Positron Production by Runaway Electrons in Lorentzian Plasmas

Abstract

We study electron–positron pair production by runaway electrons in fusion plasmas. The analysis uses a Coulomb logarithm derived for a Lorentzian plasma, which captures the high-energy “tail” in the non-Maxwellian runaway-electron distribution (κ-distribution). The Coulomb logarithm decreases as the distribution departs further from a Maxwellian. A lower kappa (κ) therefore reduces the Coulomb logarithm and yields a faster decline of the differential positron generation rate with increasing electron energy. Future work will explore how the electron–positron charge asymmetry manifests in their synchrotron radiation.

1. Introduction

Runaway electrons (REs), high-energy electrons accelerated by the electric field, represent a major challenge for magnetic confinement fusion devices [1,2]. These electrons can reach energies of several tens of MeV and may cause severe damage to the plasma-facing components. A particularly interesting consequence of the presence of such ultra-relativistic electrons is electron–positron pair production through their collisions with plasma particles. Early theoretical works predicted that interactions of multi-MeV REs with thermal plasma could result in the generation of a substantial number of positrons [3,4,5,6]. Being the antiparticles of electrons, positrons can either become runaways—accelerated in the opposite direction due to their positive charge—or rapidly annihilate, thus adding a new direction to tokamak plasma physics.

Helander and Ward (2003) [3] were among the first to highlight the possibility of significant positron generation in large tokamaks such as JET and JT-60U. Their study predicted that post-disruption RE collisions with thermal particles could yield up to $\sim {10}^{14}$ positrons. Two scenarios for positron behavior were identified: (i) if the loop voltage remains sufficiently high after the disruption, the positrons are accelerated to form a beam of long-lived runaway positrons propagating opposite to the RE current; (ii) if the electric field is too weak, the positrons decelerate to near-thermal energies ($\sim 10 \mathrm{e}\mathrm{V}$) within a few hundred milliseconds and subsequently annihilate. Thus, two distinct positron populations—fast (runaway) and slow (thermalizing)—were predicted, with potential implications for plasma diagnostics via their annihilation signatures.

Nearly a decade later, Fülöp and Papp (2012) [4] provided a detailed kinetic description of positron production under avalanche RE conditions typical of tokamak disruptions. They computed the energy spectrum of generated positrons, the fraction that become runaways, and the characteristics of their synchrotron radiation. Their results show that positron generation is most efficient for RE Lorentz factors ${\gamma }_e$ of several tens (corresponding to energies of several tens of MeV), and that the resulting synchrotron spectrum peaks in the infrared range (few μm). The authors emphasized that the spectral shape and power of this emission are highly sensitive to plasma parameters such as density and magnetic field, making positron radiation a promising diagnostic tool.

Several attempts have been made to experimentally detect positrons in tokamak plasma. Attention has been focused on indirect diagnostic signatures—primarily on the radiation emitted by electrons and positrons, where asymmetry between electrons and positrons can be utilized. Recently, a research group in China [7] proposed a diagnostic technique whose key element is the coincident detection of pairs of 511 keV photons emitted by electron and positron in opposite directions. In addition to annihilation photons, synchrotron photons generated by energetic positrons also have diagnostic potential. From a theoretical standpoint, due to opposite direction of motion, the synchrotron radiation from positrons is expected to be oriented differently from that of electrons.

In Refs. [3,4], expressions for positron generation rates involve the Coulomb logarithm, which is often assumed to be constant. However, in plasma it may vary significantly and depend on the electron distribution function.

It should be noted that in many plasmas—especially astrophysical and space plasmas—deviations from the familiar Maxwellian velocity distribution of particles are observed. Instead of an exponential decrease in the number of particles at high velocities, empirical data often show “long tails” in the distribution, described by a power-law dependence. Historically, the first indication of such a distribution shape came from magnetospheric measurements: it was found that the electron distributions in the solar wind and magnetospheric plasma contain an excess of fast particles compared to the Maxwellian distribution [8,9]. Vasyliunas [8] and Olbert [9] demonstrated that such distributions with enhanced high-energy tails are well described by the family of kappa-distributions [10,11]. Livadiotis and McComas [11] noted that the κ-distribution can be regarded as a natural generalization of the Maxwellian distribution for non-equilibrium stationary states commonly found in space plasmas.

