Phonons and the Heat Capacity of Solids
Lattice vibrations quantise into modes whose occupation follows the Bose distribution, so a solid's heat capacity falls away at low temperature instead of staying constant. Counting modes up to a cutoff gives the cubic law that measurements show.
What a learner can do afterwards
- Explains why equipartition fails for a solid at low temperature
- Counts vibrational modes to reach the low-temperature power law
- Distinguishes acoustic from optical branches on a dispersion curve
1 · Read
Cooling electronics or storing liquid gases needs heat capacity, which starts with the atoms themselves. Warming a solid feeds the jiggles of atoms around their lattice places. Each atom shakes in three directions, each acting as a spring with motion plus stretch energy: six shares per atom. Equipartition grants every share about 1 over 2 kT, so one mole holds about 3RT and the heat capacity sits near 3R, about 25 joules per mole per kelvin, the Dulong-Petit rule that fits many metals at room temperature.
Cool the solid and the rule breaks: heat capacity falls away toward zero instead of holding steady. Nature hands vibrational energy in packets of size hf, and when kT drops below a mode packet that mode stays empty and frozen out. Equipartition assumed smooth energy flow, which fails once packets matter.
Counting only modes below a cutoff gives the famous low-temperature law: capacity grows like T cubed, as experiments show. Halving the temperature gives an eighth of the capacity. Only slow long-wavelength vibrations still soak energy down there, so the classical rule is a high-temperature limit.
Read a dispersion curve in two branches. In an acoustic branch neighbours swing together the way sound moves, while in an optical branch neighbours swing against each other in a pattern light can excite. Counting branches correctly is how the mode census starts.
Heat feeds lattice jiggles, cold freezes high packets out, and the surviving slow modes give the T-cubed fall.
2 · Watch
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Where it sits
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Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
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.