22}\)Na or \(^{44}\)Sc, decay to daughter nucleus in excited states and emit prompt gamma with a well defined energy. In plastic scintillators gamma quanta interact mostly via the Compton scattering. Figure 3 shows the energy loss spectrum expected for the gamma quanta from the \(e^+e^ \rightarrow 2 \gamma \) annihilation compared to the spectra expected from the deexcitation quanta from \(^{22}\)Na and \(^{44}\)Sc isotopes.
In further considerations we will focus on sodium isotope, which is commonly used as a source of positrons for various experiments and tests of detectors. Pictorial representation of the studied \(\text{ oPs }\rightarrow 3\gamma \) process is shown in Fig. 4.
In the conducted simulations we took into account the description of positron properties after thermalisation. Its energy was simulated according to the distribution presented in Fig. 5 [29].
The distribution of the initial positron kinetic energy depends only on thermalisation processes. This distribution is taken into account in the transformation of gamma quanta fourmomenta from the rest frame of orthopositronium to the laboratory frame. In addition, the small distance traveled by positron in matter was taken into account. Positron range depends on material properties and can be generated from profiles known in the literature [30] provided by many simulation packages, such as GATE [31] or PeneloPET [32]. In this work the positron range distribution obtained by PeneloPET was adopted. Abovementioned effects introduce additional smearing of oPs annihilation position (see Fig. 6) and are included into performed simulations.
2.2 \(\text{ oPs }\rightarrow 3\gamma \) process
Details of simulated layers of the JPET geometry. JPET detector has been already built [1]. The mechanical construction for the next phases JPET+1 and JPET+2 is also prepared and the hardware upgrade is planned within the next 2 years
Layer number  Layer radius with respect to the center of scintillator (cm)  Number of scintillators in the layer  Angular displacement of \(n_i\) scintillator 

JPET  
1  42.50  48  \(n_i \times 7.5^{\circ }\) 
2  46.75  48  \(n_i \times 7.5^{\circ } + 3.75^{\circ }\) 
3  57.50  96  \(n_i \times 3.75^{\circ } + 1.875^{\circ }\) 
JPET+2  
1  42.50  48  \(n_i \times 7.5^{\circ }\) 
2  46.75  48  \(n_i \times 7.5^{\circ } + 3.75^{\circ }\) 
3  50.90  96  \(n_i \times 3.75^{\circ }\) 
4  53.30  96  \(n_i \times 3.75^{\circ } + 1.875^{\circ }\) 
5  57.50  96  \(n_i \times 3.75^{\circ } + 1.875^{\circ }\) 
JPET+1  
1  42.50  48  \(n_i \times 7.5^{\circ }\) 
2  46.75  48  \(n_i \times 7.5^{\circ } + 3.75^{\circ }\) 
3  53.30  96  \(n_i \times 3.75^{\circ }+1.875^{\circ } \) 
4  57.50  96  \(n_i \times 3.75^{\circ }+1.875^{\circ }\) 
JPETfull  
1  43.0  400  \(n_i \times 0.9^{\circ }\) 
2  45.0  437  \(n_i \times 0.82^{\circ }\) 
3  47.0  473  \(n_i \times 0.76^{\circ }\) 
4  49.0  508  \(n_i \times 0.71^{\circ }\) 
3 Simulated geometries
 JPET
corresponds to the already built detector [1] with 3 layers of scintillators (from in to out: 48 + 48 + 96 scintillators).
 JPET+1
the JPET geometry extended by an additional layer filled by 96 scintillators.
 JPET+2
geometry assumes complete fulfillment of all available layers in the JPET detector (48 + 48 + 96 + 96 + 96).
 JPETfull
detector with fully coverage of four plastic scintillator layers.
4 JPET detector properties
The multipurpose detector (JPET) constructed at the Jagiellonian University of which novelty lies in using large blocks of plastic scintillators instead of crystals as detectors of annihilation quanta, requires the usage of the time of signals, instead of their amplitude, and allows to obtain time resolution better than 100 ps [28].
4.1 Determination of hit and time position at JPET
4.2 Spectra of deposited energy
The probability of incident gamma quanta registration is a function of the attenuation coefficient \(\mu \) and distance that gamma quantum travels through the material. In the simulations the attenuation coefficient was parametrized as a function of incident gamma quanta energy (see Fig. 11, left panel).
4.3 Background rejection

