Simulations of stand-off runaway electron beam termination by tungsten particulates for tokamak disruption mitigation

Figure 2 shows volumetric energy deposition profiles for runaways incident on tungsten particulates with incident energies from 1 to 10 MeV. For 6.3 and 10 MeV incident energies, the energy deposition profiles are close to uniform, tapering off towards the far corner (i.e. towards the maximal radial and depth coordinates). This tapering effect reflects the fact that runaways incident near the outer edge of the particulate tend to scatter out from the sides before traversing the full depth. For 4 MeV and lower incident energies, the profiles shift as energy deposition drops to zero beyond some terminal distance. This reflects the fact that lower-energy runaways cannot penetrate through the entire particulate, and will either deposit their full energy before being absorbed inside the material or will scatter backwards.

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Nonuniform energy deposition from incident runaways implies that tungsten particulates will ablate from front to back, rather than by uniform vaporization and expansion. To confirm this, the thermal diffusion time can be computed by

$$\begin{equation} \tau_\mathrm{td} = \frac{\rho c_\mathrm{p} L^2}{k_\mathrm{c}} \end{equation} \tag{ 1 }$$

where density $\rho = 19.25~\mathrm{g~cm^{-3}}$, specific heat capacity $c_\mathrm{p} = 0.134~\mathrm{J~(g~K)^{-1}}$, particulate depth $L = 0.1~\mathrm{cm}$, and thermal conductivity $k_\mathrm{c} = 173~\mathrm{W~(m~K)^{-1}}$ yield a thermal diffusion time $\tau_\mathrm{td} = 16.5~\mathrm{ms}$. As shown below, this timescale is much greater than those for particulate melting or vaporization, thus the ablation will proceed front-to-back rather than by fast heat conduction and uniform vaporization. This means that in the event of particulate melting or vaporization, the size of the particulate may change with time, potentially affecting the termination efficacy (although effects such as vapor shielding could mitigate this). Additionally, since a smaller particulate would absorb less energy per incident runaway the particulate may survive longer under the runaway flux, which could also mitigate the effect on termination efficacy.

Table 1 gives the major statistical quantities relating to runaway energy loss and deposition in tungsten particulates. The first key quantity is the average exit energy of a runaway electron, $E_\mathrm{out}$, from which the average energy lost per incident runaway is $E_\mathrm{loss} = E_0 - E_\mathrm{out}$. Runaway energy losses are classified as radiation losses ($E_\mathrm{rad}$), to secondary electron emission ($E_\mathrm{see}$) and to gamma radiation (E_γ ), and as energy deposited into the tungsten itself ($E_\mathrm{dep}$). The majority of energy lost at all incident energies is deposited into the tungsten particulate. However, while the absolute energy deposition increases with increasing incident energy, the relative magnitude (i.e. as a fraction of E₀) decreases with incident energy, from 60% deposition at 1 MeV down to only 20% deposition at 10 MeV. Radiation losses are the minority component at all energies, but increase in both absolute and relative magnitude with increasing incident energy which partially offsets the relative decrease in energy deposition.

Table 1. Key statistics for runaway energy loss and deposition in tungsten particulates. E₀: incident runaway energy. $E_\mathrm{out}$: average runaway exit energy. $E_\mathrm{see}$: average energy lost to secondary electron emission. E_γ : average energy lost to gamma radiation. $E_\mathrm{rad} = E_\mathrm{see} + E_\gamma$: total average radiative energy loss. $E_\mathrm{dep}$: average energy deposited into the tungsten particulate. $\Delta\epsilon_\mathrm{max}$: Peak volumetric energy deposition. $\tau_\mathrm{erg}$: time to absorb 99% of energy from runaways with initial energy E₀. $\tau_\mathrm{m}$ and $\tau_\mathrm{v}$: estimated melting and vaporization times under ablation by runaways incident with uniform energy E₀. Statistical uncertainty in the last one or two digits is given in parentheses for each reported value.

