Effect of bremsstrahlung radiation emission on fast electrons in plasmas

Bremsstrahlung radiation emission is an important energy loss mechanism for energetic electrons in plasmas. In this paper we investigate the effect of spontaneous bremsstrahlung emission on the momentum-space structure of the electron distribution, fully accounting for the emission of finite-energy photons. We find that electrons accelerated by electric fields can reach significantly higher energies than what is expected from energy-loss considerations. Furthermore, we show that the emission of soft photons can contribute significantly to the dynamics of electrons with an anisotropic distribution.

Energetic electrons are ubiquitous in plasmas, and bremsstrahlung radiation is one of their most important energy loss mechanisms [1,2]. At sufficiently high electron energy, around a few hundred megaelectronvolts in hydrogen plasmas, the energy loss associated with the emission of bremsstrahlung radiation dominates the energy loss by collisions. Bremsstrahlung emission can also strongly affect electrons at lower energies, particularly in plasmas containing highly charged ion species.
An important electron acceleration process, producing energetic electrons in both space and laboratory plasmas, is the runaway mechanism [3]. In the presence of an electric field which exceeds the minimum to overcome collisional friction [4], a fraction of the charged particles can detach from the bulk population and be accelerated to high energies, where radiative losses become important. Previous studies of laboratory plasmas [5,6] and lightning discharges [7] have shown that the energy carried away by bremsstrahlung radiation is important in limiting the energy of runaway electrons. The effect of bremsstrahlung radiation loss on energeticelectron transport has also been considered in astrophysical plasmas, for example in the context of solar flares [8]. However, only the average bremsstrahlung friction force on test particles has been considered in these studies. In this paper, we present the first quantitative kinetic study of how bremsstrahlung emission affects the runawayelectron distribution function.
Starting from the Boltzmann electron transport equation, we derive a collision operator representing bremsstrahlung radiation reaction, fully accounting for the finite energy and emission angle of the emitted photons. We implement the operator in a continuum kinetic-equation solver [9], and use it to study the effect of bremsstrahlung on the distribution of electrons in 2D momentum-space. We find significant differences in the distribution function when bremsstrahlung losses are modeled with a Boltzmann equation (referred to as the "Boltzmann" or "full" bremsstrahlung model), compared to the model where only the average friction force is accounted for (the "mean-force" model). In the former model, the maximum energy reached by the energetic electrons is significantly higher than is predicted by the latter. In previous treatments which considered average energy loss [5][6][7] or isotropic plasmas [2], the emission of soft (low-energy) photons did not influence the electron motion. We show that in the general case, emission of soft photons contributes significantly to angular deflection of the electron trajectories.
We will treat bremsstrahlung as a binary interaction ("collision") between two charged particles, resulting in the emission of a photon [1]. We shall describe the effect of such collisions on the rate of change of the distribution function f a (t, x, p) of some particle species a at time t, position x and momentum p, defined such that n a (t, x) = dp f a (t, x, p) is the number density of species a at x. In what follows we suppress the time-and space dependence of all functions, as the collisions will be assumed local in space-time, and we shall consider only spatially homogeneous plasmas.
The collision operator C B ab {f a , f b } describing the rate of change of the distribution function due to bremsstrahlung interactions between species a and b is given by C B ab = (∂f a /∂t) c,ab = (dn a ) c,ab /dtdp, where the differential change (dn a ) c,ab in the phase-space density due to collisions in a time interval dt is given by [10,11] dn a c,ab = f a (p 1 )f b (p 2 )ḡ ø dσ ab dp 1 dp 2 dt Here, dσ ab = dσ ab (p 1 , p 2 , k; p, p ′ ) is the differential cross-section for a particle a of momentum p and a particle b of momentum p ′ to be taken to momentum p 1 and p 2 , respectively, while emitting a photon of momentum k/c. We have also introduced the Møller relative speed The barred quantities dσ andḡ ø are defined likewise, but with (p, p ′ ) and (p 1 , p 2 ) exchanged. Eq. (1) accounts only for the effect on the distribution of the spontaneous emission of photons; interactions with existing photons by absorption and stimulated bremsstrahlung emission will be neglected here. The correction to the collision operator by these processes is described in [12]; the effect is negligible when φ(x, p) ≪ 2/h 3 , where h is Planck's constant and φ is the distribution function of photons. An estimate of the photon distribution function shows that the corrections are important for sufficiently dense, or large, plasmas; however, for the special case of electron runaway during tokamak disruptions, which is of particular concern, the corrections may be safely neglected. In other scenarios it is primarily bremsstrahlung processes involving low-energy photons that may be affected. The collision operator then takes the form where σ ab = dp 1 (∂σ ab /∂p 1 ) is the total cross-section. A significant simplification to (2) occurs if (i) target particles can be assumed stationary, f b (p) = n b δ(p); and (ii) the plasma is cylindrically symmetric (and spin unpolarized), f a (p) = f a (p, cos θ), where cos θ = p /p and p is the Cartesian component of p along the symmetry axis. Then the differential cross-section ∂σ ab /∂p, for an electron to scatter from momentum p into p 1 with the emission of a photon, depends only on p, p 1 and cos θ s = p 1 · p/p 1 p. The resulting operator can be conveniently expressed in terms of an expansion in Legendre , and obtain The integration limits in p 1 are determined by the conservation of energy, giving m e c (γ + k/m e c 2 ) 2 − 1 < p 1 < ∞. In this work we use the differential cross-section ∂σ/∂p for scattering in a static Coulomb field in the Born approximation, integrated over photon emission angles. This expression was first derived by Racah [13], with a misprint later corrected in [14]. For the Boltzmann model this full cross-section is employed, while for the meanforce model we use the high-energy limit as in [5][6][7].
