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Two-surface multipactor between parallel plates, with surface impacts from simgeometry.

Multipactor is a resonant electron avalanche in vacuum RF devices (cavities, couplers, waveguide windows, gaps between accelerator grids): an electron emitted from one wall is accelerated across the gap by the RF field, strikes the opposite wall just as the field reverses, and knocks out more than one secondary electron, which repeat the trip in the next half period. The classic setting is two parallel plates a distance d apart with a voltage V0 sin(omega t) between them, and the classic result is the susceptibility chart: the zones in the (f d, V0) plane where the electron number grows.

This example builds that chart from the max-models building blocks rather than from the analytic theory, and then compares it with the theory:

  • the two plates are simgeometry Rectangle electrodes, panelled by :func:tree_d_3d.conductors_from_shape and solved once at unit voltage by :class:tree_d_3d.BoundaryIntegralElectrodes; the RF field is that static solution times V0 sin(omega t), the quasi-static approximation (the gap, at most a few millimetres, is far smaller than the RF wavelength of tens of centimetres). It is tabulated in the gap by :meth:~tree_d_3d.BoundaryIntegralElectrodes.tabulate, accurate up to the plate surfaces since panels near a point enter with their exact field; * electrons are pushed nonrelativistically (impact energies stay below a few keV, well under the 511 keV rest energy) by a velocity-Verlet step in the RF phase, and wall impacts come from simgeometry’s shape.crossings(start, end), which tests the whole step, so no electron steps through a plate, and returns the impact point and the surface normal, hence the impact energy and angle; * each impact emits one secondary with the yield of Vaughan’s formula (J. R. M. Vaughan, “A new formula for secondary emission yield”, IEEE Trans. Electron Devices 36, 1963 (1989)), carried as a statistical weight: the weight is multiplied by the yield delta(E, theta), so the population grows where delta > 1 at the resonant impact energy. The secondary leaves along a cosine (Lambert) distribution about the normal, with energies from the flux of a 1 eV Maxwellian (mean 2 eV). The yield parameters, delta_max = 2.0 at E_max = 300 eV with threshold E_0 = 12.5 eV, are illustrative values of the order quoted for air-exposed copper, not a measured set; electrons arriving below E_0 are absorbed (backscattered primaries are left out).

The theory overlaid on the chart is the parallel-plate return map with a constant emission velocity (the setting of Hatch and Williams, Phys. Rev. 112, 681 (1958)): an electron leaves a plate at a fixed speed and phase, crosses a uniform field, and its impact phase becomes the next emission phase, a map iterated to its fixed point. Its closed-form resonances (odd orders N = 1, 3, 5: transit in N half periods) are drawn as lines; the zone where the map’s fixed point has delta > 1 is drawn as a contour. The simulation has none of the map’s idealisations (real 2D field from the panel solve, spread of emission energies and angles, oblique impacts), so where the two agree the agreement is a check of the whole chain, and where they differ the difference is physics the theory leaves out.

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

Figures

Section titled “Figures”

the plates and the field magnitude in the gap, from the panel solve at unit voltage, against the uniform 1/d of ideal plates.

one electron at a first-order resonance crossing the gap in each half period, against the RF voltage.

Vaughan’s yield against impact energy at normal and oblique incidence, with the first crossover delta = 1 that sets the lower voltage edge of the zones.

the weighted electron number against RF cycles for a voltage inside the first-order zone and one below it.

the growth rate per RF cycle over (f d, V0), with the theory’s resonance lines and multipactor zone overlaid; the printed overlap counts how many simulated growth points the theory’s zone contains and vice versa.

--quick uses a coarser chart and fewer electrons (under a minute instead of about five); --smoke is the tiny configuration the test suite runs. Needs Plotly (pip install 'tree-d-3d[examples]').