E0 (MeV)	$E_\mathrm{out}$ (MeV)	$E_\mathrm{see}$ (MeV)	Eγ (MeV)	$E_\mathrm{dep}$ (MeV)	$\Delta\epsilon_\mathrm{max}$ (MeV cm−3)	$\tau_\mathrm{erg}$ (µs)	$\tau_\mathrm{m}$ (µs)	$\tau_\mathrm{v}$ (µs)
1.0	0.38(4)	0.0021(2)	0.026(3)	0.5963(2)	$1.2(2) \times 10^4$	4.48(2)	9.6(14)	150.76(4)
1.6	0.61(7)	0.0041(4)	0.065(7)	0.9200(2)	$1.16(15) \times 10^4$	4.53(4)	9.9(12)	97.72(3)
2.5	1.03(11)	0.0079(8)	0.15(2)	1.3120(3)	$1.2(6) \times 10^4$	4.77(6)	9.(4)	68.53(2)
4.0	1.9(2)	0.016(2)	0.35(4)	1.7217(5)	$1.1(4) \times 10^4$	5.42(11)	10.(4)	52.220(14)
6.3	3.6(4)	0.034(3)	0.72(8)	1.9308(6)	$1.1(7) \times 10^4$	6.7(2)	10.(6)	46.565(14)
10.0	6.5(7)	0.072(6)	1.37(15)	2.0102(8)	$1.0(4) \times 10^4$	8.4(4)	11.(4)	44.72(2)

Given the average energy loss, we can immediately assess the runaway termination in energy terms,

$$\begin{equation} F_\mathrm{term} = 1 - \left(1 - f_\mathrm{area}\,f_\mathrm{erg}\right)^{t / \tau_\mathrm{re}} \end{equation} \tag{ 2 }$$

where $f_\mathrm{area}\approx 20\%$ is the fraction of the runaway beam cross-sectional area intersected by tungsten particulates, $f_\mathrm{erg} = E_\mathrm{loss} / E_0$ is the average fraction of incident energy lost by a colliding runaway, and $\tau_\mathrm{re} = 2\pi R_0 / c = 0.13~\mathrm{\mu s}$ is the runaway orbital period in ITER ($R_0 = 6.2~\mathrm{m}$), indicating that runaways orbiting the tokamak will have multiple chances to collide with a particulate, or may strike a particulate multiple times (for this estimate of $\tau_\mathrm{re}$, we neglect the helicity of the runaway orbits, on the presumption that the injected particulate cloud is spread across the cross-sectional area of the runaway beam, and we take the velocity of all runaways as the vacuum speed of light c, which is a reasonable approximation since the velocity of even a 1 MeV electron is 0.94c). We find that the time to remove 99% of runaway energy, denoted as $\tau_\mathrm{erg}$ in table 1, is less than 10 µs for all simulated incident energies. In other words, 99% of runaway energy will be removed in fewer than 10² runaway toroidal transits. Therefore, tungsten particulates are effective for removing nearly all runaway energy in a very short time.

As an aside, the total volume of tungsten needed to cover $f_\mathrm{area}\approx 20\%$ may be estimated by

$$\begin{equation} V_\mathrm{tot} = f_\mathrm{area}\frac{I_\mathrm{re}}{j_\mathrm{re}}\frac{V_\mathrm{par}}{A_\mathrm{par}} \end{equation} \tag{ 3 }$$

