1]Moscow Institute of Physics and Technology - 1 “A” Kerchenskaya st., Moscow,
117303, Russian Federation2]HSE University - 20 Myasnitskaya ulitsa, Moscow
101000 Russia

# Simple reactor model of relativistic runaway electron avalanche development

Egor Stadnichuk  [  [<EMAIL_ADDRESS>Moscow Institute of
Physics and Technology - 1 “A” Kerchenskaya st., Moscow, 117303, Russian
Federation
HSE University - 20 Myasnitskaya ulitsa, Moscow 101000 Russia  Daria
Zemlianskaya<EMAIL_ADDRESS>Moscow Institute of Physics and
Technology - 1 “A” Kerchenskaya st., Moscow, 117303, Russian Federation
Institute for Nuclear Research of RAS - prospekt 60-letiya Oktyabrya 7a,
Moscow 117312  Ekaterina Svechnikova<EMAIL_ADDRESS>Institute of
Applied Physics of RAS - 46 Ul’yanov str., 603950, Nizhny Novgorod, Russia
Eduard Kim  Moscow Institute of Physics and Technology - 1 “A” Kerchenskaya
st., Moscow, 117303, Russian Federation
Institute for Nuclear Research of RAS - prospekt 60-letiya Oktyabrya 7a,
Moscow 117312<EMAIL_ADDRESS>Alexander Sedelnikov  Moscow Institute of
Physics and Technology - 1 “A” Kerchenskaya st., Moscow, 117303, Russian
Federation
Lebedev Physical Institute RAS<EMAIL_ADDRESS>Oraz Anuaruly
Moscow Institute of Physics and Technology - 1 “A” Kerchenskaya st., Moscow,
117303, Russian Federation
Kurchatov Institute, Russian Research Centre - sq. Academician Kurchatov, 1,
Moscow, 123098, Russian Federation<EMAIL_ADDRESS>
###### Abstract

High-energy gamma radiation in the Earth’s atmosphere is associated with the
bremsstrahlung of Relativistic Runaway Electron Avalanches (RREA) developing
in thunderstorm electric fields. In this paper, RREA development is studied in
the system of two strong electric-field regions within thunderstorms, which
accelerate runaway electrons toward each other. Such a system is called the
simple reactor. It is discovered that the propagation of gamma rays and
runaway electrons from one region to another leads to positive feedback. This
feedback called the reactor feedback can make RREA self-sustaining, thus
effectively multiplying high-energy particles inside thunderstorms containing
the simple reactor. The spectrum and characteristic time scale of the simple
reactor gamma radiation are in agreement with Terrestrial Gamma-ray Flashes
(TGFs) data. The applicability of the simple reactor model to TGF is
discussed, and the distinguishing observable properties of the simple reactor
radiation during TGF and Thunderstorm Ground Enhancement are considered.

††preprint: APS/123-QED

##  I Keypoints

  * •

RREA development in thunderstorms containing the simple reactor is studied

  * •

Two feedback mechanisms that can make RREA self-sustaining in the simple
reactor are discovered: electron and gamma-ray reactor feedback

  * •

Characteristics of gamma radiation of the simple reactor are in agreement with
TGF and TGE experimental data

##  II Introduction

Atmospheric physics is rich in mysterious natural phenomena. One of the new
directions in atmospheric research is high-energy atmospheric physics. It
suddenly appeared in 1992, when the Burst and Transient Source Experiment
(BATSE) detector aboard the Compton Gamma-Ray Observatory experiment
discovered short and intensive bursts of gamma rays originating in the
atmosphere of Earth [1]. These energetic bursts are called Terrestrial Gamma-
ray Flashes (TGFs). It is established that the source of TGFs are
thunderstorms [2]. The characteristic duration of a TGF is 100 $\mu$s [3],
energy of detected TGF gamma-rays is up to 40 MeV [4, 5]. Thunderstorm gamma
radiation is also detected on the Earth’s surface. It is called Thunderstorm
Ground Enhancement (TGE) [6] or gamma-ray glows [7], and its characteristic
duration is up to tens of minutes. It is important to note that high-energy
processes within thunderstorms are closely related to lightning. TGE precede
lightning and are always terminated by lightning discharges [8]. TGFs are
established to occur at the early stage of the lightning leader propagation
[9, 10, 11]. Moreover, many other interesting bright phenomena were registered
in connection with the high-energy radiation from thunderstorms [12, 13, 14].

The underlying physics of high-energy atmospheric radiation is the
acceleration of electrons in thunderstorm electric fields [15, 2, 16, 17, 18,
19]. In strong thunderstorm electric fields, relativistic electrons obtain
more energy from the acceleration in the electric field than they on average
lose on interactions with atmosphere air molecules. Such electrons are called
runaway electrons [20]. When the electric field strength exceeds the critical
value, $E_{c}=276\frac{kV}{m\cdot atm}$, runaway electrons are Møller
scattered by air molecules, which leads to the appearance of additional
runaway electrons [21]. In this way, runaway electrons multiply in the process
of their propagation along the thunderstorm electric field, forming the
Relativistic Runaway Electron Avalanche (RREA) [16, 17]. To start a RREA an
initial seed energetic particle is needed to appear within the thunderstorm
electric field [16, 22]. For example, it can be a secondary cosmic ray
particle [23] or a seed particle generated inside the thunderstorm [24, 25,
26]. Characteristic RREA particle energies range from tens of keV to tens of
MeV [16]. Thus, runaway electrons naturally produce bremsstrahlung gamma rays
in collisions with air molecules, which is detected as TGF and TGE [17, 4, 10,
8, 22, 7].

