Possible mitigation of tokamak plasma–surface interactions using thermionic divertor plates with inverse sheaths

Reducing the divertor plate heat flux and sputtering to a tolerable level in future tokamaks likely necessitates operating in a “detached” regime where most of the power exhausted into the scrape-off-layer (SOL) dissipates through radiation instead of on the divertor surface.^1–4 Achieving detachment requires reducing the target plasma temperature to a few eV or less in which case radiation rates are high enough to yield an effective energy sink. Conventional detachment scenarios⁴ establish a cold target plasma through injection of neutral fuel atoms or radiating impurities, but injection methods are known to compromise the core fusion plasma. It is therefore worthwhile to explore alternative methods of taming the plasma-wall interaction. In this paper, we consider using thermionic electrons.

Although emission from divertor plates has been considered in the tokamak literature for decades,^5–10 previous modeling works assumed that the surface potential remains below the plasma potential for all emission intensities. Recent studies found that under sufficiently strong emission, conventional models break down and an inverse sheath forms where the surface is above the nearby plasma potential.¹¹ Basic simulations comparing conventional and inverse regimes suggest that inverse sheaths cause sharp reduction of the nearby plasma temperature.¹² This motivated an idea that inverse sheaths could be used to establish a cold detached target plasma in tokamaks. We encourage readers to see the very recent work by Masline et al.^13,14 on inverse sheath effects in divertor plasma fluid simulations. Although the particular cases considered in their paper did not show an improvement in heat flux or detachment due to inverse sheaths over classical sheaths, inverse sheaths would alter other aspects of the SOL plasma (e.g., prevent acceleration of ions into the targets), possibly offering other advantages over conventional operating regimes including suppression of sputtering and prevention of arcs.

This paper will give a broad overview of how divertor plasmas will change in the presence of inverse sheaths and establish groundwork for further study. In Sec. II, we will review past works predicting how electron emission affects the sheath and the operation of tokamak divertors. Then in Sec. III, we derive a new electron heat flux boundary condition for the recently discovered inverse sheath. From that, we show that inverse sheaths yield a lower target plasma temperature than conventional sheaths for given SOL input power, thereby facilitating detachment. In Sec. IV, we discuss significant advantages that thermionic inverse sheath detachment may have over conventional techniques of inducing detachment. In Sec. V, we address the questions of how much emission is needed for inverse sheath detachment and what materials might be used to provide it. Concluding remarks and suggestions for further study are presented in Sec. VI.

The likelihood of intense electron-induced electron emission from plasma-facing surfaces in tokamaks has long been recognized.^5–10 Electron-induced emission consists of plasma electrons elastically or inelastically reflecting off the surfaces, plus secondary electrons ejected from the solid.¹⁵ Many plasma-facing component materials give substantial electron-induced emission when the temperature of incident plasma electrons is tens of eV or more.¹⁶ For some materials including lithium, oxide layers accumulated during plasma exposure can increase the emission yield well beyond the yields predicted based on measurements with clean samples.^17–19 Tungsten divertor plates in tokamaks can be heated by plasma bombardment to temperatures sufficient for thermionic emission, whose flux might exceed the influx of plasma electrons under certain conditions in ITER.²⁰ The radiation released within the SOL in attached or detached divertor conditions is capable of inducing substantial photoelectron emission from the target.²¹ 

The conventional view of electron emission is as follows. For a floating surface, which is often assumed for a divertor plate, the condition of zero current requires that Γ_e,in(1 − γ) = Γ_ion. The electron influx to the target Γ_e,in is typically calculated assuming a Maxwellian electron source filtered by the sheath potential Φ_sh, defined as the potential of the surface relative to the sheath edge. The Bohm velocity is used to estimate the ion flux, which is (T_e/m_ion)^1/2 assuming cold ions (zero ion temperature). The emission coefficient γ is defined as the average number of electrons emitted per plasma electron striking the surface. From these basic assumptions, one can derive a common expression⁸ used by fusion researchers for Φ_sh in terms of γ,

Note that Φ_sh is negative. Under zero emission, q_eΦ_sh = −3.2T_e for deuterium ions. The sheath weakens in amplitude as γ increases, as shown in Fig. 1. Although Eq. (1) appears valid for γ > 1, it is not because the original term Γ_e,in(1 − γ) becomes negative and Γ_ion must be non-negative. In an influential paper, Hobbs and Wesson²² calculated that for some critical γ_cr close to unity, see Eq. (2), the classical sheath transitions to “space-charge limited” (SCL) with a virtual cathode VC, sketched here in Fig. 2. They predicted that any supplemental increases of emission are suppressed by the virtual cathode, saturating the effective γ at γ_cr. Hence, the SCL sheath solution provides a viable way to maintain zero current even if γ ≫ 1.