A useful approximation to the collision operator is obtained by neglecting the deflection of the electron in the bremsstrahlung reactions, formally achieved by the re- This takes the form of a one-dimensional integral operator acting only on the energy variable, and involves the integrated cross-section which is well known and given analytically for example in Eq. (14) of [1]. Physically, this approximation is motivated by the strong forwardpeaking of the cross-section, with typical deflection angles being of order θ s ∼ 1/γ due to relativistic beaming.
The bremsstrahlung cross-section has an infrared divergence; for low photon energies k, it diverges logarithmically as dσ ∝ 1/k. The total energy loss rate is however finite, indicating that a large number of photons carrying negligible net energy are emitted. A consequence of this behavior is that the two terms in the Boltzmann operator (2) are individually infinitely large, necessitating the introduction of a photon cut-off energy k 0 , below which the bremsstrahlung interactions are ignored in (3) and (4). We can however proceed analytically to evaluate the effect of the low-energy photons. While they carry little energy, they may contribute to angular deflection, analogously to the small-angle collisions associated with elastic scattering. Taylor expanding (3) in small photon energy k = γ 1 − γ yields to leading order Since P 0 (cos θ s ) ≡ 1, the angle-averaged electron distribution (represented by the L = 0 term) is not directly affected by the low-energy photons, reflecting the fact that the photons carry negligible energy, consistent with the description by Blumenthal & Gould [2] for the isotropic case. Due to the logarithmic divergence of the crosssection, however, a significant contribution to angular deflection (represented by the L = 0 terms) is possible. Inspection of the integrand in (5) further reveals that significant contributions originate from large-angle scatterings, indicating that a Fokker-Planck approximation is inappropriate. While it may seem counter-intuitive that low-energy photon emissions contribute to largeangle collisions, note that due to the large mass ratio between electron and ion, large momentum transfers to the nucleus is allowed even without energy transfer. When the electron energy exceeds the ion rest energy, however, ion recoil effects will modify (5). We can quantify the importance of the low-energy photons by calculating the L = 1 term of (5) -giving the loss rate of parallel momentum -and comparing it to the corresponding term of the elastic-scattering collision operator given in [9]. Carrying out the integration, one obtains the ratio with a relative error of magnitude O(m 2 e c 2 /p 2 ) + O(k 0 /pc), and where α = e 2 /4πε 0 c ≃ 1/137 is the fine-structure constant. We have introduced a bremsstrahlung logarithm ln Λ B = ln(k 0 /k c ), which arises in a way similar to the Coulomb logarithm ln Λ for elastic collisions, and is due to cutting off the logarithmic divergence at some lowest photon energy k c . This energy corresponds to photons emitted at the plasma frequency ω p , at which point polarization of the background will dampen the bremsstrahlung interactions [15], and is thus given by k c = ω p . This gives a bremsstrahlung logarithm ln Λ B ≈ 21 + ln k 0 /(m e c 2 √ n 20 ) , where n 20 = n e /(10 20 m −3 ) is the electron density in units of 10 20 m −3 . Assuming a plasma with ln Λ = 15, n 20 = 1 and choosing k 0 = 0.01p, the ratio (6) is of order 10% at 30 MeV, 50% at 2 GeV and 100% at 30 GeV, demonstrating that angular deflection caused by the emission of low-energy photons can contribute significantly to the motion of highly energetic electrons.
The bremsstrahlung collision operator has been implemented in the initial-value continuum kinetic-equation solver CODE (COllisional Distribution of Electrons) [9]. For this study we use CODE to solve the equation which in a magnetized plasma represents the gyroaveraged kinetic equation, with the parallel direction given by the magnetic field B. The equation is also valid for an unmagnetized plasma which is cylindrically symmetric around the electric field E. Elastic collisions are accounted for by the linearized relativistic Fokker-Planck operator for Coulomb collisions C FP , and C B is the bremsstrahlung operator C B eb summed over all particle species b in the plasma. Both thermal and fast electrons are resolved simultaneously, allowing runaway generation as well as the slowing-down of the fast population to be accurately modeled.