The mystery of TGFs is that a large number of high-energy particles appear
almost instantly inside a thundercloud [1, 5, 25]. There are two possible ways
to explain such phenomena. The first possible scenario is the generation of a
large number of seed electrons within the thunderstorm super strong electric
fields, possibly created by the lightning leader propagation [18, 24, 25, 27,
28, 29]. Also it is considered that lightning leader itself can radiate
synchrotron gamma-rays [30]. These ideas are supported by the fact that x-rays
are observed in association with lightning leader propagation [31, 32]. The
second possible scenario is the multiplication of RREAs by positive feedback
mechanisms [21, 33, 34, 35, 26, 36, 37]. The relativistic feedback works in
the following way. Bremsstrahlung gamma ray radiated by runaway electrons can
produce electron-positron pairs within thunderstorm supercritical electric
field region. Positrons are accelerated by the electric field in the direction
opposite to the runaway electrons acceleration direction. In this way,
positrons reach the beginning of the supercritical region, where they produce
seed runaway electrons by the Bhabha scattering [33]. Thus, relativistic
positron feedback multiplies RREAs and, moreover, can make RREA self-
sustaining [21]. Similarly bremsstrahlung gamma rays can Compton backscatter
and thus produce seed runaway electrons at the beginning of the supercritical
region, which is the relativistic gamma ray feedback [38]. RREA models based
on the positive feedback are supported by the fact that their characteristic
time and spectrum coincide with the characteristic time and spectrum of TGF
[25, 2, 4, 3]. Nevertheless, the relativistic feedback requires strong large-
scale electric fields, which have never been directly experimentally observed
in thunderstorms [34, 39, 36, 33].

It has been discovered that non-uniform thunderstorm electric field geometry
leads to another feedback mechanism called the reactor feedback [26, 40]. Let
a thunderstorm consist of several separate electric-field regions with
electric-field strength sufficient for RREA production. Such regions, for
simplicity, are called cells [26]. If a seed electron starts a RREA within one
of the cells, the following processes occur. A RREA radiates bremsstrahlung
gamma-rays. Gamma-rays have a large attenuation length at thunderstorm
altitudes. Therefore, gamma-ray photons propagate through the thunderstorm and
can reach other cells. There is a probability that a gamma-ray photon will
interact with air molecules by compton scattering, photoelectric effect of
electron-positron pair production within a cell, which can result in runaway
electron generation. A runaway electron can produce a RREA. In this way the
reactor gamma-ray feedback works: separate thunderstorm cells irradiate each
other with gamma radiation, which results in RREA multiplication. Another
reactor feedback mechanism - runaway electron transport between cells. If the
cells are close to each other, runaway electrons are able to penetrate the air
layer between them. In this way, runaway electrons propagate from one cell to
another and, thus, multiply RREA. In general, reactor feedback is defined as
the multiplication of RREA by high-energy particle exchange between separate
thunderstorm RREA-accelerating regions. Distant cells amplify each other
mostly by gamma-ray photon exchange because of their high penetrating power in
the air. For cells located close to each other, the reactor feedback works
mainly by runaway electron exchange, since RREA consists mainly of runaway
electrons. It has been established that reactor feedback can lead to self-
sustaining development of RREA, and, moreover, requires lower electric field
strength in comparison with the relativistic feedback [26].

In this paper, the reactor feedback is studied in the simplest case of non-
uniform thunderstorm electric field, when thunderstorm consists of two cells,
oriented in the way that they accelerate runaway electrons towards each other.
The system is called the simple reactor. The research is motivated by
observations of thunderstorm electric structures, part of which can be
described as the simple reactor electric structure [41, 42, 43, 44, 45, 46,
47]. In this paper, it has been discovered that both gamma-ray reactor
feedback and runaway electron transport feedback amplify RREA in the simple
reactor, and these feedback mechanisms require a smaller electric field
strength to provide self-sustaining RREA development in comparison with the
relativistic feedback. In Section III the feedback mechanisms of the simple
reactor are described. In Section IV the reactor feedback is theoretically
described. Section V provides Monte Carlo simulations of the simple reactor
using GEANT4. In Section VI the simple reactor is discussed as a possible
mechanism for TGF and TGE, the distinguishing properties of this model that
are experimentally observable are considered.

##  III Simple reactor model

The reactor feedback is the intensification of RREA development in a
thunderstorm supercritical electric field region (cell) by the radiation of
other cells [26]. The simplest system capable of demonstrating the reactor
feedback is the system of two cells with oppositely directed electric fields
accelerating electrons toward each other. This system is called “the simple
reactor” and we consider it as the next step from uniform electric field
models to the description of RREA in the electric field of a real
thundercloud. In the simple reactor, the distribution of the electric field
corresponds to the system of two flat capacitors placed one on top of the
other. It can be considered as the approximation of the electric field
distribution in the region of the cloud with three charge layers: a positive
middle layer and two negative layers. The described electric field
distribution can be a part of a natural thunderstorm [41, 42, 43, 44, 45, 46,
47].

