The simulations were performed using the general-purpose 3D Monte Carlo particle transport simulation tool MCNP6 . To estimate the fluences and spatial distributions of particles generated by TGFs at various altitudes, the transport of accelerated particles through the atmosphere was simulated. The calculations were conducted using the NRLMSISE-00 atmosphere model , with parameters set for 1 January 2022, at a longitude of 15° and three different latitudes: 0, 30, and 60° (representing tropical, temperate, and high-latitude regions, respectively). The model accounts for variations in air composition and density as a function of altitude.

The Monte Carlo model of the atmosphere was constructed up to an altitude of 50 km in discrete steps of 100 m, with different densities and compositions. The air nuclides considered in the model were ¹⁴N, ¹⁶O, and ⁴⁰Ar. Water content in the atmosphere was not considered in the calculations. The primary particles used as input in the model were electrons with an energy spectrum calculated by and can be seen in Fig. . This spectrum was derived based on the relativistic runaway electron avalanche theory. The point source emitted the accelerated electrons upward in a vertical direction. The source altitudes were 5 km for a latitude of 60° (high-latitude region), 10 km for a latitude of 30° (temperate region), and 15 km for latitude of 0° (tropical region) . The numbers of primary particles in these calculations were 2.78×109, 2.64×109, and 2.63×109, respectively. These values differ because each simulation was subject to varying compute time limits on a shared computing cluster, where jobs were queued and executed as resources became available. The particles produced from the interactions of primary electrons were scored by various tallies, which are constructs within the MCNP6 software used to record properties of the radiation field (such as fluence and flux) or the radiation itself (such as energy and direction). Properly defining these tallies in the simulation allows for the extraction of meaningful information. In this study, a cylindrical mesh tally of type 1 was used to evaluate fluences, while standard F2 tallies were employed to score the energy spectra of ionizing radiation particles. The results of fluences scored by cylindrical mesh tallies were normalized to 10¹⁷ particles of primary electrons (>1 MeV) to estimate the flux for a TGF. Many studies normalize results to the number of observed photons as they can be observed. Such normalization would also make comparisons with other studies more straightforward. However, we chose to normalize to primary electrons because the efficiency of bremsstrahlung production depends on altitude and chemical composition of the air . In denser air, electrons undergo more frequent bremsstrahlung interactions, leading to a higher probability of gamma ray emission compared to lower-density regions at higher altitudes. Therefore, normalizing to the number of primary electrons provides a more direct way to account for these altitude-dependent effects and to assess the TGF intensity. The results show fluences and the spatial distribution of electrons, photons, and neutrons.

MCNP6 incorporates a comprehensive set of libraries for simulating the interactions of electrons, photons, and neutrons with matter. These include major processes such as Møller scattering, elastic scattering, Compton scattering, pair production, bremsstrahlung, photoelectric absorption, and photonuclear reactions. These libraries provide detailed cross-section data for a wide range of interactions, enabling accurate simulations of electromagnetic radiation transport and particle interactions in the atmosphere. While MCNP6 has not been specifically benchmarked in the studies , it is a widely used and well-established tool for Monte Carlo simulations in various research fields. Given that it contains the necessary libraries and interactions, we assume it is an appropriate tool for this type of analysis. The absorption of neutrons in the atmosphere was considered in our analysis. Photonuclear reactions were modeled using established nuclear data libraries, specifically the 7-N-14 and 18-Ar-40 photonuclear reaction data from ENDF/B-VII.0, as well as the O-16 photonuclear data from the LA150 Photonuclear Data Library. These datasets provide cross-sections for photon-induced nuclear reactions relevant to neutron production in the atmosphere.

The energy spectra of particles were tallied during their passage through surfaces at different altitudes. The simulated cylindrical volume (40 km radius and 50 km altitude) was divided into smaller cylinders to allow the segmentation of the tally surfaces (with a segmentation step of 1 km in cylinder radius).

The cylindrical mesh tallies for scoring of particle fluences had a different division than the tally used for scoring the energies of particles. Mesh tallies had radii of 40 km, with steps of 40 m; altitude from 50 m up to 50 km, with steps of 50 m; and one angle, from 0 up to 360°.

