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A plasma sheath between two grounded (Dirichlet) walls.

Runs a two-species (electron/ion) particle simulation confined in the strip 0 < x < L between two grounded conducting plates, using GroundedStripField: its image-charge Green’s function enforces phi = 0 exactly at x = 0 and x = L (a Dirichlet boundary condition), not approximately. A quasi-neutral plasma starts uniform and Maxwellian; electrons, being lighter and faster, are lost to and repelled from the grounded walls more than the slower ions, leaving a thin layer of excess positive space charge against each wall — the classic plasma sheath — while the bulk stays quasi-neutral. Saves one figure per claim:

Source: examples/plasma_sheath.py. Run it yourself with python examples/plasma_sheath.py [--output examples/output] [--quick] (needs pip install "tree-d-3d[examples]").

Animations

Section titled “Animations”

an (x, v_x) phase-space animation of both species, from the initial uniform state to the developed sheath.

Figures

Section titled “Figures”

the initial electron and ion positions, uniform and quasi-neutral, with the two grounded walls marked.

electron and ion density against x, time-averaged over a late window once the sheath has formed; quasi-neutral in the bulk, with an excess of ions over electrons in thin layers against both walls.

the electrostatic potential against x, same time average; it returns to (near) zero at both grounded plates — the Dirichlet condition the image charges enforce exactly — while the bulk floats higher, with most of the drop concentrated in the sheath layers.

the field E_x against x, same time average; close to zero in the quasi-neutral bulk and pointing from the bulk toward each wall inside the sheath layers.

electrons and ions remaining against time; the walls are a real particle sink, with the bulk only partly depleted over the plotted window.

--quick uses fewer particles and a shorter run (well under a minute instead of roughly ten). Needs Plotly (pip install 'tree-d-3d[examples]').