There are two reactor feedback mechanisms in the simple reactor. The first
mechanism, reactor gamma-ray feedback, works in the following way. Let a seed
electron form a RREA within one of the cells. RREA grows towards the opposite
cell and radiates bremsstrahlung gamma rays [16]. Gamma rays have a
significant penetration power through the atmosphere at thunderstorm
altitudes. Thus, gamma rays reach the opposite cell and propagate through it.
Interactions of gamma rays with air molecules of the opposite cell generate
seed runaway electrons. These electrons start a new RREA in the opposite cell.
Further, the new RREA propagates towards the initial cell, generating gamma
rays, which similarly produce RREA in the initial cell. In this way, the
process loops, and thus gamma-ray reactor feedback makes RREA self-sustaining
in the simple reactor. The second mechanism, runaway electron transport
feedback, works in the case when cells are close to each other. In this case,
runaway electrons can penetrate the gap between cells, reaching the opposite
cell. When a runaway electron reaches the opposite cell, it penetrates inside
the cell until the electric field reverses it. After reversal runaway
electrons are accelerated toward the initial cell. Thus, in the simple
reactor, runaway electrons oscillate near the border of the cells. During the
oscillation, runaway electrons are multiplied by the Møller scattering, which
also leads to a self-sustaining process. It is established in this paper, that
both feedback mechanisms not only can make RREA self-sustaining but also can
significantly multiply the number of relativistic particles in the
thunderstorm containing the simple reactor (Figure 1). It should be noted that
the relativistic feedback naturally impacts RREA development in the simple
reactor, however, further it is shown that the influence of the relativistic
feedback is negligible compared to the influence of the reactor feedback.

Figure 1: The physics of the simple reactor in the GEANT4 simulation [48].
Green lines - gamma-ray photon tracks, red lines - runaway electron tracks.
Blue arrows - electric field lines, yellow dots - particle interaction points.
The simulation is started with a single seed electron. The simple reactor
consists of two supercritical electric field regions accelerating runaway
electrons towards each other. High-energy particles exchange between these
regions makes the process self-sustaining. The picture resembles the Eye of
Sauron from the Lord of the Rings trilogy: runaway electrons oscillating near
the cell boundary form a pupil, the halo of gamma-ray photons and RREAs formed
by gamma-ray reactor feedback resembles the cornea of the eye.

##  IV Analytical simple reactor model

###  IV.1 Gamma-ray reactor feedback

To describe the simple reactor gamma-ray feedback theoretically, it is
necessary to study the response of a cell to gamma radiation falling into it
[26]. Let $N$ gamma-ray photons enter the $i$-th cell ($i=$1,2) from above,
along the cell’s electric field vector. In order to find the feedback
coefficient, it should be calculated how much gamma will fly back from the
cell towards the other cell. Let $\lambda_{RREA}^{i}$ be the growth length of
an avalanche of runaway electrons, $\lambda_{-}^{i}$ be the decay length of
gamma radiation, $\lambda_{\gamma}^{i}$ be the path of a runaway electron
before the emission of a gamma-ray photon with supercritical energy,
$\lambda_{e^{-}}^{i}$ be the path length of a gamma before the birth of a
runaway electron, and $P^{i}$ is the probability of a turn of an electron with
further development of the runaway avalanche. These parameters depend on the
magnitude of the electric field and the density of the air. In the first
approximation, the values for these parameters can be retrieved from the
article [34]. In general, the cells of the simple reactor are assumed to have
different field strengths, air density, and cell lengths.

The gamma entering the cell along the electric field will generate electrons
with supercritical energy, while losing its energy, which leads to an
exponential decay of the primary gamma flux. On the segment [z, z + dz], the
flown gammas will give birth to the following number of avalanches of runaway
electrons:

$\frac{df_{e^{-}}^{i}(z)}{dz}dz=NP^{i}e^{-\frac{z}{\lambda_{-}^{i}}}\frac{dz}{\lambda_{e^{-}}^{i}}$
(1)

The dynamics of the number of bremsstrahlung gamma-ray photons during RREA
propagation along the z axis is described by the following equation [34]:

$dN_{\gamma}=e^{\frac{z-z_{0}}{\lambda_{RREA}}}\frac{dz}{\lambda_{\gamma}}-N_{\gamma}\frac{dz}{\lambda_{-}}$
(2)

The first term in 2 describes the production of bremsstrahlung gamma-ray
photons by runaway electrons, and the second term describes a decrease in the
number of gamma-rays due to its interaction with air molecules. The solution
for 2 is 3:

$N_{\gamma}(z,z_{0})=\frac{\lambda_{RREA}\lambda_{-}}{\lambda_{\gamma}(\lambda_{-}+\lambda_{RREA})}\cdot\left(e^{\frac{z-z_{0}}{\lambda_{RREA}}}-1\right)$
(3)

Each RREA grows according to the well-known exponential law [20], spreading
toward the initial gamma rays entering the plane. In this case, depending on
the point of birth of the avalanche, the amount of secondary gamma rays that
will reach the end of the cell will be as follows (since the avalanche born at
the point z will travel a distance equal to z):

dF_γ^i(z) = dfe-i(z)dz dz λRREAλ-λγ(λ-\+ λRREA) ⋅(e^zλRREA \- 1)

(4)

Thus, the gamma-ray local multiplication factor [26] can be calculated
(Formula 5):

$\nu^{i}=\frac{\int_{0}^{L^{i}}\frac{dF_{\gamma}^{i}(z)}{dz}dz}{N}$ (5)

Integration leads to the following formula for the gamma-ray multiplication
factor:

$\begin{split}\nu^{i}=\frac{P^{i}\lambda_{RREA}^{i}\lambda_{-}^{i}}{\lambda_{e^{-}}^{i}\lambda_{\gamma}^{i}(\lambda_{RREA}^{i}+\lambda_{-}^{i})}\\\
\left[\frac{\lambda_{RREA}^{i}\lambda_{-}^{i}}{\lambda_{-}^{i}-\lambda_{RREA}^{i}}\left(e^{L^{i}\frac{\lambda_{-}^{i}-\lambda_{RREA}^{i}}{\lambda_{RREA}^{i}\lambda_{-}^{i}}}-1\right)-\lambda_{-}^{i}\left(1-e^{-\frac{L^{i}}{\lambda_{-}^{i}}}\right)\right]\end{split}$
(6)