The idea of applying κ-distributions to tokamak plasmas is of considerable interest, since runaway electrons themselves represent a significant deviation from thermal equilibrium (see Refs. [12,13,14,15,16,17,18,19,20,21]). During plasma disruptions (thermal quenches) in a tokamak, the so-called “hot-tail runaway” mechanism may occur—when the Maxwellian distribution does not have time to re-establish, leaving an elongated high-energy tail in the electron population. In essence, this situation qualitatively resembles a κ-distribution: most electrons are rapidly cooled, but a fraction remains at high energies, forming a tail from which runaway electrons are generated. Modern models of runaway electron formation, for instance those that include the “hot-tail” mechanism, in fact already go beyond the purely Maxwellian approximation and consider the evolution of a non-equilibrium distribution. As a result, they predict earlier and more intense formation of the runaway electron beam, which agrees with experimental observations.

If a population of electrons with a κ-distribution is present in the plasma, it can also have a significant effect on positron generation. In this work, we investigate the influence of κ-distribution on the dynamics of electron–positron pair production in tokamak plasma. An expression for the Coulomb logarithm for a κ-distributed plasma was derived in Ref. [21] and is employed in the present work. Using this formulation, we calculate the positron production rate, taking into account the effect of the non-Maxwellian electron distribution.

Although symmetry aspects are not analyzed in this work, our future studies will address the asymmetry between the electron and positron in their synchrotron radiation characteristics.

The paper is organized as follows. Section 2 describes the method for calculating the positron production rate, the distribution functions of runaway electrons and positrons in fusion plasmas. Section 3 presents the results, and Section 4 summarizes the conclusions.

2. Theory and Methods

2.1. Positron Production and Distribution Functions

Electron–positron pair formation can occur in collisions between runaway electrons and plasma particles. If the typical energy of ultra-relativistic runaway electrons exceeds the electron rest mass by a factor of three, pair formation takes place in collisions with plasma ions. If the runaway electron energy exceeds their rest mass by more than a factor of seven, electron–positron pairs are born in collisions with thermal electrons. The cross sections for the production of electron–positron pairs for JET and JT-60U were estimated and analyzed in [3]. The following expression gives the cross section for electron–positron pair production in the ultra-relativistic limit [22,23,24]:

$${\sigma }_{p({\gamma }_e\gg 1)}^s\left[28{\left(Z_s\alpha r_e\right)}^2/27\pi \right]{ln}^3{\gamma }_e,$$

(1)

where $\alpha =e^2/4\pi {ϵ}_0\hslash c\cong 1/137$ is the fine-structure constant, $Z_s$ is the charge of the target particle (ion or electron, $s=i$ or $e$), $r_e=e^2/4\pi {ϵ}_0{m}_e{c}^2$ is the classical electron radius, $c$ is the speed of light. For incident electron energies ranging to 100 MeV, a fit was presented in the works [4,25] that describes numerical calculations for the cross section of the electron–positron pair production [25]:

$${\sigma }_{tot}=a{Z_s}^2{\mathit{ln}}^3\left({\displaystyle \frac{{\gamma }_e+x_0}{3+x_0}}\right),$$

(2)

where $a=5.22 \mu \mathrm{b} \left(1 \mathrm{b}=10^{-28} {\mathrm{m}}^2\right),x_0=3.6$. As energy increases, cross section (2) approaches cross section (1). In [4], it was shown that exact cross section ${\sigma }_{tot}$ tends to ultra-relativistic cross section with increase in electron momentum and when Lorentz factors ${\gamma }_e=100$ both cross sections coincide.

It should be noted that in the toroidal configuration of a tokamak, the field geometry has a significant effect on the motion of runaway electrons and positrons. Due to the toroidal curvature of the magnetic field lines and the magnetic field gradient, particles experience transverse drifts that shift their orbits relative to the magnetic surfaces. For high-energy runaway electrons, this leads to the finite orbit width effect, in which their trajectories cross neighboring magnetic surfaces, thereby increasing wall losses. In addition, particles with small parallel velocity can be reflected in regions of enhanced magnetic field, forming traps (magnetic mirroring/trapping). Taking into account the toroidal curvature, trapping, and drift orbits significantly enriches the description of runaway electron and positron dynamics and provides better agreement between models and experimental observations [4]. In our work, we used the expressions for the distribution function of runaway electrons obtained with consideration of both the influence of the electric field and the effect of the magnetic field on the components of the electron momentum parallel and perpendicular to the magnetic field. The expression for the distribution function of relativistic runaway electrons $f_e^{RE}$ has the following form [4]:

$$f_e^{RE}(p_{e\parallel },p_{e\perp })={\displaystyle \frac{n_r\widehat{E}}{2\pi p_{e\parallel }{c}_z\mathit{ln}\Lambda }}exp\left(-{\displaystyle \frac{p_{e\parallel }}{c_z\mathit{ln}\Lambda }}-{\displaystyle \frac{\widehat{E}{p}_{e\perp }^2}{2p_{e\parallel }}}\right),$$