The physics settings for the calculation were set such that the production of bremsstrahlung and photonuclear reactions was on. The energy cut-offs for electrons, photons, and neutrons were 10, 10, and 10⁻⁶ keV, respectively. The Earth's magnetic fields were not considered in the simulations.

The spatial distribution of photon fluences is depicted in Fig. . Figure  shows fluences for three different altitudes of TGF origin: 5, 10, and 15 km. It can be observed that the photon beam has the shape of a cone. The opening angle of the cones increases with the altitude of the TGF origin. When the altitude of the TGF origin is at 5 km, the fluence of at least 1 cm⁻² extends up to an altitude of nearly 25 km, with a radial spread of up to 6 km from the origin. For a TGF origin at 10 km, the irradiated cone rapidly elongates and widens; the 1 cm⁻² threshold region reaches up to an altitude above 50 km (beyond the simulation boundaries) with a maximum radial spread of 1 cm⁻² isofluence of nearly 20 km. In the case of a TGF at an altitude of 15 km, the region with a fluence higher than 1 cm⁻² stretches beyond the simulation boundaries of 50 km in altitude and a radial distance of over 40 km. At an altitude of 20–35 km (reachable by weather balloons), the radial distance of the detectability region exceeds 30 km. It can be seen that for narrow geometry the detectability region rapidly broadens at higher altitudes. Another implication is that TGFs originating at higher altitudes can be measured at larger distances. Therefore, platforms such as weather balloons which can climb up to 35 km are more suitable for high-altitude TGFs. It can be seen that the TGF altitude is a crucial parameter influencing the irradiated volume by the bremsstrahlung photons and the beam shape in the observed simulation boundaries. This has significant consequences for all measurements performed above the thunderstorms as it defines the detectability regions together with the brightness of the TGF and its energy spectrum.

Figure  shows a comparison of the obtained photon fluences with results published in . The comparison is not straightforward due to differences in simulation methodologies. In particular, directly simulated photon propagation, whereas the present study is based on simulations of electrons, where bremsstrahlung photons are produced as secondary particles. Since the number of generated photons per electron was not explicitly recorded in this work, a scaling factor is required for direct comparison. We refer to this scaling as the electron-to-photon conversion factor, which represents the average number of electrons required to produce one bremsstrahlung photon. Here, we assume that, on average, 10 electrons are needed to produce 1 bremsstrahlung photon. This value was taken from as an upper bound value for TGFs produced in strong electric fields. For weaker electric fields ∼ 284 kV m⁻¹ the authors estimated the value to be closer to one runaway electron to one bremsstrahlung photon, which would result in better agreement between the comparison.

Another key difference lies in the energy thresholds considered in the simulations. only accounted for photons with energies above 300 keV, while the present study includes photons down to 10 keV. This distinction leads to differences in the total fluence values, as lower-energy photons contribute significantly to the overall photon population. Additionally, the beaming geometries differ: the present study employs a narrow beam geometry, whereas used a wider beam. These factors collectively influence the observed fluence distributions and must be taken into account when comparing results.

The photon energy spectra were recovered for various radial distances from the TGF origin. Figure  shows the energy spectra of photons at three different altitudes – 25, 20, and 15 km, – as a function of radial distance from the axis of the TGF originating at an altitude of 10 km. It can be seen that for various radial distances, the ratio between low-energy (<100 KeV) and high-energy (>1 MeV) components changes. The high-energy component is more substantial near the axis of the TGF. It can also be observed that the maximum energy of photons decreases as the radial distance increases. Hence, there is a softening of the gamma ray energy spectra at the edges of the irradiated cones. This effect has been observed by and . This effect is especially pronounced at higher altitudes. To better visualize and analyze the spectral hardness, Fig.  shows the normalized energy spectra of photons (normalization to its maximum). By normalizing the spectra, the change in the ratio between low-energy (<100 keV) and high-energy (>1 MeV) components becomes more apparent. This normalization highlights the softening of the gamma ray energy spectra at the edges of the irradiated cones. The calculated figures of merit, defined as the ratio of low-energy to high-energy gamma rays, are 34.9, 50.95, 88.39, 237.74, and 1649.85 for radial distance ranges of 0–2, 2–5, 5–10, 10–20, and 20–40 km, respectively, indicating progressively softer spectra with increasing distance. Notably, immediately above the TGF, a distinct sharp edge appears in the energy spectrum (approximately at 50 MeV), which corresponds to the maximum energy of primary electrons. However, at a higher radial distance from the center of the TGF, this sharp edge moves to lower energies. The visible peak between 497–599 keV is caused by 511 keV photons from the annihilation of positrons. The positrons are created mainly by the pair production of high-energy photons. The energy interval of 497–599 keV corresponds to the bin size of the tally used to score the energy spectrum; the true energy of the annihilation photons is 511 keV. The 511 keV peak is more pronounced at higher altitudes and at distances closer to the axis of the TGF. To quantify the annihilation peak we calculated the equivalent width of the peak for a detection altitude of 25 km and an altitude of TGF origin of 10 km. The resulting equivalent widths are 0.16, 0.15, 0.13, 0.11, and 0.08 MeV for radial distances of 0–2, 2–5, 5–10, 10–20, and 20–40 km, respectively. The equivalent width is a measure used to quantify the strength of a spectral line or peak. It represents the width of a feature in a spectrum, normalized by the intensity of the peak itself.