The system with positive feedback can be characterized by the feedback
coefficient [33, 26, 34]. For the simple reactor, the feedback coefficient
shows how many times the number of high-energy particles will increase in one
full reactor feedback cycle, and it is found with the following formula:

$\Gamma=\nu^{1}\cdot\nu^{2}$ (7)

The number of particles in the simple reactor grows exponentially with each
feedback generation: $N(n)=\Gamma^{n}$, where n is the number of feedback
generation [26]. Therefore, the criterion for self-sustaining RREA development
in a simple reactor is as follows:

$\Gamma=\nu^{1}\cdot\nu^{2}\geq 1$ (8)

With the obtained criterion, the thunderstorm conditions necessary for self-
sustaining RREA development by the reactor gamma-ray feedback can be
calculated. These conditions are presented in Figure 2. Conditions are
presented for 3 types of feedback: relativistic positron feedback [21], simple
reactor feedback, and multicell reactor feedback [26]. The coordinates in the
figure are chosen so that the conditions are invariant with respect to
altitude [33, 34]. It can be seen from the figure that both the multicell and
the simple reactor feedback mechanisms require significantly lower electric
field strength in comparison to the relativistic feedback discharge model.
This important property of the reactor feedback comes with a price of the
thunderstorm electric field geometry complexity. The most complex electric
field geometry, the multicell reactor, requires the lowest electric field
strength for the self-sustaining RREA development, while the simplest reactor
structure, the simple reactor, requires the electric field strength lying in
between the multicell reactor and the uniform electric field. It should be
noted that if thundercloud electric field parameters lie above the curve in
Figure 2 then the number of energetic particles within the thundercloud grows
exponentially [33, 26, 34]. Otherwise, provided that there is no external
source of seed particles, the number of energetic particles decays, and the
decay rate depends on the feedback coefficient [26]. Thus, even if the
feedback does not make the RREA development self-sustaining, it still
increases its duration.

Figure 2: The comparison of self-sustaining positron feedback necessary
conditions [34] and simple reactor self-sustaining feedback necessary
conditions (Formula 6, 8). RREA accelerating region length is normalized to
$\lambda_{RREA}$, electric field strength is normalized to critical electric
field strength (the electric field required for RREA development [16]). The
simple reactor is also compared with necessary conditions for self-sustaining
gamma-ray feedback in multicell reactor model [26]. It can be seen that
reactor models require significantly lower thunderstorm electric field
strengths for self-sustaining RREA development than the relativistic feedback.
This important property of the model comes at expense of the complexity of the
electric field geometry. The more complex electric field geometry is, the
lower electric field strength is required for the feedback to be effective. It
should be noted that the conditions for the reactor models are presented
without taking into account runaway electron transport between cells.
Moreover, it has been discovered that thundercloud hydrometeors can amplify
RREA [49]. Thus, the exact self-sustaining RREA conditions can be lower than
the conditions presented on this picture.

###  IV.2 Runaway electron transport between cells in the simple reactor

GEANT4 simulations of the simple reactor showed the importance of runaway
electron transport between cells for the reactor feedback (Figure 1). In this
section, it is shown that the oscillations of runaway electrons between cells
in the simple reactor can become self-sustaining and even lead to runaway
electron multiplication. At first glance, this effect may seem paradoxical and
contrary to the law of energy conservation: While a single runaway electron
move from one cell to another, the total energy it receive from the electric
field in the full circle of its oscillation is zero, and, thus, this runaway
electron, on average, loses energy in interaction with air. Therefore, a
single runaway electron will inevitably lose its energy and stop. However, it
is seen in the simulations that runaway electrons oscillate and multiply in
the strong electric field of the simple reactor. Therefore, the following
question arises: Where do runaway electrons take energy when the feedback
becomes self-sustaining? Moreover, the total length of runaway electron motion
between cells back and forth cannot be longer than its energy divided by
$eE_{c}$, where $E_{c}$ \- critical electric field, e — elementary electric
charge. This is not more than several tens of meters.

It turns out that the effect of runaway electron transport between cells can
be physically explained and that the energy conservation paradox is resolved
by runaway electron multiplication. If a runaway electron multiplies by Møller
scattering [21], the result is that the initial and generated runaway
electrons receive twice as much energy from the electric field compared to the
single initial runaway electron. When the initial runaway electron stops, the
secondary electron continues to oscillate and multiply. The reactor feedback
in the simple reactor caused by runaway electrons can even become self-
sustaining. If the thunderstorm electric field is much stronger than the
critical electric field, the runaway electron interaction with the air becomes
negligible. Moreover, by the interaction with air, runaway electrons will
multiply, which leads to an enormous growth of the number of relativistic
particles. Thus, when the electric field strength decreases to values
comparable to $E_{c}$, there is a point where the multiplication of runaway
electrons compensates for the energy losses in the air interaction. At this
point, the runaway electron transport feedback becomes self-sustaining.