where the ratio of runaway current to runaway current density, $I_\mathrm{re} / j_\mathrm{re}$, gives the cross-sectional area of the runaway beam. For typical values $I_\mathrm{re} = 5~\mathrm{MA}$ and $j_\mathrm{re} = 1~\mathrm{MA~m^{-2}}$ gives a cross-sectional area of 5 m². Multiplying by $f_\mathrm{area}$ yields a total particulate cross section of 1 m². Dividing this by the particulate cross-sectional area, $A_\mathrm{par} = \pi D^2/4 = 7.85\times 10^{-3}~\mathrm{cm^2}$, gives the total number of particulates to inject, and multiplying this by the particulate volume, $V_\mathrm{par} = A_\mathrm{par}L = 7.85\times 10^{-4}~\mathrm{cm^3}$, gives the total volume. For the given conditions, we find $V_\mathrm{tot} = 10^3~\mathrm{cm^3}$ of tungsten is required to achieve $f_\mathrm{area} = 20\%$, which is 19.25 kg after multiplying by density. This highlights the advantage of stand-off termination by particulates rather than terminating runaways at the wall, since to achieve the same volume of tungsten at a runaway final impact spot size of ∼100 cm² would require a rather impractical armor thickness of 10 cm. Note that $f_\mathrm{area}$, along with the particulate size, may be considered optimizable parameters to minimize the required volume of material, as discussed later. Additionally, delivery of $10^3~\mathrm{cm^3}$ of tungsten into the tokamak chamber will require designing new particulate injection devices, as noted in the Introduction, to enable the stand-off termination scheme.

To assess if the energy removal time, $\tau_\mathrm{erg}$, is short enough, we also estimate the melting and vaporization times for the tungsten particulates. The characteristic time for onset of melting is

$$\begin{equation} \tau_\mathrm{m} = \frac{e}{A_\mathrm{par} j_\mathrm{re}}\times\frac{\Delta H_\mathrm{m}^\circ N}{\Delta\epsilon_\mathrm{max}} \end{equation} \tag{ 4 }$$

where e is the electronic charge, $j_\mathrm{re}\sim 1~\mathrm{MA~m^{-2}}$ is the runaway current density, $\Delta H_\mathrm{m}^\circ = 1.594~\mathrm{eV}$ is the per-atom enthalpy of melting, $N = 6.306\times 10^{22}~\mathrm{cm^{-3}}$ is the atomic density of tungsten, $A_\mathrm{par}$ is the particulate cross-sectional area given previously, and $\Delta\epsilon_\mathrm{max}$ is the peak volumetric energy deposition (in MeV cm⁻³), given in table 1. Physically, since the peak rate of energy deposition is the product of runaway-particulate collision frequency ($A_\mathrm{par} j_\mathrm{re} / e$) and peak volumetric energy deposition per runaway ($\Delta\epsilon_\mathrm{max}$), dividing the volumetric melting enthalpy ($\Delta H_\mathrm{m}^\circ N$) by the peak rate of energy deposition yields the time before onset of melting. Using the peak energy deposition yields the earliest possible onset time for particulate melting, providing a conservative lower bound on the particulate lifetime as shown in table 1. Inspecting table 1 shows that for all simulated runaway energies, the energy removal time is less than the onset-of-melting time, i.e. $\tau_\mathrm{erg} \lt \tau_\mathrm{m}$. Thus, tungsten particulates appear capable of removing nearly all runaway energy before any tungsten can melt and splatter on the first wall.

For completeness, we also estimate the particulate lifetime before total vaporization,

$$\begin{equation} \tau_\mathrm{v} = \frac{e}{A_\mathrm{par} j_\mathrm{re}}\times\frac{E_\mathrm{coh} NV_\mathrm{par}}{E_\mathrm{dep}} \end{equation} \tag{ 5 }$$

where $E_\mathrm{coh} = 8.90~\mathrm{eV}$ is the cohesive energy of tungsten, $E_\mathrm{dep}$ is the energy deposition from table 1, and $V_\mathrm{par}$ is the particulate volume given previously. Physically, this is similar to (4), giving the ratio of the energy required for complete vaporization ($E_\mathrm{coh} NV_\mathrm{par}$) over the average energy deposition rate per runaway ($E_\mathrm{dep} \times A_\mathrm{par} j_\mathrm{re} / e$). This provides an approximate upper bound for runaway termination by tungsten particulates. In the event that runaways are not completely terminated before the onset of melting, the particulates should at least remain in a condensed state until time $\tau_\mathrm{v}$ and thus remain effective. Note that this only a rough estimate, neglecting important material effects such as vapor shielding which can also contribute to runaway termination.