relation between position of the individual detectors and the time difference between registered hits,

angular correlation of relative angles between the gamma quanta propagation directions,

the distance between the origin of the annihilation (position of the annihilation chamber) and the decay plane.

\(N_{3\gamma \ pick{\text {}}off} / N_{oPs}< (1\frac{\tau _{matter}}{\tau _{vacuum}}) / 370 \approx 2\cdot 10^{4} (\text{ IC3100 }) < 10^{3} (\text{ XAD4 })\);

\(N_{2\gamma \ pick{\text {}}off} / N_{oPs}< 0.07 \cdot 10^{9} (\text{ IC3100 }) < 0.36\cdot 10^{9} (\text{ XAD4 })\);

\(N_{3\gamma \ conv} / N_{oPs}< 0.07 \times 2.8 \cdot 10^{6} (\text{ IC3100 }) <0.36 \times 2.8 \cdot 10^{6} (\text{ XAD4 })\);

\(N_{2\gamma \ conv} / N_{oPs}< 0.07 \cdot 10^{9} (\text{ IC3100 }) < 0.36\cdot 10^{9} (\text{ XAD4 })\).
5 JPET performance in \(\text{ oPs }\rightarrow 3\gamma \) decay measurements
In order to determine the angular and energy resolution we have performed simulations of “pointlike” \(^{22}\)Na source surrounded by water and localized in the geometrical center of the JPET detector. The conducted simulations accounted for positron emission and thermalisation in the target material, angular and energy distributions of gamma quanta originating from orthopositronium annihilation and Compton interactions of emitted gamma quanta in the JPET detector. Details were presented in the Sect. 2. In the next step, based on the simulated data, we reconstructed hittime and hitposition of the registered gamma quantum interaction in the detector, taking into account the experimentally determined resolutions. Based on obtained informations the reconstruction of angles between gamma quanta and of their energies is performed, as described in the next paragraph.
5.1 Angular and energy resolution
Incident gamma quantum transmits energy as well as momentum to an electron in the plastic scintillator via Compton effect. Due to that, registered signals at the end of the scintillator strips cannot give information about the energy of the incident gamma quantum on the eventbyevent basis. However, registration of three gamma quanta hitposition from \(\text{ oPs }\rightarrow 3\gamma \) annihilation allows reconstruction of their energies based on the energy and momentum conservation.
5.1.1 Pointlike positronium source
5.1.2 Spatially extended positronium source
The angles (\(\theta _{12}\), \(\theta _{23}\), \(\theta _{13}\)) and hence a full kinematics of \(\text{ oPs }\rightarrow 3\gamma \) decay can be also reconstructed in the case of the extended positronium target. For example a target of a cylindrical shape with the diameter of 20 cm was proposed for the production of a linearly polarized positronium [1]. Polarisation can be determined provided that positron emission and positronium formation (approximately the same as annihilation) position are known.
A new reconstruction algorithm that allows reconstruction of orthopositronium annihilation position for an event by event basis was recently reported [9, 18]. The method based on trilateration allows for a simultaneous reconstruction of both location and time of the annihilation based on time and interaction position of gamma quanta in the JPET detector. The reconstruction performance strongly depends on detector time resolution (\(\sigma (T_{hit})\)). Using aforementioned reconstruction algorithm, current JPET spatial resolution for annihilation point reconstruction is at the level of 1.5 cm along the main detector axis and 2 cm in the transverse plane [18].
5.1.3 Performance studies
Since the angular and energy resolution strongly depend on hittime resolution registered in the JPET detector, the studies of resolution were made for \(\sigma (T_{hit}^0)\) in the range from 0 ps to 190 ps. Comparison between obtained resolutions for the “pointlike” and extended positronium source is shown in Fig. 16. In both cases energy and angular resolutions are improving with decreasing \(\sigma (T_{hit}^0)\), and for presently achieved time resolution of \(\sigma (T^0_{hit})\), and well a localized “pointlike” positronium source, they amount to \(\sigma (\theta ) = {0.4^{\circ }}\) and \(\sigma (E_{hit}) = 4.1\,{\mathrm{keV}}\), respectively. In case of the extended positronium source, when the reconstruction of the annihilation point is needed both resolutions increases to \(\sigma (\theta ) = {4.2^{\circ }}\) and \(\sigma (E_{hit}) = 30\,{\mathrm{keV}}\), respectively.
5.2 JPET efficiency studies with Monte Carlo simulations
Expected rate of registered signal events in different geometries and target materials assuming \(10^6\) annihilations per second and requiring energy deposition above 50 keV for all three gamma quanta from \(\text{ oPs }\rightarrow 3\gamma \) decay
Target material  Rate of registered \(\text{ oPs }\rightarrow 3\gamma \) events \((\mathrm{s}^{1})\)  