An interesting property of the runaway electron transport feedback is its
spatial scale. A runaway electron after hitting an adjacent cell cannot
propagate within the cell deeper than its kinetic energy divided by
$e(E+E_{c})$, where E is the electric field strength of the cell. Therefore,
runaway electron transport feedback coefficient depends only on the electric
field strength for cell lengths longer than runaway electron maximum energy
divided by $e(E+E_{c})$, which is about 100 meters for 10 km altitude and 40
MeV maximum energy [15, 50]. Thus, runaway electron transport occurs near the
cell interface, which softens the conditions required for self-sustaining RREA
development in the simple reactor, because long cells are not needed as in
other types of feedback 2. Nevertheless, it should be noted that runaway
electron feedback works effectively only for cells located close to each other
since electrons are quickly absorbed by air unless they are accelerated by the
electric field.

To theoretically analyze the runaway electron transport feedback, it should
first be understood how runaway electrons are decelerated in the electric
field of the adjustment cell. Decelerated runaway electron attenuation length
can be found by substituting negative electric field strength into the
empirical formula for the RREA e-folding length:

$\lambda_{decay}=\frac{7300[kV]}{-E-E_{c}}$ (9)

In this way, number of runaway electrons in the beam will decrease
exponentially:

$N_{beam}(z)=N_{0}e^{\frac{z}{\lambda_{decay}}}$ (10)

This analytic continuation of the RREA growth law [21] can be justified in the
following way. Normalized runaway electron spectrum can be described with the
function:

$\frac{df_{RREA}}{d\varepsilon}=\frac{1}{\varepsilon_{0}}e^{-\frac{\varepsilon}{\varepsilon_{0}}}$
(11)

$\varepsilon_{0}=7.3$ MeV - runaway electron mean energy [16]. An electron
with energy $\varepsilon$, on average, stops at the coordinate:

$z(\varepsilon)=\frac{\varepsilon}{e(E+E_{c})}$ (12)

Number of runaway electrons leaving the beam in the interval $(z,z+dz)$ per
one primary electron is:

$\frac{N_{beam}}{dz}dz=-N_{0}\frac{df_{RREA}}{d\varepsilon}\frac{d\varepsilon}{dz}dz=N_{0}\frac{E+E_{c}}{7300[kV]}e^{-\frac{E+E_{c}}{7300[kV]}z}$
(13)

Thus, the formula 9 is obtained.

The runaway electron transport feedback coefficient can be defined as the
number of runway electrons leaving the cell per one runaway electron entering
the cell (analogically to the gamma-ray reactor feedback). The number of
runaway electrons, which entered the cell, decreases according to the
exponential law derived above as these runaway electrons propagate into the
cell (formula 9). When a runaway electron leaves this beam it can stop or it
can reverse and form a RREA, which then propagates to the entry plane of the
cell. If a RREA starts at the point z, the number of runaway electrons within
this RREA reaches $e^{\frac{z}{\lambda_{RREA}}}$ when RREA leaves the cell
[34]. Therefore, if the reversal probability of runaway electrons from the
primary beam is equal to P, the runaway electron transport feedback
coefficient can be obtained as follows:

$\widetilde{\nu}_{e^{-}}=\int_{0}^{L}dzPe^{\frac{z}{\lambda_{RREA}}}\frac{dN_{beam}}{dz}=\frac{P}{\widetilde{\lambda}}\int_{0}^{L}dze^{\frac{1}{\lambda_{RREA}}-\frac{1}{\widetilde{\lambda}}}$
(14)

Here $\widetilde{\lambda}=-\lambda_{decay}>0$. Thus, the following formula is
obtained:

$\widetilde{\nu}_{e^{-}}=\frac{P\lambda_{RREA}}{\lambda_{RREA}-\widetilde{\lambda}}\left(1-exp\left(L\left(\frac{1}{\lambda_{RREA}}-\frac{1}{\widetilde{\lambda}}\right)\right)\right)$
(15)

This formula can be simplified using the empirical formula for
$\lambda_{RREA}$ [21]:

$\frac{\lambda_{RREA}}{\lambda_{RREA}-\widetilde{\lambda}}=\frac{E+E_{c}}{2E_{c}}$
(16)

Therefore:

$\widetilde{\nu}_{e^{-}}=P\frac{E+E_{c}}{2E_{c}}\left(1-exp\left(-L\frac{2E_{c}}{7300[kV]}\right)\right)$
(17)

This formula can be further simplified for cells with cell length
$L\gg\frac{7300[kV]}{2E_{c}}$, which works for cells larger than 100 m:

$\widetilde{\nu}_{e^{-}}=P\frac{E+E_{c}}{2E_{c}}$ (18)

Generally, there is some space between cells within a thunderstorm. Runaway
electrons lose energy by interacting with air molecules while propagating
through the gap between cells. A fraction of runaway electrons become
undercritical and leave the beam. This fraction can be estimated with the
decay length from formula 9 for $E=0$ as
$exp\left(-l\frac{E_{c}}{7300[kV]}\right)$, where $l$ is the gap between cells
in the simple reactor. Since, in the first approximation, all runaway
electrons lose the same amount of energy in the gap, the shape of their
spectrum remains the same. Thus, the formula for runaway electron transport
feedback coefficient, taking into account the gap between cells, simply
modifies in the following way:

$\begin{split}\widetilde{\nu}_{e^{-}}=P\frac{E+E_{c}}{2E_{c}}\left(1-\exp\left(-\frac{2E_{c}L}{7300[kV]}\right)\right)\cdot\\\
\exp\left(-\frac{E_{c}l}{7300[kV]}\right)\end{split}$ (19)

##  V GEANT4 simulation

The Monte Carlo simulation of the simple reactor was carried out using Geant4,
version 4.10.06.p01. Geant4 is recognized as a good tool to model RREA [37,
51]. The physics list G4EmStandardPhysics_option4 was chosen as the reliable
physics list for RREA simulations [51]. This list includes all interactions of
electrons, gamma-rays and positrons for energies characteristic for RREA
processes [48, 26]. The energy cut for the particles was chosen 50 keV based
on the fact that low-energy particles will quickly decay, as they do not run
away [16], and will not contribute to the feedback. The simulated geometry is
a large world volume filled with air, within which a child volume is
specified, also filled with air. A simple reactor by definition consists of
two child volumes: both volumes are filled with air with a density of 0.414
$kg/m^{3}$, corresponding to altitude 10 km, and contain electric field in the
way that both volumes accelerate runaway electrons towards each other (Figure
1).