(3)

where $\widehat{E}=(E-1)/(1+Z_{eff})$, $E={\displaystyle \frac{e\left|E_{\parallel }\right|\tau }{m_{e}c}}$ is the normalized parallel electric field, $p_e={\gamma }_e{v}_e/c$ is the normalized momentum of electron. Here, $\parallel \mathrm{and} \perp$ denote components parallel and perpendicular to the magnetic field, respectively. Value $c_z=\sqrt{3\left(Z_{eff}+5\right)/\pi }$, with $Z_{eff}$ the effective ion charge [26]. If $E\gg 1$, the runaway tail is beam-like, so the parallel momentum greatly exceeds the perpendicular one, $p_{e\perp }\ll p_{e\parallel }\cong p_e$. Collision time for relativistic positrons and electrons can be written as $\tau =1/4\pi r_e^2{n}_{e}c\mathit{ln}\Lambda$, where $ln\Lambda$ is the Coulomb logarithm.

Positron production rate $S_p\equiv d{n}_{prod}^{+}/dt$ can be obtained based on the electron distribution function and total production cross section:

$$S_p=n_i{\int }_{p>p_{min}}{f}_e^{RE}{\sigma }_{tot}{v}_e{d}^3{p}_e=2\pi n_i{\int }_{p>p_{min}}{f}_e^{RE}{\sigma }_{tot}{v}_e{p}_{e\perp }d{p}_{e\perp }d{p}_{e\parallel },$$

(4)

where $n_i$ is the ion density, the threshold momentum is $p_{min}=3$.

To illustrate the number of positrons generated per second per unit volume by the runaway electrons with momentum $p_e$, we calculate the differential production rate using the distribution function from Equation (3), evaluating the integral over $p_{e\perp }$ in Equation (4):

$${\displaystyle \frac{d{S}_p}{d{p}_e}}\simeq {\displaystyle \frac{n_i{n}_{r}c}{c_{z}ln\Lambda }}exp\left(-{\displaystyle \frac{p_e}{c_{z}ln\Lambda }}\right){\sigma }_{tot},$$

(5)

where $n_r$ is the runaway electron density.

In some studies, the Coulomb logarithms were taken rather arbitrarily, as a characteristic value in a given system. In Ref. [4], calculations $d{S}_p/d{p}_e$ were presented for two values of the Coulomb logarithm ($\mathit{ln}\Lambda =10$ and $\mathit{ln}\Lambda =15$), showing a substantial difference. In this paper, we calculate the Coulomb logarithm using theoretical expressions that explicitly incorporate plasma parameters.

Also, in [4], a detailed description of the method for calculating the positron distribution function $f_{+}(p_{+},t)$ in tokamak plasmas is given. We present here the basic formulas for the calculations. The positron distribution function can be determined using the kinetic equation, which takes into account the processes of positron production and annihilation:

$${\displaystyle \frac{\partial f_{+}}{\partial t}}={\displaystyle \frac{1}{\tau p_{+}^2}}{\displaystyle \frac{\partial }{\partial p_{+}}}\left[\left(1+p_{+}^2\right)f_{+}\right]-n_e{v}_{+}{\sigma }_{an}{f}_{+}+ s_p(p_{+}),$$

(6)

where the first term describes the slowing down of positrons. $p_{+}={\gamma }_{+}{v}_{+}/c$ is the normalized momentum of the positrons, and ${\gamma }_{+}$ is the Lorentz factor of the positrons. The second term accounts for annihilation with the annihilation cross section:

$${\sigma }_{an}={\displaystyle \frac{\pi r_e^2}{1+{\gamma }_{+}}}\left[{\displaystyle \frac{{\gamma }_{+}^2+4{\gamma }_{+}+1}{{\gamma }_{+}^2-1}}\mathit{ln}\left({\gamma }_{+}+\sqrt{{\gamma }_{+}^2-1} \right)-{\displaystyle \frac{{\gamma }_{+}+3}{\sqrt{{\gamma }_{+}^2-1}}}\right]{\displaystyle \frac{2\pi \alpha /p_{+}}{1-e^{-2\pi \alpha /p_{+}}}}.$$