We will compare the effect of bremsstrahlung radiation losses on the momentum-space distribution of fast electrons using several models. The contribution from the emission of large-energy photons (with k > k 0 ) are accounted for by either the Boltzmann operator in (3) or its approximation without angular deflection (4), while the low-energy photon contribution (k < k 0 ) is described by (5). For the numerical solutions we choose an energydependent cut-off k 0 = m e c 2 (γ − 1)/1000. The Boltzmann models will be compared to the mean-force model where the bremsstrahlung losses are accounted for by an isotropic force term in the kinetic equation, defined as dk k∂σ eb /∂k, which is chosen to produce the correct average energy-loss rate [1].
To characterize the effect of bremsstrahlung on the electron distribution, we investigate quasi-steady-state numerical solutions of the kinetic equation (7). These are obtained by evolving the distribution function in time until an equilibrium is reached, typically after a few seconds at density n 20 = 1 if an initial seed of fast electrons is provided. We investigate a range of electric-field values near the minimum field E c = 4π ln Λn e r 2 0 m e c 2 /e to overcome collisional friction [4], using plasma parameters characteristic of tokamak-disruption experiments with massive gas injection.  Figure 1 shows the electron distribution function in momentum space, calculated using CODE, with full Boltzmann bremsstrahlung effects included (black, solid); neglecting angular deflections in the large-k contribution (yellow, dash-dotted); also neglecting the smallk contribution (blue, dashed); and finally using the mean-force model (red, solid). Non-monotonic features (bumps) form in the mean-force as well as the Boltzmann models, but their characteristics are significantly different. With the Boltzmann models, an extended tail forms in the electron distribution. In contrast, the mean-force model produces a sharp feature, located where the energy gain due to the electric-field acceleration balances friction and bremsstrahlung losses. The addition of lowk scatterings (5), which lead to large-angle deflections, causes a subpopulation of fast electrons with significant perpendicular momentum to form. Furthermore, (3) and (4) appear to generally produce the same qualitative features, indicating that scatterings involving large-energy photons are well approximated by neglecting the angular deflection of the electron.
Inclusion of synchrotron radiation losses associated with the gyromotion of electrons in a straight magnetic field has been shown to be an important energy-loss mechanism [16][17][18][19][20]. Figure 1(b) shows that, in conjunction with bremsstrahlung losses, synchrotron losses (modelled as in [16]) shifts the distribution towards lower energies but does not change its qualitative features. The difference between the Boltzmann and mean-force models is therefore reduced in such cases, as the extent of the distribution when full bremsstrahlung effects are included is reduced by the synchrotron effect. Angle-averages of the electron distribution functions in figure 1 are shown in figure 2 as a function of electron kinetic energy W = m e c 2 (γ − 1). When there are no synchrotron losses present, the difference between the Boltzmann models for bremsstrahlung losses is insignificant.
In the presence of effects which are sensitive to the angular distribution of electrons, such as synchrotron radiation losses (which are proportional to p 2 ⊥ ), the difference is somewhat enhanced as angular deflection amplifies the dissipation.
To quantify the width in energy of the fast-electron tail, figure 3 shows the fraction of total plasma kinetic energy carried by electrons with energy greater than W , for a range of plasma compositions and electric fields. Again, the steady-state solutions are considered, and the energy ratio is calculated as ∞ W dW W (dn e /dW )/W tot . When normalized to the energy W 0 which solves the energybalance equation eE − eE c + F B = 0 (accounting for collisional and bremsstrahlung energy loss), the behavior is seen to be insensitive to electric field and effective charge. The Boltzmann model consistently predicts that a fraction of the electron population reaches significantly higher energies than in the mean-force model, where all electrons have energy near W 0 . For instance, in the Boltzmann model 5% of the plasma energy is carried by electrons with energy more than 2W 0 . Summary -We have developed a kinetic description of the effect of spontaneous bremsstrahlung emission on energetic electrons in plasmas. A computationally efficient representation of the bremsstrahlung collision operator has been obtained using an expansion in Legendre polynomials, with which the operator is reduced to a set of 1D energy integrals. This allows for rapid evaluation of the self-consistent electron distribution function in the presence of bremsstrahlung losses derived from the full Boltzmann operator.
By treating bremsstrahlung emission as a discrete process, we have shown that electrons may be accelerated to significantly higher energies than would be predicted by energy balance alone, with a significant fraction of particles reaching at least twice the expected energy. The explanation for this can be intuitively understood in the single-particle picture, where the new model allows some electrons to suddenly lose a large fraction of their energy in one emission, whereas others may be accelerated for a long time before a bremsstrahlung reaction occurs, thereby allowing higher maximum energies to be reached. This has important implications for the interpretation of experimental observation of fast electron beams in plasmas where bremsstrahlung losses are important, such as in magnetic-fusion plasmas. Furthermore, new effects are revealed in our treatment, as the emission of soft photons is found to contribute to angular deflection of the electron trajectory at a rate that increases with electron energy. This effect shifts part of the momentum-space distribution function towards higher perpendicular momenta, which in turn has implications for e.g. the destabilization of kinetic instabilities or the level of synchrotron radiation loss in magnetized plasmas.
The authors are grateful to G. Papp for fruitful discussions. This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement number 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. This work was supported by the Knut and Alice Wallenberg Foundation (PLIONA project) and the European Research Council (ERC-2014-CoG grant 647121).