Over the years, researchers extended the Hobbs-Wesson model to treat emitting sheaths in situations with non-Maxwellian plasma electrons,²³ nonzero thermionic temperature,²⁴ and biased surfaces.²⁵ Ahedo modeled the effect of emission on the presheath-sheath matching.²⁶ Other authors considered the influence of the oblique B field in tokamaks.²⁷ A substantial portion of the emitted electron flux from a flat surface is predicted to return to the surface by Larmor gyration^28–33 when the field is grazing, thereby reducing the effective γ. However, it has also been noted that roughness can make the actual angle of incidence on most of the surface more normal, thereby suppressing the suppression.^7,8 Even if the divertor plate is flat when installed, an intricate microscopic structure develops during exposure.^34,35

The Hobbs-Wesson paper remains the primary work cited by fusion researchers when predicting how electron emission affects the sheath, both in tokamaks³⁶ and magnetic mirrors.³⁷ Their prediction—weakening of the classical sheath and transition to a SCL sheath with increasing emitted flux—was verified in many 1D planar geometry particle-in-cell (PIC) simulations^38–41 starting in the 1990s. Recently, Komm et al. developed an advanced 2D-3V PIC model and demonstrated SCL sheaths in an oblique B field.⁴² 

However, all theoretical models and simulation demonstrations of SCL sheaths have omitted collisions. Campanell and Umansky showed in 1D-1V unmagnetized plasma simulations that when charge-exchange collisions are included, the virtual cathodes accumulate cold ions and SCL sheaths are destroyed, transitioning to inverse sheaths.¹¹ Kinetic simulations of inverse sheaths in multidimensional geometries or tokamak-relevant models with an oblique B field have not yet been demonstrated. Even with one spatial dimension, realistic demonstration of the emissive plasma-wall transition with oblique B would be computationally expensive, requiring three velocity dimensions, ion and electron collisions, large plasma and thin sheath, and high enough resolution to capture the fine structures in physical space and the velocity space distributions. We encourage readers to pursue such a demonstration.

It is important to consider how emission from the divertor plate affects not just the sheath but the overall tokamak. It is often thought to be harmful. Since the influx of plasma electrons to the surface increases with γ as Γ_ion/(1 − γ), many researchers⁴³ developed an impression that emission worsens the heat flux. However, that argument assumes that the T_e for incident electrons is constrained (independent of γ). If instead the power entering the SOL is a global constraint and the target is the dominant heat sink, that concern breaks down (we will revisit this issue later). Emission is more likely to increase the heat flux at surfaces too small to affect the global power balance such as probes, dust grains, and localized hot spots on the target. A feedback loop of thermionic emission weakening the sheath, causing enhanced heating, and thus more thermionic emission results in vaporization of tungsten dust in tokamaks.^44,45

Weakening of the sheath by electron emission has some benefits for divertors. It reduces the impact energies of ions, thereby reducing several problems⁴⁶ related to ion impacts including surface erosion, impurity generation, and tritium implantation. Decades ago, researchers even considered deliberate use of thermionic divertor plates to mitigate the plasma-surface interactions.⁴⁷ Schwager et al.⁴³ predicted that for certain operating conditions, emissive plates could bring ion impact energies below the sputtering threshold of tungsten. They verified experimentally in a low-temperature plasma device the reduction of ion impact energies with increasing thermionic emission from a thoriated tungsten surface. They reported an unexpected result that the measured ion impact energies approached zero under strong emission, in contrast to their SCL sheath theory. This anomaly could now be thought of as evidence of an inverse sheath. Takamura et al. put an externally heated thermionic LaB₆ target in a linear divertor simulator.⁴⁸ They showed that the emission has a strong influence on the target potential and that surface heating provided by the plasma enhances the thermionic emission.

Studies of using thermionic divertor plates to reduce sputtering that were popular in the 1990s did not continue through today. That is probably because most researchers believed that it could not solve the overall heat flux problem and is therefore not a practical divertor solution. The new idea proposed in the present paper is that if the emission is intense enough to form an inverse sheath, it can mitigate most of the plasma–wall interaction challenges including ion-impact damage, total heat flux, and arcs.