While the calculations in table 1 give a favorable picture for runaway termination from an energy perspective, we note that the large uncertainty in $\tau_\mathrm{m}$ as well as the possibility for larger values of $j_\mathrm{re}$ do raise the possibility that particulate melting could occur nevertheless. For example, if the runaway current density is doubled to $j_\mathrm{re} \sim 2~\mathrm{MA m^{-1}}$, then $\tau_\mathrm{m}$ and $\tau_\mathrm{v}$ will be halved, which could lead to the onset of particulate melting dependent on the detailed runaway energy distribution function. In this case, there are a few considerations. First, since our estimates for $\tau_\mathrm{m}$ and $\tau_\mathrm{v}$ assume a constant runaway current, while in reality the runaway current will decrease as runaways are terminated, the actual time before onset of melting could be significantly longer; in this work we have endeavored to provide the most conservative estimate. Second, it would be possible to increase the amount of tungsten particulates injected to increase $f_\mathrm{area}$ and thus reduce the termination time $\tau_\mathrm{erg}$.

Finally, in a case where particulate melting is probable, complete vaporization of the particulates may be preferable, since the broadly uniform deposition of vaporized tungsten on the first wall is preferable to droplets splashing on the wall which can pose significant plasma-material interactions problems during later operation. In this case, an optimal particulate size would be desired such that $\tau_\mathrm{erg} \gtrsim \tau_\mathrm{v}$ (counting on the vaporized tungsten to finish the runaway termination process). This latter consideration is also valuable since size optimization for complete vaporization can reduce the mass of tungsten required. Investigations to determine an optimal particulate size for both complete vaporization and runaway termination are therefore a potential avenue for future studies.

While the energy removal time calculated above is a convincing metric for the runaway termination efficacy of tungsten particulates, it may not give the full picture. Runaways which lose energy on collision with a tungsten particulate may be accelerated by the toroidal electric field and regain some energy, complicating the picture. Runaway orbit simulations which include this acceleration can provide further confidence, but are beyond the scope of the present work and will be carried out in a separate study using the MCNP data produced in current study. Therefore, here we present runaway collisional data to qualitatively assess the ability of tungsten particulates to reduce the runaway population from a collisional kinematics perspective. In this perspective, tungsten particulates terminate colliding runaways by (1) pitch-angle scattering of runaways into magnetically trapped orbits or directional reversal (backscattering), leading to rapid deceleration, or by (2) directly absorbing runaways into the particulate.

Figure 3 shows energy and angular-resolved fluxes of runaways exiting from simulated particulates. Strong pitch-angle scattering is visible as the spread of data away from the 0^∘ angular mark. While significant scattering through the full angular range occurs for all incident runaway energies, there is a slight tendency toward forward scattering at 6.3 and 10 MeV, whereas for 4 MeV and lower energies backscattering is more prevalent. The shift towards backscattering arises because the lower-energy runaways are unable to penetrate through the tungsten particulate (see corresponding energy deposition plots in figure 2). In the same figure, the runaway energy attenuation is visible as the spread of the distribution along the energy (radial) axis below the incident energy. The energy distribution is not strongly correlated with the angular distribution in any case. It is visually evident that while most exiting runaways lose up to a decade (90%) of the incident energy, these remain relativistic electrons (i.e. $E \unicode{x2A7E} 100~\mathrm{keV}$).

