Interpretation of runaway electron synchrotron and bremsstrahlung images

Aside from bringing the computational advantages of not having to resolve the full gyro-orbit, the cone model also provides a simple framework for qualitative reasoning about the region from which synchrotron radiation can be detected. Since synchrotron radiation is only detected when the emitting particle is moving directly towards the detector, the overall shape of the observed synchrotron spot is mainly determined by the magnetic field geometry [24, 27]. Neglecting drift velocities, the cone model gives a very simple condition for a particle to be seen by the detector:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\thetap}{\theta_{{\rm p}}} \newcommand{\bb}[1]{{\bf #1}} \displaystyle \label{eq:sov} \left\vert \hat{\bb{b}}(\bb{x})\cdot\hat{\bb{r}}(\bb{x})\right\vert = \vert \cos\thetap\vert . \nonumber \end{align} \tag{ 2 }$$

Here $\newcommand{\bb}[1]{{\bf #1}} \hat{\bb{b}}(\bb{x})$ is the magnetic field unit vector in the point $\newcommand{\bb}[1]{{\bf #1}} \bb{x}$ in space, $\newcommand{\bb}[1]{{\bf #1}} \hat{\bb{r}} = (\bb{x}-\bb{x}_0) / \vert \bb{x}-\bb{x}_0\vert$ is the unit vector pointing from the detector, located at $\newcommand{\bb}[1]{{\bf #1}} \bb{x}_0$, to $\newcommand{\bb}[1]{{\bf #1}} \bb{x}$ and $\newcommand{\thetap}{\theta_{{\rm p}}} \thetap$ is the pitch angle at $\newcommand{\bb}[1]{{\bf #1}} \bb{x}$ under consideration. For a fixed value of $\newcommand{\thetap}{\theta_{{\rm p}}} \thetap$, the solution to this equation is a surface in real space, which we call the surface-of-visibility (SOV). It is clear from equation (2) that aside from a strong dependence on the magnetic field and the particle's pitch angle, the shape of the SOV is also strongly dependent of the placement of the detector. None of the quantities involved in (2) however depends on the particle's energy, which means that the SOV is independent of the the energy. In reality, drift orbits introduce an energy dependence, but these effects are neglected in the zeroth order guiding-center model we employ. In figure 1, different projections of the surface-of-visibility for a given $\newcommand{\bb}[1]{{\bf #1}} \bb{x}_0$ lying in the midplane and runaway-electron population with $\newcommand{\thetap}{\theta_{{\rm p}}} \thetap = 0.16$ rad and beam radius $r\approx 50$ cm in DIII-D is shown, revealing its cylindrical structure. Note that the surface-of-visibility in figure 1 is the surface corresponding to particles given an initial pitch angle $\newcommand{\thetap}{\theta_{{\rm p}}} \thetap = 0.16$ rad in the outer midplane. The pitch angle then varies together with the magnetic field along the particle orbit.

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The importance of the magnetic field geometry for determining the synchrotron spot shape was emphasized already by Pankratov in 1996 [24], but the realization that what is perceived as a 'spot' in synchrotron images is in fact the projection of a surface of finite toroidal extent could be even more important when considering momentum-space distributed populations of runaway electrons. The reason for this is that SOVs for particles with small pitch angles (typically $\newcommand{\thetap}{\theta_{{\rm p}}} \thetap\lesssim 0.20$ rad) close on themselves, and the line-integrated contribution from a line-of-sight passing through the edge of such a surface will be greater due to that the line-of-sight tangents the surface. The edges of the projected SOV therefore tend to be significantly brighter than other parts of the SOV, as is exemplified in figure 1. Because of this the edges of single-particle synchrotron spots (i.e. spots corresponding to a specific set of pitch angle $\newcommand{\thetap}{\theta_{{\rm p}}} \thetap$ and momentum p) tend to dominate synchrotron spots from momentum-space distributed runaway electron populations and 'fill in' different parts of the overall spot.

The amount of radiation emitted by a particle is another important quantity affecting the appearance of a synchrotron spot. Since the total synchrotron power emitted by an electron in a magnetic field is given by [7]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle P = \frac{e^4}{6\pi\epsilon_0m_e^2c}p_\perp^2B^2 \propto p_\perp^2 B^2 \propto p^2 B^3, \nonumber \end{align} \tag{ 3 }$$

where p is momentum and $\newcommand{\thetap}{\theta_{{\rm p}}} p_\perp = p\sin\thetap\propto p\sqrt{B}$ is the component of p perpendicular to the magnetic field $\newcommand{\bb}[1]{{\bf #1}} \bb{B}$, we expect a particle to emit more radiation when it passes through the high-field side of a tokamak, both because the magnetic field is stronger there, and because of the larger pitch angle the particle will have due to the stronger magnetic field. It turns out however, that while the synchrotron emission from the high-field side is typically stronger than that from the low-field side, the amount of synchrotron radiation received by a camera from the high-field side can scale much more strongly with magnetic field than B³. In fact, as we will now show, the ratio between emission from the HFS to the emission from the LFS can be on the order of 10⁶ in DIII-D.