It is worth noting that other uses for thermionic surfaces in tokamaks have been studied. It was shown that injection of thermoelectrons into the plasma by auxiliary negatively biased filaments can weaken the sheath potential at the surfaces that collect the return current, thereby reducing sputtering.⁴⁹ Radial current driven by thermionic filaments was shown to establish a radial transport barrier (H-mode).⁵⁰ Thermionic emissive probes are used to measure plasma potential in edge plasmas.⁵¹ Filaments and probes are less likely to survive in future reactor tokamaks. To make reactors more efficient, it has been proposed that thermionic converters can be attached to the hot surfaces in order to generate some electricity from the exhaust power.⁵² 

It is useful to review conventional sheath particle and energy balance, and then discuss what changes in the presence of inverse sheath. A “conventional” (classical or SCL) sheath confines most plasma electrons near the target and lets a small high-energy tail through. The structure of the electron velocity distribution functions (EVDF's) is sketched in Fig. 2. Following Stangeby's book³⁶ Sec. 25.5, the electron heat loss through a conventional sheath Q_e,conv expressed in terms of the incident electron flux Γ_e,in is

The first term on the right hand side of (3) captures the thermal energy upon impact. The second term is the energy the plasma electrons had to give up to overcome the sheath before impacting the surface. The third is the energy injected back to the plasma by the thermoelectron flux γΓ_e,in accelerated by the sheath. The thermal energy of emission ∼T_emit is assumed small compared to |q_eΦ_sh| and is neglected. For a floating surface, Γ_e,in(1 − γ)= Γ_ion where Γ_ion is estimated by the Bohm criterion noted earlier and zero ion temperature was assumed for simplicity. The effect of emission on the Bohm velocity^26,53 is small and often neglected.

Thus, for a classical sheath with zero emission in a deuterium plasma we have Φ_sh = −3.2 T_e/q_e from Eq. (1) and then (3) reduces to,

The ion mass that normally appears in the heat flux expression was replaced with m_deut → 3672 m_e for easier comparison with the inverse sheath heat flux formula derived later. Q_e in conventional regimes is dependent on ion mass because m_ion influences the Bohm ion flux and hence the floating potential, and also γ_cr.

Conventional theory predicts a SCL sheath for any γ > γ_cr with γ_cr given by (2). For deuterium, γ_cr = 0.863 and we have Φ_sh,SCL= −1.22 T_e. The resulting heat flux is found to be

The extent to which a SCL sheath can raise the heat loss is not as severe as often feared. While the influx of plasma electrons increases about seven times from zero emission to SCL emission, via Γ_e,in= Γ_ion/(1 – 0.86), the heat transmission only ∼triples. This is because the average energy lost per escaping plasma electron is less when the sheath is weaker.

We note that heat transmission through sheaths is an intricate subject. The constant coefficients in Eqs. (4) and (5) change if more physics is included in the sheath model⁵⁴ or if the ion species is changed. We omit complexities of the grazing B field^55–58 on the presheath and sheath physics. Also note that our Q_e calculations treat the heat carried out of the plasma by electrons, which is not equivalent to the electron heat flux on the surface (a conventional sheath slows down electrons and adds energy to ions on their way to the surface³⁶). Finally, the SCL sheath model used here presumes cold (zero temperature) emission, which is reasonable for T_e ≫ T_emit. Thus, (5) cannot be used at low T_e. Sheehan's²⁴ model includes nonzero emission temperature and gives an interesting result that the SCL solution vanishes for T_e < T_emit.

The inverse regime carries a fundamentally different EVDF structure at the sheath edge, see Fig. 2. It is known from solid state physics theory⁵⁹ and experiments⁶⁰ that thermoelectrons are emitted with a half-Maxwellian distribution at a temperature T_emit equal to the surface temperature. A half-Maxwellian incident upon a decelerating potential barrier remains half-Maxwellian with the same temperature across the barrier (just the density drops).³⁶ The amplitude of an inverse sheath potential barrier, shown in Ref. 61 to be T_emitln(γ)/q_e, is immaterial to that argument. The distribution function of thermoelectrons entering the plasma therefore has to be half-Maxwellian with temperature T_cold (equal to T_emit) and some density n_cold. The incident plasma electrons do not have to be half-Maxwellian but we will approximate that they are. They have some effective temperature T_hot that must exceed T_emit in order for there to be a net heat flow into the surface. These electrons will have their own density n_hot. The lopsided two-temperature EVDF has been demonstrated in kinetic simulations of plasmas with inverse sheaths, cf. Fig. 3(d) of Ref. 11, but the heat flow was not studied in that paper.

Masline et al. were the first group to model electron heat flux through an inverse sheath in very recent work.^13,14 They used an analytical heat flux expression Q_e = n_ev_th,eγ_e(T_e – T_e,w) where γ_e is a transmission factor (not emission coefficient), v_th,e a thermal velocity corresponding to T_e, and T_e,w the wall temperature. Their expression is motivated by the fact that the heat flux is carried by two populations, plasma electrons moving into the wall and colder thermionic electrons entering the plasma. We agree with that point of view but our analysis arrives at a heat flux expression somewhat different in form from Refs. 13 and 14. Deriving the optimal boundary condition is an intricate problem because in the presence of an inverse sheath, the fluid n_e and T_e at the sheath edge are not equal to the temperature of the hotter plasma electrons approaching the wall (which we call T_hot in Fig. 2), but are strongly altered by the presence of the cold electrons. Establishing Q_e in terms of the fluid quantities requires some algebraic steps, which we consider below.

