# A feasibility study of ortho-positronium decays measurement with the J-PET scanner based on plastic scintillators

## Abstract

We present a study of the application of the Jagiellonian positron emission tomograph (J-PET) for the registration of gamma quanta from decays of ortho-positronium (o-Ps). The J-PET 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 J-PET as an axially symmetric and high acceptance scanner can be used as a multi-purpose detector well suited to pursue research including e.g. tests of discrete symmetries in decays of ortho-positronium in addition to the medical imaging. The gamma quanta originating from o-Ps 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 J-PET scanner will enable studies of the \(\text{ o-Ps }\rightarrow 3\gamma \) 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 pile-ups, thus enabling compensation of the lower efficiency of the plastic scintillators by performing measurements with higher positron source activities.

## 1 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 ortho-positronium (o-Ps), or singlet state (S = 0) referred to as para-positronium (p-Ps). Positronium, being a bound-state built from electron and anti-electron 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 ortho-positronium [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]. Ortho-positronium is symmetric in space and spin and, therefore, as a system built from fermions it must be charge symmetry odd. Para-positronium, in turn, as anti-symmetric in spin and symmetric in space, must be charge symmetry even. C symmetry conservation implies that the ortho-positronium annihilate into odd number of gamma quanta, \(3\gamma \) being the most probable, with lifetime 142 ns and para-positronium decays into even number of gamma quanta with lifetime 125 ps [5, 6, 7, 8]. Such a huge difference in the life-times enables an efficient experimental disentangling of o-Ps from p-Ps decays.

Summary of major physical characteristics of beta-plus 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 | Half-life | \(\beta ^+\) 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 ortho-positronium annihilation,

- (iii)
Compton interactions of emitted gamma quanta in the detector built from plastic scintillators,

- (iv)
determination of gamma quanta hit-position and hit-time in the detector with experimentally determined resolutions,

- (v)
multiple scattering and accidental coincidences,

- (vi)
reconstruction of registered gamma quanta four-momenta,

Section 2 gives a general introduction of positron emission and interaction with matter together with the formation of positronium and the description of ortho-positronium annihilation into three gamma quanta. Possible detector geometries are summarized in Sect. 3. Properties of J-PET 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 non-zero residual momentum of the annihilation positron-electron pair. Special emphasis is put on a proper description of available phase-space of photons from ortho-positronium annihilation and their further detection in the J-PET 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 de-excitation 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 \(\text{ o-Ps }\rightarrow 3\gamma \) 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 four-momenta from the rest frame of ortho-positronium 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 o-Ps annihilation position (see Fig. 6) and are included into performed simulations.

### 2.2 \(\text{ o-Ps }\rightarrow 3\gamma \) process

*v*denotes electron-positron relative velocity,

*e*is the elementary charge. In above formula the conservation of 4-momentum allows to eliminate one of the frequencies (\(\omega _{3}\)). Equation 1 results in the characteristic energy distribution of gamma quanta (see Figs. 7, 8).

Details of simulated layers of the J-PET geometry. J-PET detector has been already built [1]. The mechanical construction for the next phases J-PET+1 and J-PET+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 |
---|---|---|---|

J-PET | |||

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 | \(n_i \times 3.75^{\circ } + 1.875^{\circ }\) |

J-PET+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 | \(n_i \times 3.75^{\circ } + 1.875^{\circ }\) |

J-PET+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 | \(n_i \times 3.75^{\circ }+1.875^{\circ }\) |

J-PET-full | |||

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

- J-PET
corresponds to the already built detector [1] with 3 layers of scintillators (from in to out: 48 + 48 + 96 scintillators).

- J-PET+1
the J-PET geometry extended by an additional layer filled by 96 scintillators.

- J-PET+2
geometry assumes complete fulfillment of all available layers in the J-PET detector (48 + 48 + 96 + 96 + 96).

- J-PET-full
detector with fully coverage of four plastic scintillator layers.