The reason for the strong scaling of the detected radiation stems from the finite spectral range of the camera. In most present-day tokamaks, the runaway electrons emit most of their radiation at wavelengths around a few micrometers, while visible light cameras seeing wavelengths up to around 900 nm are used. In this case it can be shown that the short-wavelength ($\newcommand{\lambdac}{\lambda_{{\rm c}}} \lambda\ll\lambdac$) asymptotic expansion of equation (1) is [34]

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\lambdac}{\lambda_{{\rm c}}} \newcommand{\dd}{{\rm d}} \displaystyle \frac{\dd P}{\dd\lambda}\sim \exp\left( -\frac{\lambdac}{\lambda} \right). \nonumber \end{align} \tag{ 4 }$$

By introducing the critical radius

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle R_{\rm c} = \frac{B_0R_0}{2m_e}\left( \frac{3\gamma e\lambda\sqrt{\mu}}{\pi c^2} \right)^{2/3}, \nonumber \end{align} \tag{ 5 }$$

with $\mu = p_\perp^2 / 2m_eB$ being the magnetic moment, and assuming the magnetic field strength as a function of major radius to be $B(R) = B_0R_0/R$, where B₀ and R₀ are the magnetic field strength and major radial location respectively of the magnetic axis, we can write the detected synchrotron power as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:expscale} P\sim \exp\left[ -\left(\frac{R}{R_{{\rm c}}}\right)^{3/2} \right]. \nonumber \end{align} \tag{ 6 }$$

We can also express the ratio between maximum emission of a particle (at the innermost point of its orbit) to its minimum emission (at the outermost point of its orbit) as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{P_{{\rm max}}}{P_{{\rm min}}} = \exp\left[ \frac{(R_0+\Delta R/2)^{3/2}-(R_0-\Delta R/2)^{3/2}}{R_{{\rm c}}^{3/2}} \right] \nonumber \end{align} \tag{ 7 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \approx\exp\left[\frac{3}{2}\frac{\Delta R}{R_0}\left(\frac{R_0}{R_{{\rm c}}}\right)^{3/2} \right] \nonumber \end{align} \tag{ 8 }$$

where $\Delta R$ is the distance between the inner- and outermost points of the particle orbit. The critical radius, $R_{\rm c} = R_{\rm c}(\lambda)$, can be interpreted as the major radius at which $\newcommand{\lambdac}{\lambda_{{\rm c}}} \lambdac = \lambda$, i.e. where the particle emits most of its radiation near the wavelength λ.

The exponential scaling with particle position equation (6) is the result of observing radiation in only a narrow range of wavelengths far from the wavelengths at which synchrotron emission peaks. In a typical DIII-D scenario, where $\lambda\approx 900$ nm and $\gamma\approx 30$, we find that $R_{\rm c}\approx 0.1 R_0$. A DIII-D runaway electron which moves a distance $\Delta R = 1$ m in major radius during its orbit therefore emits more on the high-field side compared to the low-field side by a factor $P_{{\rm max}}/P_{{\rm min}} \sim 10^6$, i.e. a six orders of magnitude difference. Had we instead been able to observe at a wavelength λ closer to $\newcommand{\lambdac}{\lambda_{{\rm c}}} \lambdac$, or even all radiation, the difference would merely have been about a factor of six. This shows that synchrotron radiation can appear significantly brighter in a region of space, without the number of runaways necessarily being higher in that region.

In many previous studies, synchrotron spectra and spot shapes from single particles have been used to model experimentally observed spectra [11, 18] and spot shapes [27]. As was shown in [26] however, aside from the difficulties of interpreting the results (runaway electrons are rarely homogeneous enough for a single particle to satisfactorily approximate the population), taking the distribution function of runaway electrons into account can significantly alter the simulated spectra. For the synchrotron spot, the difference can be even more dramatic, as the brightest features of a number of single particle-spot shapes will come together and create an overall pattern which does not necessarily resemble any of the single particle spot shapes.