(7)

The third expression in (6) describes the source of positrons [4]:

$$s_p\left(p_{+}\right)=\int f_e^{RE}{\sigma }_{tot}{v}_e\delta \left(p_e-4.42p_{+}^{1.445}\right)4\pi p_e^{2}d{p}_e.$$

(8)

2.2. Kappa Distribution and Coulomb Logarithm

In rarefied plasmas, the assumption of a Maxwellian velocity distribution often breaks down, since long-range interactions and the lack of frequent collisions can give rise to pronounced high-energy tails [8]. The presence of a strong external electric field, which leads to the emergence of runaway electrons, only enhances this tendency. To account for these deviations, the so-called generalized Lorentzian distribution, commonly referred to as the kappa (or just $\kappa$)—distribution, has been widely employed [8]. A plasma with a Lorentzian particle distribution is often called a Lorentzian plasma, while a plasma with a Maxwellian velocity distribution is called a Maxwellian plasma. At low velocities (energies), the kappa distribution is close to Maxwellian, while at high velocities (the so-called tail) it may differ from Maxwellian. The parameter $\kappa$ characterizes the “tail length”: for large values of $\kappa$, the distribution becomes close to Maxwellian, while for small values of $\kappa$, the tail is long, with a high fraction of superthermal particles (significantly more energetic than thermal ones) [9,10,11,12]. Formally, in the limit $\kappa \to \infty$, the kappa distribution tends to a conventional Maxwellian distribution.

So, using $\kappa$-distribution functions allows for a more realistic description of the processes in such environments. The kappa distribution has the following form:

$$f_e\left(v\right)={\displaystyle \frac{n_e}{{\pi }^{3/2} {\xi }^3}}{\displaystyle \frac{\Gamma (\kappa +1)}{{\kappa }^{3/2}\Gamma (\kappa -\frac{1}{2})}}{(1+{\displaystyle \frac{v^2}{\kappa {\xi }^2}})}^{-\kappa -1},$$

(9)

where $\mathsf{\Gamma }\left(\kappa \right)$ is the Gamma function, $\kappa$ is the spectral index, which is larger than 3/2, $\xi =\sqrt{{\displaystyle \frac{2\kappa -3}{\kappa }}}{v}_{th}$, $v_{th}$ is the thermal velocity.

The Coulomb logarithm was taken with respect to the $\kappa$-distribution of electrons, as presented in Ref. [22]

$$ln\Lambda ={\displaystyle \frac{1}{2}}(\mathit{ln}\left(1+z\right)-{\displaystyle \frac{z}{1+z}}),$$

(10)

where value $z={\left(2m_e{v}_{th}{r}_{D,\kappa }/\hslash \right)}^2$. Explanations and justifications for the use of temperature and thermal velocity in a non-equilibrium plasma are given in Ref. [11]. In Ref. [27], an expression for the screening length based on kappa distribution was obtained and given as:

$$r_{D,\kappa }=r_D{\left[{\displaystyle \frac{\kappa -3/2}{\kappa -1/2}}\right]}^{1/2},$$

(11)

where $r_D$ is the Debye length in the traditional form:

$$r_D={\left[k_{B}T/4\pi {\sum }_{s=e,i}{n}_s{Z}_s{e}^2\right]}^{1/2}.$$

(12)

Expression (12) is valid for a Maxwellian plasma. In Ref. [28], it was shown that the screening length given by Equation (11) is less than the Debye length of Equation (12). Furthermore, if $\kappa =3/2$ the screening length (11) tends to zero, regardless of plasma temperature, while in the limit $\kappa \to \infty$, Equation (11) reduces to the Debye length (12). Deviation from the Maxwellian distribution leads to a decrease in the Coulomb logarithm. In Ref. [29], a formula for the Coulomb logarithm in tokamak plasma was also proposed:

$$ln\Lambda =14.9-0.5\mathit{ln}\left({\displaystyle \frac{n_e}{{10}^{20}}}\right)+ln{T}_{keV}.$$

(13)

However, when deriving it, the deviation of the distribution of runaway electrons from the Maxwellian distribution was not taken into account.