For each side of the inverse EVDF, the electrons are assumed to have the same temperature in the y-z plane as in the x-direction. Taking the third v_x moment of the overall distribution gives the net heat flux Q_e toward the surface produced by the hot and cold half-Maxwellian species together,

Our eventual goal is to express Q_e,inv in terms of a single density and temperature, i.e., the fluid-averaged quantities at the plasma boundary (sheath edge). The fluid density n_e,se there is

The fluid temperature T_e,se is the weighted average of the temperatures of the hot and cold electron populations

We assume that the surface is floating. The zero current condition implies that the fluxes of the hot and cold electrons at the inverse sheath edge must be equal

It is interesting that the cold electrons always have a higher density than the hot electrons via (9), as has been verified in simulations.¹¹ The emitted flux at the surface Γ_emit does not appear in the preceding Eqs. (6)–(9). The inverse sheath allows the proper fraction of cold emitted thermoelectrons into the plasma to replace the electrons lost from the plasma, see Fig. 2. The extra thermoelectrons are pulled back to the surface and play no role in the particle or energy balance. Ion contribution to the current was neglected in (9) because it will always be very small compared to the electron terms (or even zero in the limit of infinite inverse sheath). The fluid flow velocity of electrons (equaling that of ions, by zero current considerations) being small compared to the electron thermal velocity justifies our centering of the inverse sheath edge EVDF at v_x = 0 in Fig. 2.

Making use of Eqs. (6)–(9), we can express Q_e,inv in terms of the fluid quantities,

Equation (10) gives a general formula for electron heat flow through a floating inverse sheath. It can be used to patch an inverse sheath boundary condition to a fluid model of a SOL plasma. We note that the extra energy that incident electrons gain accelerating through the inverse sheath is immaterial to the heat flux on the surface because the portion of thermoelectrons that escape past the inverse sheath carry the same extra energy out of the surface.

A concern was raised by a reader that Q_e,inv in (10) appears to blow up in the mathematical limit T_emit ≪ T_e,se. We will explain why that is not a problem. In any SOL regime, T_e,se will vary if conditions are changed, in order to maintain the proper heat flow which is always finite. In the presence of an inverse sheath, if one reduces T_emit (by reducing the thermionic surface temperature), then T_e,se will be dragged down too such that both the numerator and denominator in (10) become smaller. Physically, reducing T_emit reduces the temperature of the cold electrons entering the plasma, and increases their density relative to the density of the hotter electrons, via Eq. (9). Both effects reduce the fluid-averaged quantity T_e,se at the plasma boundary facing an inverse sheath. In contrast, in the classical/SCL sheath regimes, there can be a large thermionic flux entering the plasma but due to sheath acceleration the beam has a low density at the sheath edge, cf. Figure 2, so it does not directly affect the plasma fluid quantities by much.

Figure 3 shows the heat flow as a function of T_e,se in the classical zero emission, SCL, and inverse sheath regimes. A representative value 0.2 eV is used for T_emit. Figure 3 might give a misleading impression that switching from a conventional sheath to an inverse sheath will cause an enormous increase in heat flux in a given device. Instead, the T_e,se facing the surface will generally reduce because the flow of cold thermoelectrons from an inverse sheath into the plasma has a strong cooling effect as discussed in the preceding paragraph. In Ref. 12, it was shown in kinetic plasma simulations where the T_e upstream of the surface was constrained to a constant value 20 eV, the T_e,se near the target was sub-eV when the sheath was inverse but ≫1 eV when the sheath was classical or SCL. Thermoelectrons in conventional sheath regimes do not cool the plasma as effectively because they are accelerated by the sheath to a higher energy.

According to (10), the heat flow through an inverse sheath becomes negative if T_e,se < T_emit. This means that the “evaporative cooling” (energy flow out from the surface due to loss of thermoelectrons) exceeds the heat flux from incident electrons. Such a situation is unlikely in divertors but plausible if the volumetric energy dissipation in the SOL exceeds the injected power.

In divertor plasma modeling, the power input into the upper SOL is often a control parameter set based on the power exhaust from the core in a certain experiment.³⁶ It is useful to compare the T_e,se required to transmit a given power to the surface through a classical zero emission sheath, a SCL sheath, and an inverse sheath. We calculate T_e,se as a function of Q_e/n_e,se in each sheath regime by inverting Eqs. (4), (5), and (10).