These features of the photon energy spectrum (softening of the beam at the edges and the presence of a 511 keV peak) could be used for estimating the radial distance to the TGF axis. This method may have limited practical use because it requires detecting a large number of photons, which is especially challenging for small detectors.

The spatial distribution of electron fluences is depicted in Fig. . Figure  shows fluences for three different altitudes of TGF origin: 5, 10, and 15 km. The spatial distribution of electrons is similar to the photon distribution, as most of the electrons are created by interactions of gamma rays with atoms in the atmosphere. Nevertheless, electron fluences have some specific characteristics. One of these characteristics of the electron fluence distribution is the presence of long tracks of individual high-energy electrons that are created by interactions of gamma rays with the atmosphere. These tracks are visible in the high-altitude section (a), showing TGFs with an origin at 5 km. Such electrons have large energies and can travel significant distances in the air. Another characteristic is that the largest fluences are located in smaller and narrower cones than those observed for gamma rays. To highlight these differences, the profiles of photons and electrons and the electron-to-photon ratio are plotted in Fig.  for two fixed altitudes. It can be seen that as the viewer moves further away from the central axis of the TGF the number of electrons to photons quickly decreases and then stabilizes at a value near zero.

The energy spectra of electrons caused by a TGF at different radial distances are shown in Fig. . Similar to photons, the high-energy part of the electron energy spectrum is focused mainly around the center of the TGF, whereas on the edges of the cone, the energy spectrum softens. Although the overall number of electrons decreases as they move to higher altitudes, the shapes of the energy spectra do not undergo significant changes and remain relatively stable.

Figure  also presents a comparison between our simulated electron energy spectra at an altitude of 50 km (TGF altitude of 15 km) and the spectra of escaped electrons from the Earth's atmosphere performed by . To enable a meaningful comparison, our spectrum was normalized such that its integral matches that of Dwyer's within the energy range covered by their data. Because the energy axis is logarithmic, the counts were first multiplied by the corresponding bin widths before integration. Although our spectrum extends to slightly lower energies, this extended portion was excluded from the normalization integral but retained in the final plot with the same normalization factor applied. The results show good agreement, indicating that our simulation reliably captures the energy distribution of scattered electrons. This agreement also serves as a validation of our approach, as the simulations of were conducted under similar conditions, particularly in terms of the beaming geometry.

The spatial distribution of neutron fluences is depicted in Fig. . Figure  shows fluences for three different altitudes of TGF origin: 5, 10, and 15 km. Neutrons are generated by photonuclear reactions of high-energy gamma rays with atoms of the atmosphere. These reactions have low cross-sections; therefore, the resulting neutron fluences are lower than those of gamma rays or electrons. Moreover, the regions with sufficient neutron fluences (at least 1 cm⁻²) are relatively small compared to those of electrons and photons. For these reasons, photoneutrons are likely to be missed, or their contribution in the detector would be insignificant. Nevertheless, the absence of detected neutrons can still be valuable for estimating the minimal distance between the radiation source and the detector.

The energy spectra for two different radii are shown in Fig. . It can be seen that neutron energy spectra are very insensitive to the radial distance from the axis of the TGF and altitude where it is scored. also report that the neutron energy spectra produced in TGFs are largely composed of thermal neutrons, with only a small fraction of high-energy neutrons. Additionally, they find that the neutron energy spectra are insensitive to the altitude at which they were tallied, echoing our results where neutron energy spectra do not show significant dependence on altitude. However, the low statistics cause the energy spectra to exhibit noise.