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Table 2 gives the major statistical quantities relating to runaway collisional kinematics. On first inspection, the two key quantities are the fraction of runaways which experience strong pitch-angle scattering ($f_\mathrm{scat}$) or which are absorbed into the particulate ($f_\mathrm{abs}$). We define strong pitch-angle scattering in terms of the scattering cosine ξ as follows: first, we consider a critical value $\xi_\mathrm{c} = \sqrt{2a / R_0}$, where a is the tokamak minor radius and R₀ the major radius. For ITER ($a = 2.0~\mathrm{m}, R_0 = 6.2~\mathrm{m}$) this gives $\xi_\mathrm{c} = 0.8$ or a scattering angle of 38.8^∘. This critical value is the boundary between runaway passing-through and trapped orbit zones, illustrated in figure 4. Scattered runaways with $\xi \gt \xi_\mathrm{c}$ are considered to pass through and continue along a runaway orbital trajectory. Scattered runaways with $-\xi_\mathrm{c} \lt \xi \lt \xi_\mathrm{c}$ will enter trapped orbits and will lose energy due to enhanced synchrotron radiation, and thus are considered terminated ($f_\mathrm{trap}$). Finally, scattered runaways with $\xi \lt -\xi_\mathrm{c}$ are backscattered and will be decelerated by the parallel electric field before falling into the trapped zone, and thus are also considered terminated ($f_\mathrm{back}$). Therefore, we define the fraction of runaways terminated by pitch-angle scattering as $f_\mathrm{scat} = f_\mathrm{trap} + f_\mathrm{back}$.

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Table 2. Key statistics for runaway collisional kinematics and termination by tungsten particulates. E₀: incident runaway energy. $f_\mathrm{trap}$: fraction of exiting runaways with pitch-angle scattering into the trapped zone, $|\xi| \unicode{x2A7D} \xi_c\equiv \sqrt{2a/R_0}$. $f_\mathrm{back}$: backscattered fraction of runaways, $\xi\lt-\xi_c$. $f_\mathrm{abs}$: fraction of incident runaways absorbed inside the particulate. $f_\mathrm{term}$: terminated fraction, $f_\mathrm{trap} + f_\mathrm{back} + f_\mathrm{abs}$. $\tau_\mathrm{par}$: time to terminate 99% of the runaway population by pitch-angle scattering or absorption. Statistical uncertainty in the last one or two digits is given in parentheses for each reported value.

$E_\mathrm{0}$ (MeV)	$f_\mathrm{trap}$ (%)	$f_\mathrm{back}$ (%)	$f_\mathrm{abs}$ (%)	$f_\mathrm{term}$ (%)	$\tau_\mathrm{par}$ (µs)
1.0	34.58(6)	15.68(4)	48.53(6)	98.80(9)	2.718(3)
1.6	38.34(6)	13.34(4)	46.04(6)	97.72(9)	2.752(3)
2.5	45.10(7)	10.37(3)	40.44(6)	95.92(9)	2.810(3)
4.0	56.28(7)	6.86(3)	29.31(5)	92.45(9)	2.927(3)
6.3	67.00(8)	3.55(2)	14.12(3)	84.68(9)	3.225(4)
10.0	62.38(8)	1.284(11)	5.74(2)	69.40(8)	4.005(5)

Thus, the fraction of runaway population which has been eliminated by time t may be estimated from

$$\begin{equation} F_\mathrm{term} = 1 - \left(1 - f_\mathrm{area} \,f_\mathrm{term}\right)^{t/\tau_\mathrm{re}} \end{equation} \tag{ 6 }$$

following the definitions given with (2) and with $f_\mathrm{term} = f_\mathrm{scat} + f_\mathrm{abs}$. The times required to conclusively eliminate 99% of the runaway population, $\tau_\mathrm{par}$, at each runaway energy are given in the latter column of table 2. Comparing with the melting and vaporization times in table 1, the values of $\tau_\mathrm{par}$ for collisional termination are less than the melting and vaporization times at all energies. Therefore, from the collisional kinematics perspective tungsten particulates appear capable of eliminating nearly all runaways from the population before any tungsten can melt and splatter on the first wall.

The final important phenomenon arising from runaway collisions with tungsten particulates is the emission of secondary radiation. This comes in two flavors: secondary electron emission may produce additional runaways in the form of relativistic secondary electrons, while gamma radiation may also produce additional runaways if the gamma rays interact with other tungsten particulates. We consider both relativistic electrons and gamma rays to have $E\unicode{x2A7E} 100$ keV for the purposes of this work. Interested readers may find energy and angular-resolved distributions for secondary electron and photon fluxes in the Supplementary Material.

