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A planar Pierce electron gun: a space-charge-limited sheet beam kept parallel.

In a planar diode the space-charge-limited current density is the Child-Langmuir J_CL = (4 / 9) eps0 sqrt(2 q / m) V^(3/2) / d^2, with the potential V (x / d)^(4/3) across the gap. A beam of finite width breaks that solution at its edge: outside the beam the potential would be linear, the mismatch pushes transverse fields into the beam and its edge spreads. Pierce (1940) found the electrodes that make the vacuum outside the beam continue the Child-Langmuir potential with no transverse field at the edge: for a sheet beam, the harmonic continuation Re[(x + i (y - w))^(4/3)], whose zero equipotential leaves the beam edge at 67.5 degrees from the beam and whose V equipotential is the curve r cos(4 theta / 3)^(3/4) = d. Those are the cathode’s focus electrodes and the anode.

The gun here builds both as thin simgeometry Polyline electrodes (the cathode strip with its 67.5-degree wings, and the anode: the two curves joined by a grid across the beam, which the electrons pass through, as in Pierce’s planar solution; an open anode slit would lower the limiting current), solves them with :class:tree_d_3d.BoundaryIntegralElectrodes, and finds the steady beam by the gun-code fixed point: trace the electron rays, turn them into line charges (each segment of a ray of current dI holds dI dt), whose field the tree code evaluates and to which the electrodes respond, and trace again. Rays start a small distance x0 from the cathode at the local potential; the thin layer they skip is added as the Child-Langmuir charge 3 x0 dI / v0, and Gauss’s law across it gives the cathode field E(0) = E(x0) - sigma_layer (in units eps0 = 1): the current is space-charge limited when it vanishes. Code units eps0 = m = |q| = d = V = 1, so J_CL = 0.6285.

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

Figures

Section titled “Figures”

the Pierce electrodes, the potential and the converged rays at J_CL.

the potential along the beam axis against the Child-Langmuir V (x / d)^(4/3).

the cathode field at the beam centre against the current density; it vanishes near J_CL, the space-charge limit: at about 1.07 J_CL with the --quick rays, a value that stays put when the panels are refined fourfold, so the excess is not an electrode error.

the outermost ray with the Pierce electrodes and in a flat diode of the same gap (wide cathode and gridded anode planes, emitting only over the beam width): with them the edge stays within a few percent of parallel (drawn slightly in by the discrete rays and the truncated focus electrodes), without them it spreads by about a fifth across the gap.

the change of the rays per fixed-point iteration, with under-relaxation 0.5, down to the noise of the discrete rays.

--quick uses fewer rays and current values (under a minute); --smoke is the tiny configuration the test suite runs. Needs Plotly (pip install 'tree-d-3d[examples]').