Sheaths & Space-Charge Limits

Space charge sets a ceiling on how much current a converter can pass; understanding it explains why beams must be spread and staged.

The universal current limit

Whenever charged particles of one sign are pushed across a gap, their own space charge builds a field that opposes further flow. The maximum current density that a gap of a given voltage and spacing can pass is described by the Child-Langmuir law: it rises with voltage to the three-halves power and falls with the square of the gap length. This single relation governs thermionic gaps, ion collector regions, and the beam optics of the whole train.

# Child-Langmuir space-charge-limited current density (illustrative)
# J_max = (4/9) * eps0 * sqrt(2*q/m) * V**1.5 / d**2
# Key scalings:
#   J_max grows as V**1.5   -> higher voltage passes more current
#   J_max falls as 1/d**2   -> smaller gap passes more current
# Consequences for DEC:
#   spread the beam (lower J) to stay under the limit,
#   or shrink gaps (thermionic close-spacing) to raise it.

Sheaths at every surface

Where plasma meets a solid surface, a thin sheath forms in which the potential adjusts and charge separates. The sheath sets the local electric field an arriving ion actually sees and the voltage available for collection. In a direct converter the sheath physics at each electrode determines whether ions are collected at their full potential or lose voltage to the sheath drop — a first-order efficiency effect.

How the design lives within the limit

• Spread the flow: flux expansion lowers current density so the Child-Langmuir ceiling is not hit.
• Stage the conversion: no single gap must pass the whole current at full energy.
• Shrink gaps where possible: thermionic close-spacing raises the limit at the cost of tolerances.
• Neutralize where possible: cesium ions in a thermionic gap cancel electron space charge.

Why it shapes the architecture

Space charge is the reason the burner cannot simply point its exhaust at one big collector: the current density would exceed what any gap can pass. Spreading the beam in the expander and cascading converters are, at root, ways of respecting the Child-Langmuir limit while still recovering the energy. It is a constraint that shapes nearly every geometry choice in this section.