*z*axis.

## 4 J-PET detector properties

The multipurpose detector (J-PET) 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 J-PET

*i*th plastic scintillator can be based on the time values (\(t_i^A\), \(t_i^B\)) of scintillation light registration in photomultipliers located at the ends of single plastic scintillator strip. Then the distance \((\Delta z_i)\) along the strip between its center and the hit position can be expressed as:

*v*is the light velocity in the plastic scintillator. Based on this information, in case of two-gamma quanta annihilation, the line of response (LOR) and the annihilation position along it can be determined (see Fig. 10).

### 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.

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

\(N_{2\gamma \ pick{\text {-}}off} / N_{o-Ps}< 0.07 \cdot 10^{-9} (\text{ IC3100 }) < 0.36\cdot 10^{-9} (\text{ XAD-4 })\);

\(N_{3\gamma \ conv} / N_{o-Ps}< 0.07 \times 2.8 \cdot 10^{-6} (\text{ IC3100 }) <0.36 \times 2.8 \cdot 10^{-6} (\text{ XAD-4 })\);

\(N_{2\gamma \ conv} / N_{o-Ps}< 0.07 \cdot 10^{-9} (\text{ IC3100 }) < 0.36\cdot 10^{-9} (\text{ XAD-4 })\).

## 5 J-PET performance in \(\text{ o-Ps }\rightarrow 3\gamma \) decay measurements

In order to determine the angular and energy resolution we have performed simulations of “point-like” \(^{22}\)Na source surrounded by water and localized in the geometrical center of the J-PET detector. The conducted simulations accounted for positron emission and thermalisation in the target material, angular and energy distributions of gamma quanta originating from ortho-positronium annihilation and Compton interactions of emitted gamma quanta in the J-PET detector. Details were presented in the Sect. 2. In the next step, based on the simulated data, we reconstructed hit-time and hit-position 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 event-by-event basis. However, registration of three gamma quanta hit-position from \(\text{ o-Ps }\rightarrow 3\gamma \) annihilation allows reconstruction of their energies based on the energy and momentum conservation.

#### 5.1.1 Point-like positronium source

*x*and

*y*are determined as the centre of the scintillator strip, and therefore the precision of their determination correspond to the geometrical cross section of the scintillator strip. The

*z*coordinate is determined from signals arrival time to photomultipliers at the ends of scintillator strip, and its uncertainty is equal to about \(\sigma (z) = 0.94\) cm [22, 23]. Uncertainty of \(\sigma (\mathbf {r}_{hit})\) determination gives the main contribution to estimation of angular and energy resolutions. The second order effect is an uncertainty originating from non zero boost and distance traveled by positron in matter.

#### 5.1.2 Spatially extended positronium source

The angles (\(\theta _{12}\), \(\theta _{23}\), \(\theta _{13}\)) and hence a full kinematics of \(\text{ o-Ps }\rightarrow 3\gamma \) 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 ortho-positronium 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 J-PET detector. The reconstruction performance strongly depends on detector time resolution (\(\sigma (T_{hit})\)). Using aforementioned reconstruction algorithm, current J-PET 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 hit-time resolution registered in the J-PET 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 “point-like” 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 “point-like” 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 J-PET efficiency studies with Monte Carlo simulations

*A*is the total annihilation rate (fast timing of applied plastic scintillators allows for usage of the 10 MBq positron source), \(f_{oPs\rightarrow 3\gamma }\) is the fraction of annihilations via \(\text{ o-Ps }\rightarrow 3\gamma \) process in the target material, \(\epsilon _{det}(th)\) is the detector efficiency as a function of applied detection threshold while \(\epsilon _{ana}\) denotes selection efficiency used to discriminate between \(3\gamma \) and \(2\gamma \) events.