The purpose of the GEANT4 simulation is to find the parameters of the system
necessary for the self-sustaining RREA regime (when the generation of high-
energy particles within the thunderstorm does not stop until the electric
field is discharged). The simulation was carried out by varying the cell size
and the strength of the electric field inside of it. At a certain electric
field strength, the number of gamma-ray photons and runaway electrons crossing
in both directions the boundary between the simple reactor cells will not
decrease over time. Thus, RREA within the simple reactor will not die out over
time. In this case, self-sustaining feedback is reached. Thus, by increasing
the electric field with a constant cell length, one can find the critical
point at which the reactor will become self-sustaining. In this way, the
achievement of critical values is checked, and the conditions are calculated.

The most important stage in modeling is the division of the high-energy
particles into generations. If directly two cells are created with an
oppositely directed field, then it will be quite difficult to divide the
process into feedback generations, since, under certain conditions, a self-
sustaining feedback is formed and the simulation will not stop. Thus,
analogically to the theoretical model, it was decided to simultaneously
simulate only a half of a simple reactor. This approach is possible due to the
symmetry of the simple reactor. The modeling scheme is shown in Figure 3. The
model consists of a single cell filled with air and electric field. At the
beginning of the cell an air detector is placed — the volume within which
particles are stopped and registered. In the first simulation step, seed
particles with an energy of 5 MeV are launched from the beginning of the cell
along the direction of the electric field. Seed runaway electrons are
decelerated by the electric field. Some of them penetrate into the cell,
reverse, and form RREA towards the detector. Seed gamma-ray photons propagate
through the cell and interact with air molecules. This interaction results in
runaway electrons generation, which reverse and form RREA, also moving and
growing toward the beginning of the cell. All particles that reach the
detector are stopped and registered and the simulation stops. In this way, the
first feedback generation is modeled. In subsequent simulations, the particles
registered in the previous iteration are launched into the cell accordingly
(thus imitating propagation of the high-energy particles from one cell into
another in the simple reactor). These particles interact with the cell, which
results in new particles generated that reach the detector. For each feedback
generation this process repeats. Figure 4 shows the number of gamma-rays
reaching the detector in each simulated feedback generation. The graph shows
that depending on the thunderstorm conditions the number of high-energy
particles can decay from generation to generation or vice versa. The
thunderstorm conditions when the number of particles does not change are the
necessary conditions for the self-sustaining development of RREA in the simple
reactor.

Figure 3: The design of a simple reactor has been simplified in the GEANT4
simulation to consider only one cell as in the figure. Runaway electrons and
gamma-ray photons are launched from the right side of the cell along the
direction of the electric field. The interactions of launched particles lead
to RREA formation, which is accelerated by the electric field to the right
side of the cell. In the result, generated particles reach the detector and
registered. In the next stages of the simulation, registered particles are
launched and new generated particles are similarly registered. In this way,
each reactor feedback generation is studied separately, thus, allowing the
analysis of the model. Figure 4: The dependence of the logarithm of the number
of gamma-ray photons that propagates from one cell to another in the simple
reactor depending on the number of feedback generation. The number of
generation is the number of an iteration of a simple reactor simulation
(Figure 3). It can be seen that the number of gamma-rays produced by the
simple reactor exponentially grows or exponentially decays depending on the
electric field strength.

To calculate the feedback coefficient, the simulation was launched with seed
runaway electrons. Number of generated gamma-ray photons and runaway electrons
for each feedback generation was registered, thus forming plots similar to
Figure 4. Each plot was fitted with an exponential function. The feedback
coefficient 6, 19 is obtained from the coefficient in the exponent by adding 1
to it [26]. The obtained dependence of the exponent parameter on the electric
field strength is shown in Figure 5. When the exponent parameter is positive,
number of high energy particles in the simple reactor thunderstorm self-
sustainably grows until the electric field is discharged.

Figure 5: The dependence of the feedback generations exponent parameter on the
electric field in the simple reactor for the cell length 400 m. Negative
exponent parameters means the decay of RREA in the simple reactor, while
positive exponent parameter means self-sustaining RREA development with high
energy particles generation. The exponent parameter includes both simple
reactor feedback processes: gamma-ray reactor feedback and runaway electron
oscillations (Figure 1). The conditions necessary for self-sustainable regime
(when exponential parameter equals 0) are in agreement with theoretical
predictions (Figure 2).