We see in Fig. 4 that the target T_e rises much more slowly with input power when the sheath is inverse. Remarkably, the same power responsible for an ∼100 eV target plasma in the classical sheath regime leads to ∼5 eV target plasma in the inverse sheath regime, assuming that the density is unchanged. In a SOL, the density is coupled to the target plasma temperature through the force balance, radial flows, and volume recombination. More physics needs to be modeled to show the effect of an inverse sheath on the SOL power balance including interspecies collisions, radial (2D) transport, and radiation. This calculation is only intended to show that inverse sheaths have a strong cooling effect on the nearby electrons. When Q_e is small, the target plasma temperature approaches the surface temperature. This result was corroborated in 1D-1V unmagnetized plasma simulations in Ref. 12. We predict the existence and general cooling effect of inverse sheaths to hold under strong enough emission in magnetized plasmas as well.

These calculations suggest that for a given tokamak, the target plasma temperature will stay at a few eV or less over a much wider range of operating conditions if an inverse sheath is present than a conventional sheath. It follows that inverse sheaths might be used as part of an innovative divertor detachment scenario.

The common concern about electron emission raising the heat flux does not threaten the viability of inverse sheath detachment. Here, we are proposing to employ thermionic surfaces with inverse sheaths across the entire plasma-wetted area. The global heat exhaust out of the SOL is fixed inasmuch as the power injected into the SOL is fixed.³⁶ Therefore, whatever changes to the sheath occur while varying the emission intensity cannot be expected to change the total heat flux on the plasma-wetted area, unless volumetric dissipation of power changes. The benefit of operating with a low target plasma temperature is that radiative power dissipation becomes strong at a few eV or less,³⁶ and volume recombination can dissipate plasma particles at sub-eV temperature.² 

We emphasize that the cooling effect of an inverse sheath is not just inside the sheath, which is too thin to produce a useful dissipating volume. In essence, an inverse sheath sets a low-temperature boundary condition on the quasineutral plasma, ensuring that a certain length of the plasma will be cold even if the upstream plasma is hot.

Understanding the properties of SOL plasmas with inverse sheaths requires more modeling than will be attempted in this paper. For example, it is necessary to simulate the entire SOL plasma to make a quantitative calculation of the radiated power and volume recombination possible in a given regime. It is not calculable from two-point models or back-of-the-envelope estimates. Here, we will give a broad overview of the potential benefits of inverse sheath detachment in order to motivate future modeling work.

Tokamaks tend to operate in an “attached” regime with target plasma temperatures of tens-of-eV or more if no special effort is undertaken to induce detachment.³⁶ In future reactors with higher exhaust power, the heat load will be intolerable⁴ unless successful techniques are developed to mitigate the heat flux. Some techniques involve spreading the heat flux over a wider area, e.g., the snowflake divertor.⁶² Others involve finding active methods to cool the target plasma to a temperature suitable for detachment.^1,3

The two main active methods to induce detachment involve injecting gas. Injecting neutral fuel upstream raises the collisionality of the SOL plasma, which then leads to a sharper temperature gradient to conduct the exhausted heat, leading to a reduced target plasma temperature. Reduction of target plasma temperature with increased collisionality in classical sheath regimes is also shown in simulations.^12,54,63 The drawback of fuel injection is that it cools the upper SOL and counts against the density limit.⁴ Radiating impurities such as neon can be injected to directly cool the plasma near the target. But they can migrate to the core fusion plasma and degrade its performance.³⁶ Impurity radiation also exacerbates the phenomenon of MARFE.³⁶ 

Thus, a key motivator for considering detachment via thermionic emission and inverse sheaths is that the target plasma cooling can be achieved without the need to use neutrals to radiate energy or enhance collisionality. We note that at least moderate collisionality is required to sustain the requisite temperature gradients such that plasma is hot in the upper SOL and cold near the inverse sheath. If collisionality is too weak, then electron emission can cool the entire SOL, which is unfavorable.⁶⁴ In the collisionless limit envisioned in certain high temperature divertor operations with suppressed neutral recycling, emission from opposite divertor plates transits the plasma and is reabsorbed, self-canceling and having no effect on the particle or energy balance.^65,66 As long as the plasma is moderately collisional, an inverse sheath will ensure that the plasma electrons near the target become cold but does not prevent a hot plasma from existing upstream.¹² Interspecies collisions will help equilibrate the temperature profiles of electrons and ions, ensuring that ions near the target are cold enough to be confined by the inverse sheath.