3. Results and Discussion

For our calculations, we take unless specifically specified, then by default the following values of parameters: $T_e=1 \mathrm{k}\mathrm{e}\mathrm{V}$, $n_e=5\cdot 10^{13} \mathrm{c}{\mathrm{m}}^{-3}$. In works on astrophysical plasmas and laboratory plasmas (see Refs. [12,13,14,15]), kappa values are taken from 1.5 to 6. We also took kappa values in this range. For clarity, the figures are presented mainly for κ = 1.6, ∞.

Figure 1 shows graphs of the Coulomb logarithm as a function of electron temperature, obtained using Formulas (10) (for different values of kappa) and (13) for two electron concentrations. As can be seen from the graphs, the Coulomb logarithm obtained using Formula (10) as $\kappa \to \infty$ lies close to the values obtained using Formula (13). As kappa decreases, the graphs of the Coulomb logarithm lie lower.

Using Equation (3), the runaway-electron distribution in a Lorentzian plasma (LP) relevant to tokamak conditions was computed. Figure 2 presents contour plots of the electron distribution function as a function of the parallel $p_{e\parallel }$ and perpendicular $p_{e\perp }$ momenta (where the components $\perp ,\parallel$ are defined with respect to the magnetic field, and $E=E_{\parallel }$ denotes the applied electric field). It is well established that in LP with a $\kappa$-distributed electron population, the Coulomb logarithm decreases as the spectral index $\kappa$ decreases [5]. This reduction of the Coulomb logarithm results in an enhancement of the runaway-electron distribution over the parallel momentum near its maximum, accompanied by a suppression in the ultra-relativistic region. Simultaneously, a decrease in the perpendicular-momentum distribution is observed. The latter effect arises because a lower Coulomb-collision cross section promotes a more beam-like character of the distribution function along the parallel direction. This behavior is clearly visible in Figure 3, which shows the electron distribution as a function of the parallel momentum for fixed values of the perpendicular momentum.

The process of positron generation by runaway electrons was also investigated. Figure 4 shows the differential positron production rate ${\displaystyle \frac{d{S}_p}{d{p}_e}}$ as a function of the electron momentum. Panel (a) represents results obtained by using Equation (5) with the exact positron-production cross section given by Equation (2) (blue curves), as well as for the case where, in Equation (5), the cross section was replaced by its ultra-relativistic limit given by Equation (1) (red curves). Solid lines correspond to the results for MP, while dashed lines represent those for LP. In order to show noticeable differences between MP and LP cases, in this case, we took $\kappa =1.6$. Data from Fülöp and Papp (2012) [4], obtained for Maxwellian plasma (MP) and indicated by symbols, are also shown on this figure. Our MP curves and the Fülöp and Papp data are in complete agreement. In the limit of ultra-relativistic electron energies (${\gamma }_e\to 100$), the blue curves asymptotically approach the red ones. The red curves lie above the blue curves due to the larger values of the ultra-relativistic cross section. The dashed lines are lower than the corresponding solid lines, since the reduction of the Coulomb logarithm in LP leads to a smaller number of ultra-relativistic positrons. Panel (b) shows our calculations with total cross section (2) for $\kappa =3,4,5$ and $\infty$. The differences between the curves reach several percent.

The positron distribution function was calculated and investigated on the basis of expressions (6)–(8). The results obtained for two values of the electron temperature are presented in Figure 5. It can be seen that with a decrease in the temperature of the electrons, as well as with an increase in the spectral index kappa, the maximum of the distribution function decreases.

4. Conclusions

We investigated the runaway-electron distribution and positron generation in Lorentzian plasmas relevant to modern tokamaks. A modified Coulomb logarithm was used to account for the κ-distributed electrons.

Decreasing κ increases the peak of the distribution over the parallel momentum and reduces the population of ultra-relativistic electrons. The distribution thus becomes more beam-like.

We also analyzed the differential positron production rate. In Lorentzian plasmas, the reduced Coulomb logarithm lowers the high-energy positron generation rate. Calculations using the exact positron-production cross section agree well with its ultra-relativistic limit at large electron energies.

These results refine the kinetic picture of runaway electrons and associated positrons in tokamak plasmas. They are relevant for predicting disruption consequences and for developing diagnostics and control techniques for high-energy particle fluxes.

While the present work does not focus on symmetry considerations, it outlines prospects for investigating the charge-related asymmetry between the electron and positron in their synchrotron emission behavior.