It is important to note that while the spatial and energy distributions of neutrons have been analyzed in this study, the timing of neutron production and their arrival at detectors was not included in the simulations. As such, the temporal characteristics of neutron emissions such as their reduction in speed in the atmosphere or the time distribution of their arrival are not addressed here. Future studies could incorporate timing analysis to explore these factors, which would provide a more complete understanding of the neutron dynamics in relation to TGF events.

There are other studies that are related to the current work, but due to key differences in focus and methodology, direct comparison is not feasible. examine afterglows produced by neutrons some time after the TGF event. However, the present work only considers neutrons generated during the TGF itself and does not include time scoring, making this type of afterglow analysis outside the scope of the current research. present simulations of downward TGFs and measures of neutron spectra at ground level, which includes thermalization of neutrons upon reaching the Earth's surface. In contrast, the present work focuses on upward TGFs and neutron propagation at altitudes relevant to balloon and aircraft measurements. Therefore, any direct comparison would be difficult to justify.

The obvious option is to use airplanes, which are easily maneuverable, can stay airborne for a long period, and can reach relatively high altitudes of 12–13 km. Since the cumulonimbus clouds at the top of the thunderstorm in tropical regions are around 18 km, this might not always be sufficient to fly above the thunderstorm. However, it could be adequate for temperate and high-latitude regions where the tops of the clouds are at altitudes of 13 km and 8 km, respectively . Another possibility for using aircraft is during winter thunderstorms, when thunderstorms typically reach altitudes of only several kilometers in temperate regions . The advantage of winter thunderstorms is also the possibility of performing various mutual radiation measurements on the ground . The drawback of using aircraft for observing winter thunderstorms is that the detectors need to be located relatively close radially to the axis of the TGF, as shown in Figs. , , and . Hence, the TGF might miss the detectors even when the aircraft is in the vicinity of an active thunderstorm. Additionally, there are radiation risks to the aircraft and crew, which might be significant, particularly in the immediate vicinity (400 m) of the avalanche region, as noted by . However, outside of this region, the radiation risk is considerably lower. It is also important to note that pilots generally avoid flying directly over low-altitude thunderstorms due to the potential for turbulence and other safety concerns, as this presents a greater risk to flight operations. To avoid the most severe radiation effects of the TGFs, the plane should always stay out of reach of the primarily accelerated electrons .

The potential of using planes for TGF research was demonstrated by the ALOFT campaign, which measured a large number of TGFs when the NASA-owned ER-2 aircraft, flying at an altitude of approximately 20 km, was positioned above thunderstorms. Another option is to use weather balloons as carriers of detectors. Weather balloons are usually filled with helium and can ascend up to 35 km, but they cannot be maneuvered, their payload mass is limited, and their use is usually strictly regulated by law. For these reasons, it would likely be difficult to use weather balloons for tracking TGFs above thunderstorms. Nevertheless, measurements with weather balloons would have the advantage of being able to fly above the cloud tops of even the tallest thunderstorms (summer thunderstorms in temperate and tropical regions). Hence, TGFs could be detected at larger radial distances from the TGF axis than for TGFs originating at lower altitudes, as is shown in Figs. , , and . When weather balloons are launched, they drift with the wind. Although there are online tools for predicting the weather balloon trajectory, it would be very challenging to time the launch so that it flies over a fast-moving, active thunderstorm cell. This would require in-depth experience and expertise in meteorology, weather forecasting, and weather balloon launching. The last suggested option is the use of high-altitude platform stations (HAPSs), which could combine the advantages of aircraft and weather balloons, as some HAPS concepts offer the capability to ascend to altitudes of 20 km while also providing maneuverability and a large payload capacity . The disadvantage of HAPSs is that they are not currently widely available.

It was shown that the fluxes and energies of different particle types vary significantly based on the location of the TGF origin, its brightness, and the measurement location. Other variables influencing the fluxes and particle energies include the energy spectrum of the primary electrons and the tilt of the TGF. We have only considered TGFs with an upward direction, one energy spectrum, and one brightness. These variables also change, as described above, and therefore, the calculated particle fluxes originating from TGFs should be considered to be estimates. Nevertheless, these estimates should be sufficient to define the properties of ionizing radiation detectors for TGF detection.