Table 3 gives the major statistical quantities related to secondary radiation emission from tungsten particulates. Generally, gamma radiation dominates over secondary electron emission with an order of magnitude greater emission rate and modestly greater average energy per emitted particle. However, both types of secondary radiation carry away only a fraction of the incident runaway energy, which means that even if secondary radiation leads to runaway reseeding, the overall energy of the runaway population is still reduced.

Table 3. Key statistics for secondary radiation emitted from tungsten particulates. E₀: incident runaway energy. $N_\mathrm{se} / N_\mathrm{re}$ and $N_\gamma / N_\mathrm{re}$: number of secondary electrons and gamma rays emitted per incident runaway, respectively. $\langle E_\mathrm{se}\rangle$ and $\langle E_\gamma\rangle$: average emitted energy of secondary electrons and gamma rays, respectively. Statistical uncertainty in the last one or two digits is given in parentheses for each reported value.

$E_\mathrm{0}$ (MeV)	$N_\mathrm{se} / N_\mathrm{re}$	$\langle E_\mathrm{se}\rangle$ (MeV)	$N_\gamma / N_\mathrm{re}$	$\langle E_\gamma\rangle$ (MeV)
1.0	0.0081(9)	0.26(4)	0.0755(3)	0.34(4)
1.6	0.0128(10)	0.32(4)	0.1563(4)	0.42(5)
2.5	0.0191(10)	0.41(5)	0.2890(5)	0.53(6)
4.0	0.0303(11)	0.53(6)	0.4932(7)	0.71(8)
6.3	0.0456(12)	0.74(7)	0.7201(8)	1.00(11)
10.0	0.0631(12)	1.14(11)	0.9370(10)	1.5(2)

The angular distributions of relativistic secondary electrons ($E\unicode{x2A7E} 100$ keV) emitted from tungsten particulates are shown in figure 5, in terms of the exit cosine $\xi = \cos\theta$. At every runaway energy, the angular distribution is extremely broad, with a rather uniform distribution for 10 MeV runaways and a gradual shift away from forward emission and toward backward emission as runaway energy decreases. This implies that the vast majority of secondary electrons will not fall into the passing cone and be accelerated by the parallel electric field. Even for those born inside the passing cone, their pitch is usually large enough that synchrotron radiation can overwhelm the parallel electric field acceleration, which is the primary cause for the runaway vortex flow in momentum space [42]. In other words, the broad pitch distribution of the secondary electrons will lead very few of them to be promptly accelerated as new runaways. Combined with the low rates of secondary electron emission seen in table 3, we conclude that secondary electron emission from tungsten particulates will have a minimal impact on overall termination efficacy.

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Gamma rays emitted from one tungsten particulate may produce relativistic secondary electrons through collisions with other nearby particulates. These secondary electrons result chiefly from bremsstrahlung interactions, with minority contributions from fluorescence and pair production collisions. We carried out additional simulations to assess relativistic electron production by gamma radiation, with the same setup as in figure 1 but with photons instead of electrons as the source particles. Table 4 summarizes the key results from these simulations. Note that both the emission rate and emitted energy for secondary gamma rays are fairly low for all incident energies, with most of these secondary gamma rays resulting from bremsstrahlung interactions of secondary electrons, with minority contributions from fluorescence and positron annihilation. Even for 10 MeV incident gamma rays, the average energy of a secondary gamma ray is less than 1 MeV. Since the production of relativistic secondary radiation by 1 MeV gamma rays is extremely low, secondary gamma radiation is not a significant consideration for runaway reseeding. In other words, gamma radiation chains leading to secondary electron emission (e.g. $\gamma \rightarrow \gamma \rightarrow \mathrm{e^-}$) are negligible as a source of relativistic secondary electrons.

