[Paper Review] The exact form of the Bohm criterion for a collisional plasma

Tsanko Tsankov, Uwe Czarnetzki|arXiv (Cornell University)|Mar 1, 2016

Plasma Diagnostics and Applications55 references17 citations

TL;DR

This paper resolves a long-standing debate on the kinetic Bohm criterion in collisional plasmas by deriving an exact form free of divergence through solution of the Boltzmann equation with charge-exchange collisions. It introduces a new collisional and geometric term that corrects the classical criterion, enabling accurate reconstruction of ion and electron distributions, electric fields, and plasma parameters from wall-measured ion velocity distribution functions (IVDFs), with excellent agreement between simulation, experiment, and theory.

ABSTRACT

A long-standing debate in the literature about the kinetic form of the Bohm criterion is resolved for plasmas with single positive ion species when transport is dominated by charge exchange collisions. The solution of the Boltzmann equation for the ions gives the exact form free of any divergence and contains an additional term that is not included in the classical result. This term includes collisional and geometric effects and leads to a noticeable correction. Further, the question is addressed whether the space charge argument at the bottom of the Bohm criterion can actually lead to a meaningful definition of the transition point between bulk and sheath. The analysis is supported by a numerical model and experiments, showing excellent agreement throughout. As a novelty in diagnostics, the theoretical results allow from the ion velocity distribution function (IVDF), measured at the wall, a reconstruction of the IVDF and the electric field at any point in the plasma. This property is used to reconstruct non-intrusively also the ion density, flow velocity, mean energy and effective temperature and the electron density and temperature as functions of the spatial coordinate and potential. Finally, the fluid equation for ion momentum balance is verified.

Motivation & Objective

• To resolve the longstanding divergence in the kinetic form of the Bohm criterion for collisional plasmas dominated by charge-exchange collisions.
• To establish a physically consistent criterion for the sheath edge that accounts for collisional and geometric effects.
• To develop a diagnostic method that reconstructs plasma parameters (density, velocity, temperature, electric field) from measured ion velocity distribution functions at the wall.
• To verify the fluid ion momentum equation in the presence of strong gradients and finite collisionality.
• To demonstrate that the classical Bohm criterion with $ u = c_i $ is no longer a strict consequence of the corrected kinetic criterion, though still a useful approximation.

Proposed method

• Solves the Boltzmann equation for ion velocity distribution functions (IVDFs) including charge-exchange collision operators with constant mean free path $ \lambda $, using a spatially dependent electric field $ E(r) $.
• Applies an ansatz $ f_i(v,r) = g(v,r) \Theta(v) \Theta(v_{\text{max}}(r) - v) $ to derive the exact IVDF solution $ g_i(v,r) \propto Q(r') \exp(-(r - r')/\lambda) $, where $ r' $ is the ion creation position.
• Incorporates ionization source terms $ Q(r) $ from electron-impact ionization using the Biagi-v8.9 cross-section database.
• Uses the measured IVDF at the wall to reconstruct the IVDF and electric field at any spatial point via inverse solution of the transport equation.
• Applies the reconstructed IVDF and measured electron energy distribution function (EEPF) to determine spatial profiles of ion and electron densities, flow velocity, mean energy, and effective temperature.
• Validates the fluid momentum equation by comparing kinetic and fluid flow velocities, showing small but measurable deviations due to non-uniform fields and collisional effects.

Experimental results

Research questions

• RQ1 What is the exact kinetic form of the Bohm criterion in a collisional plasma with charge-exchange collisions, and how does it differ from the classical divergence-prone version?
• RQ2 Can the ion velocity distribution function measured at the wall be used to reconstruct the full spatial profile of ion and electron plasma parameters non-intrusively?
• RQ3 Does the classical Bohm criterion $ u = c_i $ still define the sheath edge when the kinetic criterion is corrected for collisional and geometric effects?
• RQ4 How do collisional and geometric effects influence the momentum balance and the validity of fluid approximations near the sheath transition?
• RQ5 To what extent does the weak form of the Bohm criterion $ \partial n_e / \partial \varphi \geq \partial n_i / \partial \varphi $ define a meaningful transition point between bulk and sheath?

Key findings

• The exact kinetic Bohm criterion contains an additional term arising from collisional and geometric effects, which removes the divergence present in the classical formulation.
• The corrected criterion shows that the equality $ \partial n_e / \partial \varphi = \partial n_i / \partial \varphi $ is never strictly satisfied in finite-collisionality plasmas, indicating the sheath edge cannot be defined by equality in the weak form.
• The classical condition $ u = c_i $ remains a useful approximation for the sheath edge but no longer follows from the corrected kinetic criterion.
• The reconstructed ion velocity distribution function from wall measurements agrees excellently with both simulation and experimental data, validating the diagnostic method.
• The fluid momentum equation is verified, but a small deviation between fluid and kinetic flow velocities is observed due to strong field gradients and non-uniform collisionality, with the CX friction contributing ~13% to momentum balance.
• The ratio of ion flow velocity to sound speed $ u/c_i $ reaches ~0.95 at the Bohm point, and the Debye length $ \lambda_D = 0.3 $ mm is much smaller than the mean free path $ \lambda $, confirming the validity of the kinetic approach in the transition region.