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Instability and Disappearance of Debye Sheaths

Due to Secondary Electron Emission (SEE)

10/5/12 LTP Teleseminar by Mike Campanell, Grad Student, PPPL

Φ(x) (V)

plasma interior

wall

wall

wall

Sheath

plasma interior

No Sheath?

x (m)

Figure: Potential profiles in a plasma slab bounded by floating walls with moderate emission (black plot) and very strong emission (gray plot).

Figure: Animation of the abrupt change of the electrostatic potential function during sheath instability.

- General treatments of SEE effects on PSI [1,2] usually assume two things:
- A familiar Debye sheath exists with surface potential below the plasma potential
- The sheath is static in time

- In this teleseminar, we will present simulations and basic theory showing that both assumptions can easily break down in practice.
- Outline of Presentation:
- Quick background on SEE effects
- Study of the inverse sheath effect (violation of assumption 1).
- Study of sheath instabilities (violation of assumption 2).

[1] G.D. Hobbs and J.A. Wesson, Plasma Phys. 9, 85 (1967). [2] L. A. Schwager, Phys. Fluids B 5, 631 (1993).

- SEE is critical in many plasma physics applications.
- The larger the SEE coefficient γ, the more plasma electrons must hit the wall to balance the ion flux: Γe = Γe,in – Γe,out = Γe,in(1 - γ) = Γion
- Time-independent solutions of sheaths with fixed γ implicitly assume stability!

- For hot plasmas and/or highly emitting materials, γ may exceed unity [1]. In this case, a classical sheath, see image (a), cannot maintain zero current.
- Conventional SEE theories usually assume a “space charge limited” sheath arises. A dip forms in the potential profile Φ(x) near the wall to repel the “extra” secondaries back to the wall, see image (b).
- But this is not the only solution. A Φ(x) in image (c) is possible. Ions are not drawn to the wall and more importantly, plasma electrons are unconfined, so the wall flux is thermal (extremely large)!

Fig: Qualitative differences between (a) the classical sheath, (b) SCL sheath and (c) the new inverse sheath.

[1] C.A. Ordonez and R.E. Peterkin Jr., J. Appl. Physics. 79, 2270 (1996).

Description of the Simulation Model:

- We use the EDIPIC code, written by D. Sydorenko [1], to simulate a planar xenon plasma bounded by walls made of boron-nitride.
- The plasma is heated by an ExB background field, as in a Hall thruster.
- When the ExB drift energy is low, classical sheaths form at the two walls.
- When the ExB drift energy is large, the electrons reaching the walls will have large energy parallel to the wall. So this could cause γ > 1.

We Simulate Two Cases:

Simulation A – Moderate drift energy case

Ez = 200V/cm, Bx,= 100G, Neutral Density nn = 1012 cm-3 ,Plasma Densitynp = 1011 cm-3

Simulation B – High drift energy case

Ez = 250V/cm (larger VDrift),otherwise similar to Sim. A

[1] D. Sydorenko, Ph.D. thesis, U. of Saskatchewan, 2006.

- The plasma in Sim. A has γ< 1 and features classical sheaths at the walls.
- The presheath/sheath structures in Sim. A form a potential well of large magnitude Φ = 21V to confine electrons and draw ions to the wall.
- In Sim. B, γ> 1. The sheaths vanish. Φ(x) is small throughout the simulation domain. Just fluctuations appear.
- It is found that Φ(x) always decreases from the wall outward in Sim. B. The surface charge is positive, the net space charge near the interface is negative.
- Conclusion: Sim. B is an “inverse sheath”. Its amplitude is Φ-1 = -1V.

Fig: Comparison of Sim. A and Sim. B.

(b) Potential profiles Φ(x).

(c) Φ(x) near the left wall.

(d) Charge density profiles in Sim. A.

(e) Charge density profiles in Sim. B

- The reason for the disappearance of the Debye sheath is as follows:
- The wall flux in the simulations consists of the following components:
- Γion: ion flux
- ΓCE: collision-ejected electrons (CEE’s)
- Γb: secondary “beam” electrons from the other wall
- γb and γCE denote the partial SEE yield of beam electrons and CEE’s.
- γb is determined by the ExB drift energy gained by secondaries crossing the plasma.
- It can be shown γ = γCE/(1 + γCE - γb)
- γb = 0.94 in Sim. A. But γb ≈ 1.2 in Sim. B. If γb > 1, classical sheaths cannot exist.
- The “inverse sheath” forms to suppress some emission and maintain current balance.

- Γion is negligible because ions are not drawn to the wall.
- Note in the plot that γnet = 1 at all times even as the dominant electron flux component Γb varies. Γo, the cold secondaries pulled back to the wall by the inverse sheath, self-adjusts.

Time evolution of the flux components in Sim. B.

- Conclusion:
- When the SEE yield of hot plasma electrons hitting the wall exceeds unity, a sheath is not needed to maintain current balance.
- Losses are far larger in the inverse sheath regime compared to classical and SCL regimes! It is found that the particle flux, energy flux and axial transport in Sim. B is ~10-20 times that of Sim. A.
- Possible Applications:
- The inverse sheath effect may be connected to the well known phenomenon of temperature saturation [1] in HT devices. Loss of sheaths should lead to enhanced wall losses and near-wall conductivity.
- Inverse sheath is also possible for surfaces with thermionic or photoemission.
- In applications where ion sputtering is more damaging than electron heat flux, the inverse sheath regime would be beneficial.