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 \(\text{ o-Ps }\rightarrow 3\gamma \) decay

Target material | Rate of registered \(\text{ o-Ps }\rightarrow 3\gamma \) events \((\mathrm{s}^{-1})\) | |||
---|---|---|---|---|

J-PET | J-PET+1 | J-PET+2 | J-PET-full | |

IC3100 | 15 | 70 | 130 | 10600 |

XAD-4 | 25 | 115 | 230 | 18300 |

## 6 Conclusions

We presented results of Monte Carlo simulations showing that the Jagiellonian-PET multipurpose detector constructed at the Jagiellonian University allows exclusive registration of the decays of ortho-positronium into three photons (o-Ps \(\rightarrow 3 \gamma \)) 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 J-PET 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 J-PET 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 XAD-4, (3) a measurements with a few times improved angular resolution and (4) about two times higher degree of o-Ps polarization, as shown recently in reference [18]. The limitation on the source activity can be overcome by the J-PET due to the application of plastic scintillators that are characterized by about two orders of magnitude shorter duration of signals, thus decreasing significantly the pile-ups problems with respect to the crystal based detector systems. In addition, the improved angular resolution combined with the superior timing of the J-PET 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 \(3\gamma \) pick-off 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 INNOTECH-K1/IN1/64/159174/NCBR/12 and LIDER-274/L-6/14/NCBR/2015, the Foundation for Polish Science through MPD program and the EU, MSHE Grant No. POIG .02.03.00-161 00-013/09, Marian Smoluchowski Kraków Research Consortium “Matter–Energy–Future”, and the Polish Ministry of Science and Higher Education through Grant 7150/E-338/M/2015. BCH gratefully acknowledges the Austrian Science Fund FWF-23627.