It is also interesting to calculate the spectrum of the gamma-rays produced
within the simple reactor and compare it with the spectrum of an ordinary RREA
bremsstrahlung. To obtain the spectrum, a full simple reactor with two cells
oriented towards each other was simulated. This simulation captures the
particles with their energies in a tracking action. The critical parameters of
the simple reactor were chosen — the field is 300 kV/m and the length of one
cell is 400 m for 10 km altitude air density. Similarly to the previous
simulation technique, the G4EmStandardPhysics_option4 physics list was used,
and the energy cut for particles is 50 keV. The simulation was stopped when
the number of high-energy particles reached $10^{6}$, and the spectrum of
registered gamma rays is obtained. In addition, a simulation for a single cell
with the same parameters was carried out to obtain the spectrum of an ordinary
RREA. The resulting spectra are shown in Figure 6. The graph shows that the
spectra are the same. It should be noted that single cell gamma-ray spectrum
contains more pronounced positron peak. However, when gamma rays propagate
from thunderstorm to the detector registering TGF or TGE, they interact with
the atmospheric layer and naturally produce the positron peak. Thus, this peak
will also be present when the TGF or TGE produced by the simple reactor is
measured. Nowadays it has been reliably established that the TGF and TGE
source spectrum is the RREA spectrum [50, 51]. Thus, the simple reactor can be
the mechanism for the TGF or TGE.

Figure 6: Comparison of the spectra obtained from the simulation of the simple
reactor and ordinary RREA spectrum, obtained from a single cell simulation
with a uniform electric field. The simulation of the simple reactor was turned
off when enough statistics were collected. It can be seen that the spectrum of
the simple reactor gamma-radiation is the same as the RREA bremsstrahlung
spectrum. It is established that the thunderstorm gamma-radiation spectrum
agrees with the RREA spectrum [4, 8]. Thus, the simple reactor can be one of
the mechanisms of TGF and TGE.

##  VI Discussion

The discovered mechanism called the simple reactor can be applied for a
thundercloud containing two regions with electric field exceeding the critical
value, i.e. allowing the RREA development (for simplicity, such regions are
called cells [26]), electric field is oriented in the way that cells
accelerate runaway electrons towards each other. It was established that there
is a positive feedback in this system caused by two mechanisms (besides the
relativistic feedback [21], which impact is relatively low (Figure 2)). The
first mechanism is the transport of runaway electrons from one strong field
region to another. This leads to the effective high-energy electron
multiplication and runaway electron oscillation near the edge between the
strong electric-field regions. The electron transport feedback coefficient is
very high for a small gap between cells, and is a dominant RREA multiplication
mechanism in the case of the small gap. On the other hand, in the case of a
significant gap, when the distance between strong field regions exceeds the
characteristic length of runaway electrons, too few runaway electrons
propagate through the gap between regions, thus another feedback mechanism
dominates. The second feedback mechanism is the gamma-ray reactor feedback
[26]. RREA bremsstrahlung gamma-rays have high penetration rate in the air.
Thus, in the simple reactor, gamma-rays effectively propagate from one cell to
another. When a gamma-ray photon propagates through the opposite cell, it
interacts with air, producing secondary RREAs, which is the gamma-ray reactor
feedback. Both feedback mechanisms can lead to self-sustaining RREA
development and, moreover, to rapid multiplication of high-energy particles
within a thunderstorm.

The formulas derived in this paper allow one to predict the feedback
coefficient for both feedback mechanisms without complicated modeling; the
theoretical predictions of this paper are verified by GEANT4. The discovered
feedback coefficients completely describe the behavior of the simple reactor,
e.g. allow to calculate the conditions required for the self-sustaining RREA
development (Figure 2). The limitations of the proposed analytical model are
as follows. Firstly, the model is one-dimensional and, therefore, does not
consider the transverse dynamics of the avalanche, which affects the feedback
coefficients in the case of narrow electric field regions [34]. Second, though
the description of runaway electron transport feedback qualitatively matches
Geant4 simulations, it lacks quantitative accuracy. More theoretical and
modeling research is needed to establish the exact influence of the electron
transport feedback on the RREA development.

The simple reactor geometry corresponds to the charge distribution with two
negative charge layers on both sides of the positive layer. This structure can
be a part of a more complicated charge structure of a thunderstorm [41, 42,
43, 44, 45, 46, 47]. In the simple reactor, the maximum density of runaway
electrons will be on the border between two oppositely directed cells — in the
center of the simple reactor, in the region of the positive charge (it should
be noted that the large value of a single positive charge in a region of a
cloud can be sufficient for RREA development below and above this region,
forming the simple reactor). This feature distinguishes the simple reactor
model from models assuming the development of RREA in a single cell with
maximum particle density in the cloud top or cloud base. Since in the simple
reactor the maximum runaway electron density is located in the center of the
reactor, it is harder for bremsstrahlung gamma-rays to reach detectors
registering TGF or TGE due to the greater thickness of the atmosphere that
they must penetrate. However, this does not contradict the observed gamma-ray
fluxes, since the reactor feedback increases the number of generated
bremsstrahlung gamma-rays within a thunderstorm containing the simple reactor.
This increase compensates for the decrease in gamma-ray flux by extra
atmosphere in has to penetrate.

Another distinguishing and important property of the simple reactor is that it
generates simultaneous gamma-ray radiation directed upward and downward from a
thundercloud (or in other opposite directions if the simple reactor is not
oriented vertically). This means that theoretically it is possible to
simultaneously detect a TGF or a TGE from two opposite sides of a
thunderstorm, e.g., from the top and from the bottom. Such observation can be
performed, for example, with an airplane containing particle detectors flying
over an observatory with particle detectors. Also a TGF generated by the
simple reactor can be registered simultaneously from space and ground
observatories, but the probability for the space station to be located above
the ground observatory at the moment of TGF is very low due to the TGF short
duration. It should be noted that the time profile of the gamma-ray flux in
measurements from both sides of the thunderstorm must match in order to
conclude that upward and downward gamma-ray radiation are connected by the
reactor feedback. This requires a good temporal resolution of the detectors.