Table 4. Key statistics for gamma ray collisions with tungsten particulates. E₀: incident gamma ray energy. $f_\mathrm{abs}$: fraction of incident gamma rays absorbed inside the particulate. $N^{^{\prime}}_\gamma / N_\gamma$ and $N_\mathrm{se} / N_\gamma$: number of secondary gamma rays and relativistic electrons emitted per incident gamma ray, respectively. $\langle E_{\gamma^{^{\prime}}}\rangle$ and $\langle E_\mathrm{se}\rangle$: average emitted energy of secondary gamma rays and relativistic electrons, respectively. Statistical uncertainty in the last one or two digits is given in parentheses for each reported value.

E0 (MeV)	$f_\mathrm{abs}$ (%)	$N^{^{\prime}}_\gamma / N_\gamma$	$\langle E_{\gamma^{^{\prime}}}\rangle$ (MeV)	$N_\mathrm{se} / N_\gamma$	$\langle E_\mathrm{se}\rangle$ (MeV)
1.0	3.172(7)	0.0040(5)	0.39(3)	0.01 319(11)	0.55(6)
1.6	1.812(5)	0.0133(6)	0.48(4)	0.01 736(13)	0.84(9)
2.5	2.164(5)	0.0336(6)	0.52(5)	0.0259(2)	1.22(14)
4.0	3.276(6)	0.0574(6)	0.55(5)	0.0444(2)	1.8(2)
6.3	4.734(7)	0.0770(7)	0.62(5)	0.0745(3)	2.7(3)
10.0	6.481(8)	0.0935(7)	0.81(7)	0.1156(3)	4.3(5)

The rates of relativistic electron emission by gamma ray collisions are similar to the rates of gamma radiation, with the emitted secondary electron flux ranging from 1% to 12% of the incident gamma ray flux. However, the emitted energies of secondary electrons are of the order ∼50% of the incident gamma ray energy. These energies are much greater than not only the secondary gamma radiation energies but also the secondary radiation energies from runaway collisions with the particulates. Secondary electron emission at these energies could contribute significantly to the runaway population.

In light of these large secondary electron energies from gamma ray collisions, we give two reasons for optimism. First, we note that the energy distribution of gamma radiation from tungsten particulates has a broad spread and is concentrated away from the highest gamma ray energies. Figure 6(a) shows the energy distributions for gamma radiation from tungsten particulates induced by runaways from 1 to 10 MeV. At every incident runaway energy, the majority of gamma radiation has energies less than 1 MeV, and the distribution falls off sharply (i.e. by an order of magnitude) towards the highest energy of each distribution. Thus, only a minority of gamma rays can produce relativistic secondary electrons. In the end, the majority of emitted gamma rays will produce a very small number of relativistic electrons (1 per 100 gamma rays or fewer), and most of these with energies well below 1 MeV.

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Second, as for secondary electron emission by runaway collisions, the angular distribution for secondary electron emission by gamma ray collisions is also quite broad. These distributions are shown in figure 6(b) for each incident gamma ray energy. Since these distribution are spread out over the whole angular range, relatively few emitted relativistic electrons will have the narrow pitch that can be re-accelerated by the parallel electric field. Note that while these distributions do show a preference for forward emission at higher incident gamma ray energies, the forward direction is relative to the incident gamma ray trajectory, which is not in general aligned with magnetic field lines (see the Supplementary Material for details of gamma ray angular distributions). In other words, an emitted secondary electron is rarely aligned with the magnetic field and thus rarely experiences parallel electric field re-acceleration.

To summarize, while runaway collisions with tungsten particulates can lead to relativistic secondary electron emission and gamma radiation, further MCNP simulations show that neither of these secondary radiation types pose a serious challenge to runaway termination efficacy. On one hand, the energy of secondary radiation particles are necessarily lower than the incident runaway energy, thus reducing the overall energy of the runaway population. On the other hand, secondary radiation is emitted with broad energy and angular distributions, such that the majority of secondary electrons are not aligned with the magnetic field for parallel electric field acceleration. Future studies on runaway orbit simulations will quantify the role of these secondary electrons on plasma current decay.

