The Free Electron Gas and the Fermi Energy
Electrons in a metal fill states up to the Fermi energy even at absolute zero, which explains why they contribute so little to heat capacity. Only the few near the top of the stack can be excited.
What a learner can do afterwards
- Estimates the Fermi energy of a metal from its electron density
- Explains why the electronic heat capacity is far smaller than equipartition predicts
- Connects the filled band picture to electrical conduction in metals
1 · Read
Crowd atoms into a crystal and their sharp levels smear into bands. A filled band cannot carry current, because every moving electron is cancelled by another. Conduction needs a partly filled band, where free slots let electrons shift under a field.
Model that partly filled band as particles in a box and fill the lowest states, two electrons per level. The topmost energy at absolute zero is the Fermi energy. It depends only on electron density, so denser metals pack higher.
Heat can only excite electrons near the top of the stack, since deeper ones have no empty states nearby. Only that thin skin of electrons joins in, so the electronic heat capacity is far smaller than old rules predict. Most of the stack sits out.
To judge a metal, ask two questions: is the top band partly filled, and how dense are its electrons. Partly filled means it conducts, and higher density means a higher Fermi energy.
Electrons stack two per level to the Fermi energy, and only the top skin can move or take heat.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.