The simple reactor with a large feedback coefficient can be a source of TGF.
Characteristic timescale of the simple reactor is its size divided by the
speed of light, which is in order of microsecond. Therefore, the timescale and
radiated gamma-ray spectrum satisfy the experimentally observed TGF data [3,
5, 1]. Runaway electron acceleration and its bremsstrahlung gamma-ray
radiation in the simple reactor precede the lightning leader and should
coincide with the early stage of the lightning initiation. It should be noted
that a TGF generated by positive feedback has a characteristic exponential
gamma-ray flux rise time profile. Number of high-energy particles grows
exponentially on TGF timescales as thunderstorm electric field remains almost
constant on these timescales. At the TGF peak, thunderstorm electric field
lowers, thus feedback coefficient drops and the feedback becomes finite: the
flux of high-energy particles starts to decay or even abruptly terminates, if
the electric field required for RREA development abruptly disappear. The
disappearance of the electric field can be connected either with local
discharges or with the initiation of a lightning leader. From the rise profile
of measured TGF flux the feedback coefficient can be restored. The feedback
coefficient is a good source of information on the thunderstorm electric field
during the TGF (Formula 6, 19) [34].

Another TGF model based on RREA, the relativistic feedback discharge model,
supposes significant positive feedback (the relativistic feedback) in the most
simple thunderstorm geometry - uniform electric field [38]. The disadvantage
of this model is that it requires very high values of electric field strength
extended over a large thunderstorm space [34, 39, 16]. The significant feature
of the simple reactor is that it requires smaller electric field strength for
the self-sustaining RREA development than it is in the uniform electric field
(Figure 2) [38, 34, 26]. Moreover, provided that two strong field regions are
formed by the same positive charge layer, the conditions for self-sustaining
feedback in the simple reactor are significantly more achievable than for
self-sustaining relativistic feedback.

For the simple reactor (as for any other RREA model with positive feedback
[26, 21]) the following time dependence of the gamma radiation flux measured
on the ground is possible. Usually during a TGE measurement, the gamma flux
slowly increases exponentially [8, 7]. This can be explained by the fact that
when the cloud approaches the detector at a constant speed, so the distance
from the cloud to the TGE source decreases linearly in time. The measured
particle flux decays exponentially with distance, thus, if the distance is
decreased linearly, the measured flux grows exponentially [7]. If RREAs are
self-sustaining within the thunderstorm due to the positive feedback, then
their bremsstrahlung gamma-ray flux grows exponentially within the
thunderstorm itself (it can grow slowly if the multiplication rate is slightly
higher than unity). Moreover, even if the feedback is present but the RREA is
not self-sustaining due to the low feedback coefficient, the RREA time profile
is modified and its radiation time increases [26]. Thus, with the positive
feedback, the time profile of the measured gamma-ray flux is exponent
superimposed on exponent. The time profile can be more complicated if the
electric field within thunderstorm is changing. Such time profile was measured
during winter thunderstorms gamma-ray glows [7], which supports the hypothesis
about the importance of the positive feedback in thunderstorm physics.

Lightning initiation by RREA is a widely discussed problem in the atmospheric
electricity science community [52, 53, 23, 20, 38, 8, 26, 11]. Within the
simple reactor, RREAs are directed to the center of the system, thus creating
the maximal density of RREA electrons and their products in the center. This
also leads to maximum ionization in the middle part of the simple reactor [54,
55]. The described case can be more favorable for streamer initiation when
compared to a single strong field region with RREAs directed to the top or to
the bottom base of a cloud because the ionization has its maximum at the end
of a RREA, on the edge of the strong field region. Moreover, the simple
reactor naturally contains more high-energy particles than the uniform
electric-field region because of the reactor feedback. Thus, the simple
reactor model can be a useful mechanism for lightning initiation research. It
should be noted, that if streamers are generated with the reactor feedback, it
can lead to an exponential growth of radio signal preceding the lightning
leader.

##  VII Conclusion

This paper studies RREA physics in thunderstorms containing two supercritical
electric field regions accelerating runaway electrons toward each other. Such
a system, named the simple reactor, can be a part of a natural thunderstorm.
It is discovered that RREA in the simple reactor has positive reactor
feedback. The reactor feedback enhances RREA duration and can lead to self-
sustaining RREA development. There are two mechanisms of the reactor feedback
in the simple reactor. RREA is effectively multiplied by the gamma-ray
exchange between regions even if they are far enough apart. If regions are
close to each other, high-energy particles are generated by the runaway
electron oscillations near the border between regions. In this case, the
small-scale strong electric field is sufficient for self-sustaining RREA
development. It is shown that the reactor feedback in the simple reactor
requires significantly lower electric field strength for RREA multiplication
compared to relativistic feedback.

The simple reactor in the self-sustaining regime rapidly increases the number
of high-energy particles within a thunderstorm and can hypothetically precede
or cause lightning initiation. It is established that the time scale and the
spectrum of the simple reactor gamma radiation agree with TGF data. The
distinguishing property of the simple reactor is that it radiates gamma rays
in two opposite directions. This allows simultaneous and correlated
observation of TGF or TGE gamma rays from the top and from the bottom of a
thundercloud. Moreover, the feedback coefficient can be retrieved from TGF and
TGE data, which can be a good source of information about gamma radiating
thunderstorm parameters, including electric field strength, supercritical
region length, and the electric field geometry.

## Acknowledgements

The work of E. Stadnichuk was supported by the Foundation for the Advancement
of Theoretical Physics and Mathematics “BASIS”. The work of E. Svechnikova was
supported by a grant from the Government of the Russian Federation (contract
no. 075-15-2019-1892).

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