Assuming that the plasma is cold enough and detached (for any sheath type), ion impact energies are below the physical sputtering threshold of typical plasma-facing components. In a situation where the target plasma remains attached it is worth mentioning that inverse sheaths could still offer benefits. The strong accelerating voltage of the classical sheath often brings ion impacts above the physical sputtering threshold of plasma-facing materials,^36,43 which could be avoided with an inverse sheath. Sputtering by charge-exchange (CX) neutrals, formed by ions accelerating in the Bohm presheath and classical sheath undergoing CX collisions, should also be avoided due to the lack of accelerating potentials. Finally, by reducing the ion flux, inverse sheaths can help reduce problems from chemical reactions caused when impurities and fuel particles accumulate on the surface.

Each (hydrogen) ion-electron pair that bombards the divertor plate imparts their kinetic energy and causes an additional 13.6 eV to be deposited² on the surface during recombination, matching the energy it initially took to ionize the hydrogen atom. Note that an energy equal to the work function is lost (gained) by the material to neutralize each plasma ion (plasma electron), but these cancel⁶⁷ assuming that the influxes of plasma electrons and ions are equal. Even with thermionic emission, the cancelation holds for floating surfaces because each thermoelectron that escapes is compensated by one “extra” plasma electron entering the surface. Overall, the rate of 13.6 eV surface recombination energy release events is proportional to the ion flux. If the kinetic energies of the impacting particles are only a few eV, the surface recombination heat flux becomes the dominant heat flux.⁶⁸ Even if conventional detachment is able to dissipate most of the SOL plasma's thermal energy by radiation, a surface recombination heat flux persists and might by itself present an intolerable heat load in future tokamaks.^2,69

The inverse sheath detachment regime is well-equipped to minimize surface recombination heat flux. Because an inverse sheath confines most ions, the flux of ions reaching the surface can in principle be made small compared to conventional sheath regimes. As long as the inverse sheath reduces the nearby plasma to a sub-eV temperature, volume recombination rates should become high enough to provide the necessary charge sink to dissipate the particles injected into the SOL (both the exhaust fuel and helium ash). For SOL-relevant densities, volume recombination rates as a function of temperature increase by orders of magnitude from temperature 5 eV to 0.5 eV.⁷⁰ 

If the electric field at a conducting surface with a classical sheath is strong enough to force electrons over the work function barrier, it can cause arcs that result in target damage and impurity production.^71–74 The arc problem is attracting renewed interest and concern in the tokamak community.^75–77 Arcing can even be the dominant surface erosion mechanism when using high-Z targets under certain conditions.⁷⁸ There is no guarantee that conventional detached divertor solutions for future tokamaks will be immune to arcs because even if the target plasma is made cold, it may have a high density liable to result in a strong surface electric field in the electrostatic Debye-scale region of the Chodura sheath. We note that at a sufficiently grazing B field angle^55,58 most of the floating potential drop falls across the magnetic presheath, which has a wider length scale related to the ion Larmor radius.

A key benefit of an inverse sheath operating scenario is that it will always be immune to arcs due to the electric field at the surface having the opposite sign of a classical sheath. A SCL sheath could also inhibit arcs for the same reason, but due to the destructive ion trapping effect¹¹ we believe that it would only exist in strongly emitting divertors during startup as a transient state leading to an inverse sheath.

A key question regarding inverse sheath detachment is, how much emission would be required in tokamak conditions and what realistic thermionic materials (if any) can provide it? The short answer is that it depends on the target plasma density n_se, but the range of n_se is uncertain. There are ample data on n_se in classical sheath operating regimes in current tokamaks as well as predictions for future tokamaks.⁴⁶ One may be inclined to say that an emitted flux equal to the target electron saturation current in the classical sheath regime is the critical quantity needed to form an inverse sheath. But an interesting study by Hollmann et al.⁷⁹ shows that if the existing classical sheath voltage is large, introduction of thermoelectrons causes enhanced ionization and plasma density near the target, further increasing the amount of emission that can pass before the classical sheath breaks down. An inverse regime under the same tokamak operating conditions will feature a different force balance, different sources and sinks of charge and energy, and perhaps different radial transport rates, resulting in different spatial distributions of n_e and T_e. Self-consistent modeling of the inverted detached SOL in a fluid code is necessary to determine what range of n_se values are possible in operating regimes with inverse target sheaths.

