How much of copper's electron sea a temperature can reach

At its defaults it draws how much of copper's electron sea a temperature can reach. The probability that a state of a given energy is occupied, in copper, at 4 temperatures, with energy measured in units of the Fermi energy — 7.04 eV here. At absolute zero the curve is a step: every state below the ceiling is full and every state above it is empty. Raising the temperature rounds the step, and rounds it over a range of about kT, which is the whole point — at room temperature kT is 0.0259 eV against a ceiling of 7.04 eV, so the rounding is 1.6 per cent of the way down the sea and everything deeper is untouched. An electron in the deep is not held there by a force; it simply has nowhere to go, because every state it could be promoted to is occupied. at 0 K the step is spread over 0.00 per cent of E_F, at 300 K the step is spread over 1.61 per cent of E_F, at 3000 K the step is spread over 16.13 per cent of E_F, at 20000 K the step is spread over 107.52 per cent of E_F.

fermi-sea is one function in lib/figures/atomic.js — what happens once one is bound by the other. 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 probability that a state of a given energy is occupied, in copper, at 4 temperatures, with energy measured in units of the Fermi energy — 7.04 eV here. At absolute zero the curve is a step: every state below the ceiling is full and every state above it is empty. Raising the temperature rounds the step, and rounds it over a range of about kT, which is the whole point — at room temperature kT is 0.0259 eV against a ceiling of 7.04 eV, so the rounding is 1.6 per cent of the way down the sea and everything deeper is untouched. An electron in the deep is not held there by a force; it simply has nowhere to go, because every state it could be promoted to is occupied. at 0 K the step is spread over 0.00 per cent of E_F, at 300 K the step is spread over 1.61 per cent of E_F, at 3000 K the step is spread over 16.13 per cent of E_F, at 20000 K the step is spread over 107.52 per cent of E_F.

The radius of a cold, degenerate star against its mass, obtained by integrating the equations of hydrostatic support outward from 11 different central densities with the exact degenerate equation of state, and nothing else. Heavier means smaller — the opposite of every ordinary object, and the direct consequence of a pressure that comes from counting states rather than from heat. The curve turns over and runs into a vertical asymptote at 1.452 solar masses, against 1.456 from the limiting polytrope, whose own constant 2.0182 is integrated here as well. That is Chandrasekhar's limit. It exists because the electrons become relativistic: once they are, the pressure goes as the four-thirds power of the density, and for that exponent alone the mass of a self-gravitating ball is independent of its radius — so squeezing it harder produces no more support and there is exactly one mass such a star can have. The horizontal line is the Earth's radius, which the curve crosses near a solar mass: a white dwarf of the Sun's mass is the size of a planet, and the ones close to the limit are a few thousand kilometres across. What the model leaves out is what actually happens at the top: at those densities electrons begin to be captured onto nuclei, which removes the very pressure holding the star up, so the collapse starts slightly below the line rather than at it.

Degeneracy pressure against electron density, both on logarithmic axes, over 9 decades. The straight line is the non-relativistic result, P = (2/5)nE_F, which rises as the five-thirds power of density; the other curve is the same integral done without assuming the electrons are slow. They agree wherever the electrons at the top of the sea are slow compared with light and part company where they are not. In copper the pressure is 38.3 gigapascals — 383 thousand atmospheres, at absolute zero, in a wire on a bench — and the electrons at the ceiling are moving at 0.53 per cent of the speed of light, so relativity is nowhere in it. The exponent falls from 5/3 toward 4/3 as the sea becomes relativistic, and a support whose pressure rises more slowly than the weight it is holding up has a ceiling of its own — which is a question about stars and belongs to the collection that owns them.

The probability that a state of a given energy is occupied, in copper, at 4 temperatures, with energy measured in units of the Fermi energy — 7.04 eV here. At absolute zero the curve is a step: every state below the ceiling is full and every state above it is empty. Raising the temperature rounds the step, and rounds it over a range of about kT, which is the whole point — at room temperature kT is 0.0259 eV against a ceiling of 7.04 eV, so the rounding is 1.6 per cent of the way down the sea and everything deeper is untouched. An electron in the deep is not held there by a force; it simply has nowhere to go, because every state it could be promoted to is occupied. at 0 K the step is spread over 0.00 per cent of E_F, at 300 K the step is spread over 1.61 per cent of E_F, at 3000 K the step is spread over 16.13 per cent of E_F, at 20000 K the step is spread over 107.52 per cent of E_F.

The radius of a cold, degenerate star against its mass, obtained by integrating the equations of hydrostatic support outward from 11 different central densities with the exact degenerate equation of state, and nothing else. Heavier means smaller — the opposite of every ordinary object, and the direct consequence of a pressure that comes from counting states rather than from heat. The curve turns over and runs into a vertical asymptote at 1.257 solar masses, against 1.260 from the limiting polytrope, whose own constant 2.0182 is integrated here as well. That is Chandrasekhar's limit. It exists because the electrons become relativistic: once they are, the pressure goes as the four-thirds power of the density, and for that exponent alone the mass of a self-gravitating ball is independent of its radius — so squeezing it harder produces no more support and there is exactly one mass such a star can have. The horizontal line is the Earth's radius, which the curve crosses near a solar mass: a white dwarf of the Sun's mass is the size of a planet, and the ones close to the limit are a few thousand kilometres across. What the model leaves out is what actually happens at the top: at those densities electrons begin to be captured onto nuclei, which removes the very pressure holding the star up, so the collapse starts slightly below the line rather than at it.

How many states there are at each energy — proportional to the square root of it, because the states are points in a momentum space and the number inside a sphere grows as its volume — with the electrons filled in from the bottom. copper has 8.49e+28 conduction electrons in a cubic metre, which fills the states to 7.04 eV; an electron at the top of the sea is moving at 1.57 million metres a second, at absolute zero, with nothing driving it. The shaded region beyond the ceiling is the part of the sea that 300 K has excited: it is 0.38 per cent of the electrons, and everything under it is exactly as it would be at absolute zero.

physicscheck asserts something about fermi-sea 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.