The most significant component of the TGF is bremsstrahlung, which has the largest reach and is relatively easy to detect with various ionizing radiation detectors. The most commonly used detectors are scintillators because of their large volumes and high density. Reasonably expected fluences of gamma rays might reach values up to 1000 cm⁻² per TGF (Fig. ) for the simulated conditions, whereas the background fluence of cosmic ray photons is around 30 cm⁻² s⁻¹ at an altitude of 20 km and in central Europe. Since the TGF consists of sub-millisecond pulses, during which a large number of particles are created, the TGF signal can exceed the background by several folds. Sometimes, the bursts are so intense that scintillator detectors become saturated as the decay of the signal takes hundreds of nanoseconds. In such cases, it is not possible to distinguish individual gamma rays, and information about gamma ray energy and intensity is either lost or diminished. This can be mitigated by using scintillators with faster signal decay times and lower light yield, decreasing the size of the scintillator, or using shielding of the detection element. In general, such precautions reduce the sensitivity of the detector to gamma rays, enabling it to measure high-intensity TGFs. On the other hand, insensitive detectors would not detect TGFs that are farther away or have different energy spectra. Therefore, a combination of gamma ray detectors with various sensitivities would be an effective approach to ensure that energy and intensity information can be recovered from the measurements, as demonstrated in previous TGF detection campaigns such as ADELE and ALOFT , where multi-detector setups were used to mitigate saturation effects and improve spectral reconstruction. Moreover, the energy spectrum can be estimated by analyzing the response of several detectors with different sensitivities to different energies. Such a TGF detector assembly could be similar to a detector used for spectrometry of pulsed radiation fields generated by high-power lasers .

The second particle type of interest is electrons, which are likely to be detected in the presence of stronger gamma radiation fields. Reasonably expected fluences of electrons from TGFs are up to 100 cm⁻² for the simulated conditions, while the background fluence of electrons and positrons is around 2.5 cm⁻² s⁻¹. For electron detection, a detector with a smaller volume can be used, as electrons directly interact with matter by ionizing it. Therefore, pixel or strip semiconductor detectors or diode-based detectors are suitable because their detection efficiency is much higher than for gamma rays. The advantage of such detectors is that they can be used in a telescopic configuration (multiple diodes), allowing estimation of the incident angle of the electrons if they traverse multiple layers of semiconductor. However, this method is primarily effective for high-energy electrons, which constitute only a small fraction of the total electron population. If the diodes have a sufficient volume, the telescope can also function as a calorimeter and provide information about the energy of incident electrons. A limiting factor for semiconductor diodes might be their size, as scaling diode volume is not straightforward. Therefore, detecting low fluxes of electrons with diodes could be challenging and might require an alternative approach, such as using large-surface scintillators. In a calorimetric configuration, the width of diodes would also be a limiting factor. The energy spectrum of electrons ranges from several kiloelectronvolts up to dozens of megaelectronvolts, which would require several millimeters of semiconductor material to fully stop such energetic electrons. If such detectors are placed on an aircraft, the aircraft skin would likely stop the electrons. Therefore, it might be reasonable to use telescopes in weather balloons instead, since the gondolas of such balloons are typically made of light-weight, low-density insulators (e.g., polystyrene), which would not decelerate the electrons significantly. The gondolas can also be designed so that there is a minimum amount of material to shield the electron detectors.

Another component of interest is neutrons. The production of neutrons by photonuclear reactions is inefficient, and the neutron fluxes are relatively low, even in the proximity of the TGF origin (Fig. ). Moreover, the cross-sections of neutron interactions with matter are relatively low, so neutron detectors need to be large to obtain useful information about TGFs. This makes them suitable only for platforms that can carry large payloads, such as aircraft. The neutron energies range from millielectronvolts up to megaelectronvolts, as shown in Fig. . Hence, TGF neutron detectors should measure over a very broad range of energies. A key challenge for neutron detectors is selectivity, as they are immersed in a much larger photon flux. Therefore, these detectors must be significantly more sensitive to neutrons than to photons, or the detectors must differentiate between photons, electrons, and neutrons. One of the possible methods is to use pulse shape discrimination, as was demonstrated in .