The pitch angle and energy of a particle, together with the flux surface on which the particle moves (radial position), together determine the shape of the SOV and at which wavelength most of the synchrotron radiation is emitted. In figure 2 the effect of each of these parameters on the synchrotron spot is sketched and the arrows indicate how the spot shape changes with increasing pitch angle, energy and radial location. Figure 2(a) indicates that the pitch angle mainly determines the vertical extent of the SOV, while the radial location of the particle mainly affects the horizontal extent of the SOV, as shown in figure 2(c). The particle energy is mainly tied to the finite spectral range effect described in section 2.2, as is illustrated in figure 2(b). As the particle energy is increased, the peak of the emitted spectrum increases as well, meaning that the critical radius $R_{{\rm c}}$ also increases. For the synchrotron radiation spot this means that the low-field side part of the spot (right side of image) gains intensity relative to the high-field side part (left side of the image) giving a more even distribution of radiation horizontally across the SOV. A more in-depth description of how the spot shape varies with different parameters can be found in [30, 31].

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The synchrotron spot of a certain population of runaway electrons will be the weighted average of several synchrotron spots, each corresponding to a unique set of runaway electron energy, pitch angle and radial location, i.e. individual particles. The weight is the distribution function which determines the relative importance of different particles in accordance with how likely they are to be found in the population and determines the overall spot shape.

A useful quantity that provides much information about the radiation from a distribution of runaway electrons is the density of radiation in momentum space,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:radiationdensity} F(\,p_\parallel, p_\perp) = \hat{I}(\,p_\parallel, p_\perp)\,f(\,p_\parallel, p_\perp), \nonumber \end{align} \tag{ 9 }$$

where $f(\,p_\parallel, p_\perp)$ is the runaway electron distribution function and $\hat{I}(\,p_\parallel, p_\perp)$ is the amount of radiation emitted by a particle with the given momentum, henceforth referred to as weight function (different for synchrotron or bremsstrahlung). Due to the $p_\perp^2$ scaling (or even stronger, as explained above) of the synchrotron emission, i.e. $\hat{I}_s$, the most numerous particle type is not necessarily the one that emits the most synchrotron radiation. While $\hat{I}$ tends to grow monotonically with momentum p, and f tends to decrease with $p_\parallel$ and $p_\perp$, the radiation density F generally has a global maximum in momentum space different from $p_\parallel = p_\perp = 0$. The maximum of F can be considered a 'super'-particle which will dominate emission of a particular radiation type from a given distribution of runaway electrons. Often, the single particle spectrum that best matches the distribution averaged spectrum is that of the super particle, and the observed spot shape from the distribution has often many similarities to the spot shape of the super particle. It should be noted though that the only significance of the super particle is that it is the maximum of F. It is not (necessarily) the particle with the highest energy or pitch angle, and no general statement about the most common electron momentum can be made.

Just like synchrotron radiation, bremsstrahlung is highly anisotropic and directed mainly in the particle's direction of motion for highly relativistic particles. The average angular spread of bremsstrahlung is the same $\gamma^{-1}$ as for synchrotron radiation, which means that bremsstrahlung also gives rise to a surface-of-visibility as discussed above, that is the same as that of synchrotron radiation. The cone model used in SOFT for synchrotron radiation can thus also be used for bremsstrahlung, but with the formula for received synchrotron radiation power replaced with the formula for the number of bremsstrahlung photons $\newcommand{\dd}{{\rm d}} \dd N_\gamma$ emitted per unit photon energy $\newcommand{\dd}{{\rm d}} \dd k$ (using the Born-approximation cross-section for a fully ionized plasma [35] summed over all ion species)

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dd}{{\rm d}} \displaystyle \label{eq:bremsspec} \frac{\dd N_\gamma}{\dd k} \!&= \!\frac{\alpha n_e Z_{\rm eff} e^4 v p'}{(4\pi\epsilon_0 m_e c^2)^2 k p}\Biggr\{\frac{4}{3} - 2\gamma'\gamma\frac{p^2+p'^2}{p'^2p^2} + \epsilon\frac{\gamma'}{p^3}+\epsilon'\frac{\gamma}{p'^3} - \frac{\epsilon\epsilon'}{p'p}+L\Biggr[ \frac{8}{3}\frac{\gamma'\gamma}{p'p} \nonumber \\ \!&+\!k^2\frac{\gamma'^2\gamma^2+p'^2p^2}{p'^3p^3} + \frac{k}{2p'p}\left( \epsilon\frac{\gamma'\gamma+p^2}{p^3} - \epsilon'\frac{\gamma'\gamma+p'^2}{p'^3} + 2k\frac{\gamma'\gamma}{p'^2p^2} \right) \Biggr] \Biggr\}, \nonumber \\ \epsilon &= \ln\frac{\gamma+p}{\gamma-p} \nonumber \\ \epsilon' &= \ln\frac{\gamma'+p'}{\gamma'-p'} \nonumber \\ L &= 2\ln\frac{\gamma'\gamma+p'p-1}{k}, \nonumber \end{align} \tag{ 10 }$$

where $\alpha=e^2/4\pi\varepsilon_0 \hbar c \approx 1/137$ is the fine-structure constant, the photon energy k is defined in units of $m_e c^2$, and the normalized ingoing and outgoing electron momenta $p=\gamma v/c$ and $p'$ are related through $p'=\sqrt{\gamma'^2-1}$, and $\gamma' = \gamma-k$.