### References

- 1.P. Moskal et al., Acta Phys. Polon. B
**47**, 509 (2016)CrossRefGoogle Scholar - 2.W. Bernreuther, U. Low, J.P. Ma, O. Nachtmann, Z. Phys. C
**41**, 143 (1988)CrossRefGoogle Scholar - 3.P.A. Vetter, S.J. Freedman, Phys. Rev. Lett. B
**91**, 263401 (2003)ADSCrossRefGoogle Scholar - 4.T. Yamazaki, T. Namba, S. Asai, T. Kobayashi, Phys. Rev. Lett.
**104**, 083401 (2010)ADSCrossRefGoogle Scholar - 5.M.D. Harpen, Med. Phys.
**31**, 57 (2004)ADSCrossRefGoogle Scholar - 6.A.H. Al-Ramadhan, D.W. Gidley, Phys. Rev. Lett.
**72**, 1632 (1994)ADSCrossRefGoogle Scholar - 7.R.S. Vallery, P.W. Zitzewitz, D.W. Gidley, Phys. Rev. Lett.
**90**, 203402 (2003)ADSCrossRefGoogle Scholar - 8.O. Jinnouchi, S. Asai, T. Kobayashi, Phys. Lett. B
**572**, 117 (2003)Google Scholar - 9.A. Gajos, E. Czerwiński, D. Kamińska, P. Moskal, Multi-tracer morphometric image reconstruction method and system using fast analytical algorithm for calculating time and position of positron-electron annihilation into three gamma quanta. PCT/PL2015/050038Google Scholar
- 10.M. Pałka et al., Bioalgorithms Med. Syst.
**10**(1), 41 (2014)Google Scholar - 11.G. Korcyl et al., Bioalgorithms Med. Syst.
**10**(1), 37 (2014)Google Scholar - 12.G. Korcyl et al., Acta Phys. Polon. B
**47**, 491 (2016)CrossRefGoogle Scholar - 13.K. Kacperski, N.M. Spyrou, F.A. Smith, IEEE Trans. Med. Im.
**23**, 525 (2004)CrossRefGoogle Scholar - 14.D. Kamińska et al., Nukleonika
**60**(4), 729 (2015)Google Scholar - 15.S.L. Dubovsky, V.A. Rubakov, P.G. Tinyakov, Phys. Rev. D
**62**, 105011 (2000)ADSCrossRefGoogle Scholar - 16.P. Crivelli, A. Belov, U. Gendotti, S. Gninenko, A. Rubbia, JINST
**5**, P08001 (2010)ADSCrossRefGoogle Scholar - 17.S.N. Gninenko, N.V. Krasnikov, A. Rubbia, Mod. Phys. Lett. A
**17**, 1713 (2002)ADSCrossRefGoogle Scholar - 18.A. Gajos et al., Nucl. Instrum. Methods A
**819**, 54 (2016)ADSCrossRefGoogle Scholar - 19.P. Moskal, Strip device and the method for the determination of the place and response time of the gamma quanta and the application of the device for the positron emission thomography. 2011008119, US2012112079, PL388555, JP2012533734, EP2454612Google Scholar
- 20.P. Moskal et al., Phys. Med. Biol.
**61**, 2025 (2016)CrossRefGoogle Scholar - 21.P. Moskal et al., Nucl. Instrum. Methods A
**775**, 54 (2015)ADSCrossRefGoogle Scholar - 22.L. Raczyński et al., Nucl. Instrum. Methods A
**786**, 105 (2015)ADSCrossRefGoogle Scholar - 23.L. Raczyński et al., Nucl. Instrum. Methods A
**764**, 186 (2014)ADSCrossRefGoogle Scholar - 24.Y.C. Jean, P.E. Mallon, D.M. Schrader,
*Positron and Positronium Chemistry*(World Scientific Publishing, Singapore, 2003)CrossRefGoogle Scholar - 25.P. Colombino, B. Fiscella, Nuovo Cimento
**3**, 1 (1971)CrossRefGoogle Scholar - 26.B. Jasińska et al., Acta Phys. Polon. B
**47**, 453 (2016)CrossRefGoogle Scholar - 27.National Nuclear Data Center. http://www.nndc.bnl.gov/. Access: 20.01.2016r
- 28.P. Moskal et al., Nucl. Instrum. Methods A
**764**, 317 (2014)ADSCrossRefGoogle Scholar - 29.Ch. Champion, Braz. Arch. Biol. Technol.
**48**, 191 (2005)CrossRefGoogle Scholar - 30.J. Cal-Gonzalez, J.L. Herraiz, S. Espana, M. Desco, J.M. Udias, Phys. Med. Biol.
**58**, 5127 (2013)Google Scholar - 31.S. Jan, G. Santin, D. Strul, S. Staelens, K. Assié, D. Autret, S. Avner, Phys. Med. Biol.
**49**, 4543 (2004)CrossRefGoogle Scholar - 32.J. España, L. Herraiz, E. Vicente, J.J. Vaquero, M. Desco, J.M. Udías, Phys. Med. Biol.
**54**, 1723 (2009)CrossRefGoogle Scholar - 33.V.B. Berestetskii, E.M. Lifshitz, L.P. Pitaevskii,
*Relativistic Quantum Theory*(Headington Hill Hall, Pergamon Press, Oxford, 1971)Google Scholar - 34.Saint-Gobain Crystals,
*Organic Scintillation Materials [Brochure]*(Saint-Gobain Crystals, Hiram, 2013)Google Scholar - 35.C. Leroy, P. Rancoita,
*Principles of radiation interaction in matter and detection*(World Scientific, Singapore, 2009)CrossRefGoogle Scholar - 36.V. Bettinardi et al., Med. Phys.
**38**(10), 5394 (2011)CrossRefGoogle Scholar - 37.S. Surti et al., J. Nucl. Med.
**48**, 471 (2007)Google Scholar - 38.O. Klein, T. Nishina, Z. Phys.
**52**, 853 (1929)ADSCrossRefGoogle Scholar - 39.A.P. Mills, S. Berko, Phys. Rev. Lett.
**18**, 420 (1967)ADSCrossRefGoogle Scholar

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