The same law, across twenty-eight decades

At its defaults it draws the same law, across twenty-eight decades. The mean free path 1/nσ against cross-section, for a target density of 6.83·10³⁰ targets per cubic metre — solid lead. It is a straight line of slope minus one, because there is only one thing in the law. At 10⁻²⁸ m² the path is 1.46 mm; at 10⁻⁴⁷ m² the path is 1.55 light-years. Nothing about the physics changes between those ends. Only the area does.

cross-section 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.

The mean free path 1/nσ against cross-section, for a target density of 6.83·10³⁰ targets per cubic metre — solid lead. It is a straight line of slope minus one, because there is only one thing in the law. At 10⁻²⁸ m² the path is 1.46 mm; at 10⁻⁴⁷ m² the path is 1.55 light-years. Nothing about the physics changes between those ends. Only the area does.

The absorption of gallium arsenide near its own band gap of 1.424 electronvolts, on a logarithmic axis, at 4 temperatures. Below the gap the absorption does not stop; it falls exponentially, along a straight line whose slope is an energy, and the straightness holds over several decades. Raising the temperature makes the line shallower — the tail reaches further below the gap — and the slope runs from 5.6 millielectronvolts at 10 kelvin to 7.5 at 300, a factor of 1.33. The lines pivot about a point just above the gap rather than rotating about nothing, which is what makes the slope a single number worth quoting. The band gap's own shift with temperature has been removed here, so that the fan is the tail's doing and not the gap's.

The Urbach energy of four semiconductors against temperature, each the sum of a frozen-in part that no cooling removes and a lattice part that follows the hyperbolic cotangent of the phonon energy over twice the temperature. The thermal energy itself is drawn for comparison: a material whose tail is shallower than that line is one whose band gap is better defined than the temperature of the sample, and the two best materials here are in that condition while the amorphous one is not. The curves flatten as the temperature falls rather than going to zero, because a lattice keeps its zero-point motion, and they flatten onto different floors because the frozen disorder differs by a factor of fifteen between a good crystal and a glass. Cooling a glass does not make it a crystal, and the measurement says so.

The absorption of hydrogenated amorphous silicon near its own band gap of 1.75 electronvolts, on a logarithmic axis, at 4 temperatures. Below the gap the absorption does not stop; it falls exponentially, along a straight line whose slope is an energy, and the straightness holds over several decades. Raising the temperature makes the line shallower — the tail reaches further below the gap — and the slope runs from 49.0 millielectronvolts at 10 kelvin to 50.7 at 300, a factor of 1.04. The lines pivot about a point just above the gap rather than rotating about nothing, which is what makes the slope a single number worth quoting. The band gap's own shift with temperature has been removed here, so that the fan is the tail's doing and not the gap's.

One computed absorption spectrum of gallium arsenide at 300 kelvin, plotted in the form whose straight part extrapolates to the gap and scaled to one at the top of the range, with four recipes for reading a gap off it marked along the axis. All four are in ordinary use and they span 98 millielectronvolts, which is 13.1 times this material's own Urbach energy. The straight portion is straight only well above the join; the tail bends it downward, so a fit that reaches into the tail returns a smaller gap, and a measurement that could not reach high enough in absorption has no way of knowing that is what it did. Choosing a threshold value of the absorption instead removes the arbitrariness of the window and replaces it with the arbitrariness of the threshold. A gap quoted without the recipe that produced it is quoted to a few tens of millielectronvolts however many decimals it carries.

One computed absorption spectrum of crystalline silicon at 300 kelvin, plotted in the form whose straight part extrapolates to the gap and scaled to one at the top of the range, with four recipes for reading a gap off it marked along the axis. All four are in ordinary use and they span 616 millielectronvolts, which is 56.0 times this material's own Urbach energy. The straight portion is straight only well above the join; the tail bends it downward, so a fit that reaches into the tail returns a smaller gap, and a measurement that could not reach high enough in absorption has no way of knowing that is what it did. Choosing a threshold value of the absorption instead removes the arbitrariness of the window and replaces it with the arbitrariness of the threshold. A gap quoted without the recipe that produced it is quoted to a few tens of millielectronvolts however many decimals it carries.

physicscheck asserts something about cross-section 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.