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 rms size of a cloud of 10⁵ ideal atoms in a harmonic trap, in units of the Fermi radius, against temperature in units of the Fermi temperature, for fermions, for bosons and for a classical gas. The classical cloud shrinks as the square root of half the temperature ratio, all the way to nothing. The bosons follow it down to their condensation temperature, 0.52 of the Fermi temperature, and then collapse into the trap's ground state, whose rms size is 0.054 Fermi radii. The fermions part from the classical line near the Fermi temperature and level off at one over the square root of eight, 0.354 of the Fermi radius: at a quarter of the Fermi temperature they are 1.17 times the classical size, and cooling further barely changes them. Mixed together in one trap, as lithium-7 and lithium-6 were in 2001, the two clouds start the same size and end 6.5 times apart in width for ideal atoms of these numbers — further still in volume. The fermions are held up by nothing but the rule that no two of them may share a state.

The energy per atom of the same three clouds, in units of the Fermi energy, against temperature in units of the Fermi temperature: the classical 3kT of a three-dimensional oscillator, the Bose gas falling to zero as its atoms condense, and the Fermi gas levelling off at 0.75 of the Fermi energy. At absolute zero each fermion still carries three-quarters of the Fermi energy on average, because the atoms that would have occupied the low levels have nowhere to go; the energy is the cost of stacking 10⁵ atoms one to a state. It is also what the cloud's size measures. Half of a trapped atom's energy is potential, so the energy per atom and the mean square size are one curve read twice — which is why a photograph of the cloud is a thermometer, until the curve goes flat.

How 220 identical atoms distribute themselves over the energy shells of a three-dimensional harmonic trap, shell n holding (n + 1)(n + 2)/2 states at energy (n + 3/2)ħω: as fermions at zero temperature, which fill shells 0 to 9 exactly, every state once, and as fermions at three-tenths of the Fermi temperature, whose top shells are part-filled and whose next ones are part-occupied; the outlines show each shell's capacity. Bosons at zero temperature would put all 220 in shell 0, which holds a single state — a bar more than two hundred times its capacity. The fermions' highest occupied shell sets the Fermi energy — 10.5 ħω here, against the large-N formula, the cube root of 6N times ħω, ≈ 11.0 ħω — and nothing in the atoms' interactions does.

The density of a trapped cloud integrated through two directions — the profile a camera records, summed along one more axis — against position in units of the Fermi radius: an ideal Fermi gas at zero temperature, one minus the square of the position raised to the power five-halves, the position in Fermi radii, ending sharply at x = 1; a Fermi gas at a quarter of the Fermi temperature; and a Gaussian with the same mean square width as the latter, which is what a classical cloud of that size would show. All have equal area. The Fermi profiles are flatter on top and steeper at the edge than the Gaussian: the centre is full, and the atoms that would crowd there classically are pushed outward. Fitting that shape is how such clouds are thermometers below the Fermi temperature, where their size has stopped changing: the profile at a quarter of the Fermi temperature and the one at zero differ in the shape of their shoulders, not in their width.

The mean number of atoms in the lowest cell of phase space — the state of zero momentum at the centre of the trap — against temperature in units of the Fermi temperature, on a logarithmic scale: z/(1 + z) for fermions, z/(1 − z) for bosons and z for a classical gas, with z the fugacity that fixes the number of atoms. A classical gas would put 10.7 atoms in that one state at a quarter of the Fermi temperature, which is meaningless for fermions and an underestimate for bosons; the bosons pile in without limit once they condense, at 0.52 of the Fermi temperature, drawn off the top. The fermions approach one and never reach it: at a quarter of the Fermi temperature the cell is 95.4% full. Below about the Fermi temperature the classical description fails for both kinds of atom, in opposite directions, and the size of the cloud is where the failure becomes visible.

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.