Measurements of bremsstrahlung from runaway electrons is today a standard diagnostic at most larger tokamak experiments, and the data acquired is often used to study the temporal evolution of runaway electrons and sometimes even to measure the runaway electron energy distribution function [36, 37]. Bremsstrahlung from runaways has also previously been modeled, for example in the Tore Supra tokamak [38, 39]. At DIII-D, a novel technique for measuring not only the bremsstrahlung spectrum, but also its spatial distribution, has been developed and is called the GRI [32, 33]. The GRI combines a lead pinhole camera with gamma-ray detectors, thus functioning as a camera for gamma rays with energies in the range 1–60 MeV. From a theoretical point-of-view, the GRI is an ideal tool for studying runaways since the spectrum is also measured by each gamma-ray detector, thus providing a set of images at different photon-energies rather than just one single image. The weak pitch angle dependence in the bremsstrahlung emission is also advantageous, as it avoids the finite spectral range effect experienced with synchrotron radiation, which tends to obscure large parts of the spatial information.

The temporal evolution of the 2D runaway electron momentum-space distribution function during DIII-D discharge 165826 was simulated using CODE [40, 41] by solving the spatially homogeneous kinetic equation, including electric-field acceleration, collisions modeled by a linearized Fokker–Planck operator, avalanche source and synchrotron-radiation reaction losses. Temporal profiles of electron temperature, density, toroidal electric field and plasma effective charge used in the calculation are shown in figure 4. All parameter profiles were measured at the magnetic axis, except for the electric field which was measured at the plasma edge. Because of this, the calculated momentum-space distribution function is formally only valid on the magnetic axis, but since the pitch-angles of most runaways in the resulting distribution are very small, so that trapping effects are negligible, we can use the same momentum-space distribution at all radii. We therefore take the obtained distribution function to be the distribution of runaways in the outer midplane. The electric field relaxation time is expected to be much shorter than the discharge time though, so that the radial profile of the electric field is expected to be approximately uniform.

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In figure 5, the resulting distribution function and corresponding synchrotron emission in momentum-space are shown. The relatively large number of runaway electrons with high energies causes the synchrotron emission to be dominated by runaway electrons with  ∼30 MeV energies and  ∼0.13 rad pitch angles. With bremsstrahlung we instead observe a different part of momentum-space, as the dominant runaways have energies around  ∼22 MeV and pitch-angles  ∼0.13 rad, as illustrated in figure 5(c).

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The synthetic synchrotron image resulting from the distribution function in figure 5 is shown in figure 6 with four different radial density profiles applied to it, with the additional assumption that the momentum-space distribution is the same at all radii. As is seen in figure 6, the radiation originates mainly from the HFS, which should be due to the finite spectral range effect described in section 2.2. It is clear from figure 6 that none of the assumed radial density profiles is suitable to reproduce all the details of the experimental image. Due to the smooth decrease in intensity towards the HFS in the experimental image figure 3, the radial density profile should have to decrease smoothly to zero at larger radii, as in figures 6(b)–(d).

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The rather wide spot obtained with a linearly decreasing profile, and the significant contribution from near the magnetic axis with the exponentially decreasing profile, suggests that such a rapidly decreasing profile is unlikely to explain the experimental image, at least with the momentum-space distribution used here. An off-axis peak in the radial density profile would provide even better agreement between simulation and experiment, however the two distinct, bright patches seen in the simulations would not go away with just a change in the radial distribution. Instead, the most plausible explanation is that the experimental runaway electron momentum-space distribution is different from the one predicted by solutions of the spatially homogeneous kinetic equation. There are important effects missing from this model, such as the effect of magnetic trapping, radial transport and drift-orbit losses that may be relevant to this DIII-D scenario. One further indication that it is the kinetic physics utilized that is not complete is, as mentioned, the appearance of two distinct, bright, vertically separated patches in the synthetic images. These bright patches should stem from the edges of the dominating SOV, and thus be the result of the line-integration effects described in section 1. The absence of these bright patches in experiment suggest that the experimentally observed SOV does not close on itself, which means that the dominating pitch angle must be much larger than the dominating pitch angle of the simulation.