The density of outgoing half-Maxwellian emitted electrons at a thermionic surface is n_emit ≡ Γ_emit(πm_e/2T_emit)^1/2. The maximum n_cold possible at the inverse sheath edge is n_emit, in the limit that the inverse sheath nears zero Volts and suppresses no emission. Assuming n_cold > n_hot via (9), we know that the fluid target electron density n_se must be between n_cold and 2n_cold [Eq. (7)]. It follows that the maximum target plasma density sustainable by a thermionic material with a certain Γ_emit and T_emit is

The density should be at least a few times less than this maximum to make the inverse sheath voltage strong enough for decent confinement of plasma ions. Our experience with simulations shows that if ions leak too easily, the sheath regime becomes amorphous and different from a true inverse regime. Also, if a fraction of thermoelectrons F_redep is promptly redeposited²⁹ by Larmor gyration, then the emitted flux in (11) is attenuated by a factor (1-F_redep).

The maximum n_se as a function of Γ_emit and T_emit is plotted in Fig. 5 neglecting redeposition (normal incidence target). For each material, Γ_emit is a strongly increasing function of T_emit that follows the Richardson–Dushman law.⁸⁰ Most externally heated thermionic cathodes are operated in the range 0.1–0.2 eV.⁸¹ Low temperatures yield too little emission. At very high temperatures, the materials degrade or melt. For tungsten in a tokamak, T_emit could get as high as the melting point 3695 K = 0.32 eV. Thus in Fig. 5, we use T_emit in the range of 0.1–0.32 eV. The maximum thermionic current densities for doped tungsten or LaB₆ are tens of A/cm². Based on Fig. 5, the maximum n_se that could be sustained by an inverse sheath using familiar thermionic materials ∼10 A/cm² is almost 10¹⁹ m⁻³. Some scandate cathodes⁸² can emit up to 400 A/cm² at a temperature of 1030 °C, which could sustain n_se values exceeding 10²⁰ m⁻³. But such materials have not been considered for operation in the fusion environment and are unlikely as durable. The hope is that thermionic detachment, if successful, will protect the surfaces from damaging particle impacts, such that the material would not need to be as durable as tungsten. Unfortunately, coatings formed by other material atoms sticking to the thermionic material are liable to interfere with the thermionic emissivity. Keeping the plasma-facing part of the emitter fresh might require an advanced scheme such as moving targets.⁸³ 

The development of thermionic materials with enhanced emission, stability, and durability properties has been pursued for many decades due to the many practical applications.⁸¹ If modeling efforts show inverse sheath detachment to be promising divertor solution but the predicted n_se values are too high for available thermionic materials, better materials may become available in the future. If we speculate that the range of possible n_se in inverse sheath regimes is similar to classical sheath regimes in current tokamaks, then sustaining the inverse sheath might be possible for a limited range of n_se using today's materials. Stronger emitters are likely necessary for practical reactors. Suppression of part of the emitted electron population by the grazing B field further raises the thermionic flux required to sustain an inverse sheath, although deliberate tilting of the plates can offset the suppression. One helpful effect is that if volume recombination is as effective as hoped in the inverse regime, n_se may be reduced by orders of magnitude and thereby reduce the thermionic flux required. Initiating the plasma into detachment using standard neutral puffing methods might reduce n_se and facilitate entering inverse sheath detachment.

Can other forms of electron emission help sustain an inverse sheath? Some photoelectron emission will be present,²¹ especially in detached regimes with strong radiation near the target (although photons also contribute heat flux⁸⁴). Future studies quantifying the photoemission flux would be valuable. Impacts of excited neutrals on the surface are predicted to cause substantial emission⁸⁵ in arcs and could become significant in detached divertor regimes. Since an inverse sheath detachment state will have a low target plasma temperature, it is unlikely to be self-sustainable by electron-induced emission. Electron-induced emission coefficients from metals are far below unity at low temperatures.¹⁶ Incidentally, an inverse sheath state driven by electron-induced emission was reported in Hall thruster simulation,⁸⁶ but the thruster has a unique design (strong E × B drift parallel to the ceramic surface) that acts as a heat source imparting high energies to the emitted electrons as soon as they enter the plasma.

Another challenge is that an ideal detached divertor solution must withstand the effects of the edge localized modes (ELM's).^87,88 It is known⁸⁷ in classical sheath regimes that the sharp increase in SOL input power produced by ELM's leads to a substantial increase in the target T_e. Detachment is weakened or destroyed when the target T_e goes above a few eV, leading to increased heat flux on the targets that result in melting.

Determining the response and survivability of inverse sheath thermionic detachment to ELM's requires much more analysis including time-dependent SOL simulations with inverse sheaths. We can predict that the target plasma is more likely to stay cool during a transient ELM event with an inverse sheath because as shown in Fig. 4, the inverse sheath regime keeps target plasma at a few eV or less over a much broader range of input power than conventional sheath regimes. However, an increased plasma electron influx to the target from an ELM requires more compensating thermoelectrons, and thus a higher target plasma density which could exceed the limit for inverse sheath existence in (11). The density increase could be self-mitigated by higher volume recombination rates if the target plasma temperature remains low enough. This has an added benefit of providing more neutrals near the target to dissipate more power.

