Nothing propagates below 8.98 MHz

At its defaults it draws nothing propagates below 8.98 mhz. Frequency against wavenumber for a wave in a plasma of 1.0e+12 electrons per cubic metre, in units of the plasma frequency. The curve is ω² = ωₚ² + c²k², so it starts at ωₚ with zero slope and becomes the light line far above it; the whole band below ωₚ has no real k at all, which is the shaded region. At k = 3.19e-1 m⁻¹ the phase velocity read off the curve is 1.1607c and the group velocity 0.8615c, whose product is c² to a part in 10¹⁰ — the crests outrun light and the signal does not, which is the same arrangement as any other medium with a cutoff.

plasma-response is one function in lib/figures/extremes.js — radiation, cross-sections and self-gravity. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

Frequency against wavenumber for a wave in a plasma of 1.0e+12 electrons per cubic metre, in units of the plasma frequency. The curve is ω² = ωₚ² + c²k², so it starts at ωₚ with zero slope and becomes the light line far above it; the whole band below ωₚ has no real k at all, which is the shaded region. At k = 3.19e-1 m⁻¹ the phase velocity read off the curve is 1.1607c and the group velocity 0.8615c, whose product is c² to a part in 10¹⁰ — the crests outrun light and the signal does not, which is the same arrangement as any other medium with a cutoff.

An electrode driven through a blocking capacitor by a symmetric waveform of amplitude 20 kTe, in an argon-like plasma whose electrons are collected 108 times as readily as its ions at saturation, starting uncharged. Potentials are relative to the plasma, in electron temperatures. On the first positive half-cycle the electrode draws a flood of electrons and nothing on the negative half-cycle can return the charge, because the ions arrive at a fixed, small rate; the electrode walks down, and within 6 cycles its mean potential is within half an electron temperature of the self-bias. That bias, found by integrating the charging, is −22.27 kTe, and the charge balance with a Bessel function gives −22.27. The floating potential of the same wall undriven is −4.68. At 3 eV, a 60 V drive makes a 67 V bias with no direct current supplied anywhere.

The current to an electrode through one steady cycle of a 20 kTe drive, in units of the ion saturation current, against the phase of the generator. Ions arrive at one unit the whole time. Electrons arrive only near the positive peak of the drive, where the electrode comes closest to the plasma potential, in a burst reaching 11.1 units and exceeding the ion current for 16 per cent of the cycle. The two shaded areas are the charge each species delivers per cycle and they agree to 0.08 per cent, which is the condition that sets the bias: the electrode sits exactly low enough that the brief approach to the plasma potential admits one cycle's worth of ions in electrons.

The self-bias of a capacitively driven electrode, as a magnitude in electron temperatures, against the drive amplitude in the same units, on logarithmic axes, for an argon-like mass ratio. The curve is the charge balance a mean of V(floating) − ln I0(V0), with I0 the modified Bessel function of order zero, which starts at the undriven floating potential, 4.68, and bends onto the line of equal bias and amplitude. The dots are the same quantity found by integrating the charging of the capacitor to a steady cycle: 1 kTe gives 4.92 against 4.92; 3 kTe gives 6.27 against 6.27; 10 kTe gives 12.63 against 12.63; 30 kTe gives 32.07 against 32.07; 100 kTe gives 101.46 against 101.46. For a large drive the bias approaches V0 plus the floating potential minus half the logarithm of 2πV0, so the electrode sits below minus the amplitude by a few electron temperatures. At 3 eV, a 100 V drive gives a self-bias of −106 V.

How the time-averaged voltage divides between the sheath on a powered electrode and the sheath on a larger grounded wall, against the ratio of their areas, on logarithmic axes. At high frequency both sheaths carry the same radio-frequency current, so the voltage divides in inverse proportion to their capacitances, and a sheath's capacitance is its area over its thickness — which itself grows with the voltage across it. Iterating that divider to its fixed point gives a voltage ratio equal to the area ratio raised to 1/(1 − p), for a thickness growing as the voltage to the power p: fixed thickness, p = 0 gives 5.0 at an area ratio of 5; collisional, p = 3/5 gives 56 at an area ratio of 5; Child's law, p = 3/4 gives 625 at an area ratio of 5. The fourth power usually quoted belongs to collisionless Child-law sheaths; measured discharges give exponents between about one and two and a half, shaded, because real sheaths collide, carry conduction current and are not planar. The direction is the same for all of them: the smaller electrode takes most of the voltage, which is why a wafer sits on the small powered electrode and the chamber wall is the large grounded one.

The energies ions arrive with at an electrode under a 40 kTe drive, whose sheath voltage swings about a mean of 41.9 kTe, for ions that take 0.05, 0.4, 1.3, 3.3 radio-frequency cycles to cross the sheath. Each ion is given the sheath voltage averaged over its own transit, and the distribution is taken over entry times spread evenly through the cycle. At 0.05 cycles the energies spread over 79.7 kTe, against 79.7 from averaging a sinusoid; at 0.4 cycles the energies spread over 60.5 kTe, against 60.5 from averaging a sinusoid; at 1.3 cycles the energies spread over 15.8 kTe, against 15.8 from averaging a sinusoid; at 3.3 cycles the energies spread over 6.2 kTe, against 6.2 from averaging a sinusoid. A fast crossing sees the instantaneous voltage and lands at either extreme, giving two peaks; a slow one sees the average and lands near the middle. For argon crossing a 1 mm sheath at that mean voltage and 3 eV, the transit takes 122 ns, which is 0.12 cycles at 1 MHz, 1.65 cycles at 13.56 MHz, 7.30 cycles at 60 MHz — which is why the frequency of an etching tool is chosen for the ions rather than for the electrons.

physicscheck asserts something about plasma-response that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.