The idea that kinetic processes not covered by the model employed are present in this DIII-D discharge was also suggested by [36], where radial transport or a kinetic instability was given as possible explanations. The evidence for this provided by [36] was an energy distribution function inverted from measurements with the GRI diagnostic, which did not match the distribution function obtained from kinetic simulations. The inverted distribution function in [36] was characterized by much lower maximum energies than the corresponding simulated distribution function. This should make the difference between the high- and low-field side contributions more distinct in figure 6 and more consistent with figure 3.

To test the hypothesis that figure 3 is consistent with the dominating particle having a lower energy and larger pitch angle, a toy distribution function where all particles have the same energy E  =  25 MeV but are distributed in pitch-angle so that the dominating particle has a pitch angle $\newcommand{\thetap}{\theta_{{\rm p}}} \thetap\sim 0.35$ rad. The distribution function is shown in figure 7(e), and in figure 7(f) the distribution function has been weighed with the synchrotron radiation weight function $\hat{I}_s$ (see section 2.3) to reveal from where in pitch-angle space that most synchrotron radiation will be emitted. In figures 7(a)–(d) the resulting synthetic synchrotron image is shown, with the radial density profiles of figures 6(e)–(h) applied in order.

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Of the synthetic images resulting from the toy distribution function in figure 7, it is figures 7(a) and (b) that most resemble the experimental image in figure 7. They all have a crescent shape with only one bright patch, and in both figures 3 and 7(a), (b) the bottom end of the spot extends slightly further to the LFS than the upper end. This provides further evidence for the conclusion that additional kinetic effects beyond those included in the spatially-homogeneous model need to be invoked in order to understand the measurements. An additional energy-limiting mechanism could shift the distribution to lower energies, where the higher rate of pitch-angle scattering could plausibly produce the required shape of the runaway distribution.

In figure 8, synthetic GRI images from mono-energetic and mono-pitch distributions simulated with SOFT are shown. Detectors are projected onto the poloidal plane orthogonal to the viewing direction of a central detector (no. 43 in figure 9, not considering the z component of the viewing vector). In figure 8(a), a runaway electron beam with a 50 cm radius was initialized with parallel and perpendicular momentum $p_\parallel = 20$ MeV c⁻¹ and $p_\perp = 1$ MeV c⁻¹, while in figure 8(b) the particles were given $p_\parallel = 20$ MeV c⁻¹ and $p_\perp = 3$ MeV c⁻¹. The overall shape of the bremsstrahlung 'spot' is the characteristic projection of a twisted cylinder, which results from the same SOV as synchrotron radiation, and in contrast to what we see in the synchrotron radiation images, the amount of radiation coming from the HFS and LFS seem to be roughly the same, due to the lack of a pitch angle dependence in the emitted bremsstrahlung spectrum equation (10). The pitch angle does affect the spatial distribution of the bremsstrahlung though, since it determines the size and shape of the SOV. The effect of the runaway electron distribution function parameters on a bremsstrahlung image is therefore the same as shown for synchrotron radiation in figure 2, except for the camera finite spectral range effect in figure 2(b) which is completely absent.

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The GRI was also used to determine the spatial distribution of bremsstrahlung during DIII-D discharge 165826, and the resulting measurement is shown in figure 9(a). In figure 9(b), the synthetic GRI image resulting from a SOFT simulation with the distribution function shown in figure 5(a) is displayed. Due to the few detector channels available in the measured GRI data, a quantitative comparison is not possible at this time and further analysis is left for the future when more detector channels are available.

The relation between the bremsstrahlung and synchrotron radiation images becomes more apparent by using a more idealized synthetic camera for the bremsstrahlung simulation, as done in figure 10. Both types of radiation give rise to exactly the same SOV, with the same bright edges, and the radial dependence must therefore also be the same. Bremsstrahlung, in contrast to synchrotron radiation, has a more uniform radiation intensity distribution across the SOV since it is independent of magnetic field-strength, which means that bremsstrahlung is seen roughly equally strongly on both the HFS and LFS. Based on figure 5(c), where the distribution function was weighed with the bremsstrahlung emission, we expect a wide range of pitch angles to contribute significantly to the emission. In the image, this appears as a very bright, S-shaped band in the center of the image, with gradually decreasing intensity in both vertical directions. This is the behavior seen in the simulated GRI image in figure 9(b), and becomes even more apparent in the high-resolution bremsstrahlung camera image figure 10(c).

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