As a final note, we observe that the fast (${\lt}10~\mu$s) termination of runaway current can potentially impose large forces on the vacuum vessel and magnetic field coils. We anticipate that these forces should be counteracted by runaway current reconstitution [17]. In this case, the generation of many relativistic or suprathermal electrons from tungsten particulate interactions can contribute to this reconstitution effect, helping to carry the current and slow down the overall current dissipation process to within acceptable limits. While the detailed plasma physics of runaway current reconstitution are well beyond the scope of the present work, the results presented here will form the basis to derive the collision operator necessary for future simulations to investigate this process in detail.

Besides the interactions discussed above, there are a few potential sources of additional runaways resulting from tungsten particulate injection which we have not yet considered. Here, we briefly review the most salient of these and demonstrate that these mechanisms can be safely neglected in our analysis.

Runaway generation by gamma ray Compton scattering. Gamma ray Compton scattering off of cold plasma electrons has been proposed as a mechanism for runaway seeding [3]. Since tungsten particulates can generate large amounts of gamma radiation, the relevance of this mechanism should be assessed. However, it is sufficient to note that the maximum cross section for Compton scattering from a free electron is the Thomson cross section, $\sigma_\mathrm{T} = 6.65\times 10^{-25}~\mathrm{cm^2}$. For an ITER-relevant electron density of $n\sim 4\times 10^{20}~\mathrm{m^{-3}}$ [43] this gives a mean free path on the order of $10^9~\mathrm{m}$, far exceeding the dimensions of any possible terrestrial fusion device. Therefore, Compton scattering of gamma rays from cold plasma electrons is extremely improbable, and cannot be a significant runaway seeding mechanism.

Secondary electron emission from the first wall. We have carried out additional simulations of gamma ray incidence at the tungsten first wall. These simulations follow the same procedure as in section 2, except that the incident particles are photons and the geometry is a flat tungsten surface instead of a small cylinder. The resulting secondary radiation fluxes in these simulations were three orders of magnitude less than those resulting from gamma ray collisions with tungsten particulates. This is because secondary radiation can only emerge from the first wall on a near-backwards trajectory (i.e. $\xi \approx -1$), whereas secondary radiation may emerge from a small particulate in any direction. Therefore, gamma ray interactions with the first wall cannot be a significant runaway seeding mechanism.

Runaway seeding from nuclear interactions. In principle, neutrons interacting with tungsten particulates could lead to runaway seeding directly (i.e. β decay) or indirectly (i.e. γ decay). In practice, such possibilities are irrelevant. On one hand, the neutron flux from fusion will disappear after the thermal quench (TQ), while tungsten particulate injection will not occur until well after the onset of the current quench (CQ). On the other hand, the most rapidly-decaying isotope resulting from neutron interaction, ^183mW, has a half-life of 5.3 s [44], many orders of magnitude greater than the ∼10 µs timescale of the runaway termination process by tungsten particulates. Therefore, nuclear interactions with tungsten particulates cannot be a significant runaway seeding mechanism during this late stage of current quench.

Runaway seeding from photonuclear interactions. In principle, high-energy gamma rays interacting with tungsten particulates could cause photonuclear interactions leading to runaway seeding by nuclear decay processes. In practice, the probability of photonuclear effects is rather remote. The cross section for photonuclear interactions in tungsten is less than 0.1 b (10⁻²⁵ cm²) for 10 MeV gamma rays and drops to 30 µb ($3\times 10^{-29}$ cm²) at 1 MeV [45]. By contrast, the EPRDATA14 photon total cross section used in our MCNP simulations is greater than 10 b (10⁻²³ cm²) for all photon energies, so even at 10 MeV the probability that a photon collision will be photonuclear is less than 1%. Furthermore, even if a photonuclear interaction does occur, the dominant interaction types are $(\gamma,n)$ leading to neutron emission [46], which as discussed above can be safely neglected. Therefore, photonuclear interactions with tungsten particles cannot be a significant runaway seeding mechanism.