JPET  JPET+1  JPET+2  JPETfull  
IC3100  15  70  130  10600 
XAD4  25  115  230  18300 
6 Conclusions
We presented results of Monte Carlo simulations showing that the JagiellonianPET multipurpose detector constructed at the Jagiellonian University allows exclusive registration of the decays of orthopositronium into three photons (oPs \(\rightarrow 3 \gamma \)) providing angular and energy resolution of \(\sigma (\theta ) \approx {0.4^{\circ }}\) and \(\sigma (E) \approx 4.1\,{\mathrm{keV}}\), respectively.
The achieved results indicate that the JPET detector gives a realistic chance to improve the best present limits established for the CP and CPT symmetry violations in the decays of positronium [3, 4] by more than an order of magnitude. This can be achieved by (1) collecting at least two orders of magnitude higher statistics, due to the possibility of using a \(\beta ^+\) source with higher rate (10 MBq at JPET vs 0.37 MBq at Gammasphere [3] or 1 MBq at Tokyo University experiment [4]), (2) the enhanced fraction of \(3\gamma \) events by the use of the amberlite polymer XAD4, (3) a measurements with a few times improved angular resolution and (4) about two times higher degree of oPs polarization, as shown recently in reference [18]. The limitation on the source activity can be overcome by the JPET due to the application of plastic scintillators that are characterized by about two orders of magnitude shorter duration of signals, thus decreasing significantly the pileups problems with respect to the crystal based detector systems. In addition, the improved angular resolution combined with the superior timing of the JPET detector (by more than order of magnitude improved with respect to the crystal detectors) and with the possibility of the triggerless registrations [11, 12] of all kind of events with no hardware coincidence window allow suppression and monitoring of the background, due to misidentification of \(2\gamma \) events and possible contribution from \(3\gamma \) pickoff annihilations.
Notes
Acknowledgments
We acknowledge valuable discussions with Dr. J. Wawryszczuk and technical and administrative support by A. Heczko, M. Kajetanowicz, W. Migdał, and the financial support by the Polish National Center for Research and Development through Grants INNOTECHK1/IN1/64/159174/NCBR/12 and LIDER274/L6/14/NCBR/2015, the Foundation for Polish Science through MPD program and the EU, MSHE Grant No. POIG .02.03.00161 00013/09, Marian Smoluchowski Kraków Research Consortium “Matter–Energy–Future”, and the Polish Ministry of Science and Higher Education through Grant 7150/E338/M/2015. BCH gratefully acknowledges the Austrian Science Fund FWF23627.
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