In this paper, we analyzed the effects of target plate electron emission on tokamak divertors. We considered whether inducement of an inverse sheath using thermionic plates can be used to mitigate plasma-wall interactions. Based on analytical derivation of the novel heat flux boundary condition for inverse sheaths (10), we showed that for a given SOL input power inverse sheaths will establish a target plasma with much lower temperature than a classical sheath or space-charge limited sheath. This gives hope that sufficient plasma cooling for detachment can be achieved by inducing inverse sheaths, thereby avoiding the usual need to inject neutrals. Other advantages of inverse sheaths over conventional sheaths include reduction or elimination of sputtering, arcs, and tritium implantation.

Understanding inverse sheath effects on divertor plasmas requires a comprehensive modeling of the SOL with inverse sheaths. Masline, Smirnov, and Krasheninnikov^13,14 recently made an important step forward by taking the workhorse edge code UEDGE⁸⁹ and changing the boundary conditions from the standard classical sheath to an inverse sheath. They compared a standard and inverse sheath case under a certain set of SOL conditions and found significant differences in some SOL properties. In the inverse case, they did not observe a substantial improvement in the ease of detachment, and even saw a worsening of heat flux at certain locations, but there was some mitigation of the heat flux to an already detached target. In future work, it is important to consider how the effects of inverse sheaths change as different SOL “control knobs” are varied. The case of zero ion flux (limit of infinite emission) was considered but it would be useful to consider cases with some ion flux resulting in enhanced neutral recycling which has the potential to help dissipate more power through radiation. Even without an improvement in detachment, Masline et al. concluded that the other advantages of inverse sheaths for plasma-material interactions mitigation make it worth further investigation. We note that the feasibility of inducing inverse regimes can also be addressed from fluid model solutions such as UEDGE⁸⁹ or SOLPS⁹⁰ with inverse sheath boundary conditions, by plugging the target plasma density from the simulation into our Eq. (11) to determine the thermionic flux required.

It is important for future studies to improve the understanding of electron emission effects on SOL plasmas, regardless of whether the inverse sheath is found to be a useful divertor solution. Intense secondary electron emission can occur in existing tokamaks⁷ and even more so in higher power future devices. Measurements from the Lithium Tokamak Experiment showed electron temperatures of hundreds of eV in the entire SOL.⁹¹ Meanwhile, separate measurements on exposed lithium show SEE coefficients above unity at the same temperatures.^17,18 Films that develop on tungsten during exposure lead to more secondary or thermionic emission than otherwise expected.⁹² Extreme thermionic emission fluxes (far exceeding the plasma electron saturation current) are predicted for ITER in situations when tungsten is heated toward the melting point²⁰ such as during ELM events. Most simulations of SOL heat transport and sheath evolution during ELM's have not included electron emission.^87,88 Some recent studies have analyzed thermionic emission⁹³ effects during melting events, but have not yet considered inverse sheaths. Even if the sheath remains in the classical regime the electron emission affects the energy balance and sputtering. Any amount of emission (over a large target area, not a small hot spot) can help cool the target plasma and possibly strengthen detachment. Fundamental experiments testing the effects of emissive tiles in tokamaks or linear divertor simulators are encouraged. Adding thermionic emission to a plasma already detached by neutral puffing would also be interesting.

Another application of inverse sheaths to magnetic fusion that requires further study concerns the negative ion sources used to generate neutral beams for heating. Early modeling efforts for the ITER negative ion source predicted surface-generated negative ions to cause SCL sheaths.^94–96 A recent 1D simulation study by Zhang et al.⁹⁷ showed that the same charge-exchange collisions that destroy electron-emitting SCL sheaths¹¹ force a transition to an inverse sheath. A consequence of an inverse sheath that plasma electrons become unconfined, get rapidly lost from the system, and then get replaced by emitted negative ions, resulting in an ion–ion plasma. This could be responsible for ion–ion plasmas measured in experiments.⁹⁸ We suggest a possible benefit of operating the source with inverse sheaths which is the current of (unwanted) coextracted electrons might be minimized. Simulations in multidimensional geometries are needed to understand how inverse sheaths will affect the overall negative ion source. Multidimensional kinetic simulation models of H- sources are in use.^99–101 But computational cost may make it difficult to achieve a high enough spatial resolution of the ion trapping process in the bounding sheaths that is needed to form inverse sheaths (the trapping process is shown in the 1D simulations^11,97).

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. The author acknowledges insightful discussions with S. Krasheninnikov, R. Smirnov, and R. Masline.