- The “inverse sheath” (c) is fundamentally different from both classical sheaths (a) and SCL sheaths (b). The inverse sheath has a negative plasma potential.

[1] Y. Raitses et. al., IEEE Trans. on Plasma Sci. 39, 4, (2011).

Appendix: Why does the inverse sheath not appear in familiar SEE theories?

General SEE theories usually assume ions are drawn to the wall. Poisson’s eq. is then solved with charge densities in the sheath written in terms of a potential Φ(x) that is assumed below the plasma potential.

[1] G.D. Hobbs and J.A. Wesson, Plasma Phys. 9, 85 (1967).

Sheath Instability in Weakly Collisional Plasmas: SEE theories?

Sheath instabilities can (a) trigger abrupt changes in the plasma, (b) drive oscillations and (c) dramatically increase energy loss and cross-field transport. [1]

(V)

wall

Fig: Illustration of the sheath rapidly collapsing over 5ns. Notice the abrupt drop in the sheath amplitude and the change in the near-wall structure of Φ(x)

plasma interior

x (m)

[1] M. Campanell, A. Khrabrov and I. Kaganovich, “General Cause of Sheath Instability Identified for Low Collisionality Plasmas in Devices with Secondary Electron Emission,” PRL 108, 235001 (2012).

Why Does Instability Occur?: SEE theories?

- Consider a planar plasma bounded by walls with SEE in a regime with classical sheaths.
- The arrangement of particles in energy space looks as in the phase plot to the right.
- Bulk plasma electrons (red) with Wx < eΦ are confined.

Plot of particle energy components parallel to the walls (Wy + Wz) and perpendicular to the walls (Wx).

- The loss cone (Wx > eΦ) is strongly depleted in low collisionality. However, the blue secondary electrons form “beams” that pass between the walls.

Mechanism of Instability:

A decrease of the sheath potential Φallows “weakly confined electrons” (WCE’s) to reach the walls. If the SEE coefficient of the WCE’s exceeds unity, the decrease of Φ causes reduction of the electron surface charge, which further lowers Φ, allowing more WCE’s to reach the wall etc… This leads to a runaway collapse of the sheath.

Verification of the Theory SEE theories?

- The electron flux to the wall consists of 3 components:
- (1) Collision ejected electrons (ΓCE), scattered into the loss cone by collisions with neutrals.
- (2) SEE “beam” electrons from other wall (Γb)
- (3) Weakly Confined Electrons nudged into the loss cone by fluctuations (ΓWC)
- By tracking each component and their SEE yield separately, it is seen that the system stability is dependent on the energies of the WCE’s.

- It was verified over many simulations that γWC = 1 is the critical point of instability.

Fig. Temporal evolution of Φ and partial SEE coefficients in a simulation with Ez, = 140V/cm, Bx,= 100G, Neutral Density nn = 1012 cm-3 ,Plasma Densitynp = 1011 cm-3, turbulent heating rate νturb = 5.6 MHz, H = 2.5cm

Consequence of Instability SEE theories?

- Instability triggers a rapid drop of Φand a rapid increase of ΓWC (see Fig. below).
- The instability quenches when the surface charge becomes positive and emitted secondaries start to get pulled back to the wall.
- The system then may restore to a new quasisteady state, as is discussed in more detail in our paper. The new state generally has enhanced losses.
- Note that there are 3 different types of instability effects in the data plot below.
- All of these effects are caused by γWC reaching or exceeding unity.

Ez, = 200V/cm, Bx,= 100G, nn = 1012 cm-3 , np = 1011 cm-3 νturb = 4.2 MHz, H = 2.5cm

WCE Theory Reveals Cause of Past Observed Instabilities: SEE theories?

- Relaxation Sheath Oscillations2:
- Quasiperiodic instabilities with steady behavior in between. (interval 3 of earlier fig)
- During instability, there is a rapid decrease of Φ, an increase of Γe,inand a transition to the SCL state.

- Beam Phase Instability1:
- A one-time abrupt transition from one quasisteady state to another (point 2 in earlier Fig)
- Φ drops drastically and the beam flight time switches to a new value.

[1] D. Sydorenko et al., Phys. Plasmas. 15, 053506, (2008). [2] D. Sydorenko et al., PRL 103, 14, (2009).

For More Information on Inverse Sheath: SEE theories?

If you would like a more complete discussion of the inverse sheath phenomenon, please see:

M. D. Campanell, A. Khrabrov and I. D. Kaganovich, “Absence of Debye Sheaths Due To Secondary Electron Emission,” PRL 108, 255001 (2012).

For More Information on Instabilities:

For more detail on the instability phenomena, please see our other paper:

M. D. Campanell, A. V. Khrabrov and I.D. Kaganovich, “General Cause of Sheath Instability Identified for Low Collisionality Plasmas in Devices with Secondary Electron Emission,” PRL 108, 235001 (2012).

(there was a typo in the original email announcement of the talk )

Acknowledgement:

This work was supported by the U.S. Department of Energy and the U.S. Air Force Office of Scientific Research. The PIC code used to obtain the results was written by D. Sydorenko, currently at U. of Alberta.

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