Why some solids conduct, some insulate, and some sit in between. In a crystal the electron energies bunch into allowed bands split by a forbidden zone — the gap Eg. Fill a band exactly full and leave the next empty, and only that gap decides everything: no gap → metal, a small gap (~1 eV) → semiconductor, a wide gap (>~4 eV) → insulator. Down the center, data flows: the gap and temperature go in, the engine sorts the solid and counts its thermal carriers, the class comes out. The blue team builds it; the red team tries to break it.
source A. H. Wilson, The Theory of Electronic Semi-Conductors, Proc. R. Soc. Lond. A 133, 458 (1931) — doi.org/10.1098/rspa.1931.0162; on F. Bloch's electron-in-a-lattice theorem (Z. Phys. 52, 555, 1928). AMBER: primary papers paywalled. Rendered, not quoted.
Bloch (1928): an electron in a periodic lattice has states that spread into continuous bands separated by gaps where no state exists. Fill the highest occupied band and the class follows from the gap at the Fermi level:
metal a band is only partly filled → no gap at E_F, always carriers. semiconductor a full valence band, empty conduction band, small Eg ~1 eV. insulator same, but Eg large (>~4 eV).
Live classification by gap size:
| solid | Eg (eV) | class |
|---|
One quantity, Eg, sorts every solid — and the thermal carrier density ni ∝ e−Eg/2kT is the door to the rest of the corpus. Populate that gap with impurities and you get the-semiconductor-doping (mass action n·p = ni²); butt an n against a p and the-pn-junction rectifies.
And the sign is the whole story: a semiconductor's conductivity rises with heat — the opposite of the metal in the-drude-model, whose σ = ne²τ/m falls as τ shrinks. Each sphere is the next one's premise.
The blue team's live check: re-derive the classification and the carrier-density sign from the engine and confirm them against the known physics. If red flips the sign of the Boltzmann factor, this badge is where it shows.
Two numbers feed the engine: the gap Eg (eV) between the full valence band and the empty conduction band, and the temperature T (K) that thermally kicks electrons across it. Reference solids:
| solid | Eg (eV) | class |
|---|---|---|
| copper | 0 (overlap) | metal |
| germanium | 0.66 | semiconductor |
| silicon | 1.12 | semiconductor |
| GaAs | 1.42 | semiconductor |
| diamond | 5.47 | insulator |
"Partly filled" makes a metal regardless of any gap elsewhere. That is what you feed the panel below.
Drag Eg through the forbidden zone; drag T to heat the crystal. Every number below is computed on the spot — never looked up.
ni = N·(T/300)3/2·e−Eg·q/2kT; the class is decided by Eg alone; the edge from Eg = hc/λ. Live from the constants, no faked numbers.
What the machine proves: a solid is a metal with no gap, a semiconductor at ~1 eV (Si 1.12), an insulator above ~4 eV (diamond 5.47); carriers ni ∝ e−Eg/2kT rise steeply with heat and fall exponentially with the gap; a semiconductor therefore conducts better hot — the opposite of a metal; and the gap sets the minimum photon it absorbs, Eg = hc/λ.
The blue team's witness (left) confirms the sign live; the red team (right) tries to flip it.
The whole free-band scheme also assumes independent electrons in a static lattice — Bloch's approximation. Strong correlation breaks it: NiO "should" be a metal by band-filling yet is an insulator (a Mott insulator). The gap is the first law, not the last.
"A semiconductor is a poor metal." Cut. Its σ rises with temperature; a metal's falls. Opposite sign — computed live in the panel, not asserted.
"Bigger gap, more carriers." Cut. ni ∝ e−Eg/2kT: a wider gap gives exponentially fewer carriers (Si→diamond drops ni by ~10³⁶). This is exactly the sign the tamper flips.
"The gap is fixed by the atom." Kept, corrected. Eg depends on temperature and pressure and shrinks as the crystal heats (Varshni) — small, but real.
The red team's move: flip one sign — make ni ∝ e+Eg/2kT — so a bigger gap would give more carriers. The blue team's witness (window 7) is watching.
Flip the Boltzmann sign and the "larger gap → fewer carriers" law inverts — the witness recomputes, disagrees with the known physics, and turns red. Nothing is faked; the attack is real and it is caught.