--- title: "Phonons and the Heat Capacity of Solids" description: "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 cuto" canonical: https://lightmysky.com/learn/science/phonons-and-the-heat-capacity-of-solids-mt_WXZPdz_JTd source: https://lightmysky.com/learn/science/phonons-and-the-heat-capacity-of-solids-mt_WXZPdz_JTd.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Science · Area: Matter & Materials · Ages 23 to 24 Page: https://lightmysky.com/learn/science/phonons-and-the-heat-capacity-of-solids-mt_WXZPdz_JTd ## Ready when they can - 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 ## Lesson: Why cold solids hold less heat 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. **Example.** 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. **Recap.** Heat feeds lattice jiggles, cold freezes high packets out, and the surviving slow modes give the T-cubed fall. ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Equipartition and the Heat Capacities of Gases](https://lightmysky.com/learn/science/equipartition-and-the-heat-capacities-of-gases-mt_3OjJwVWlYD) - [Crystal Lattices, the Reciprocal Lattice and Bloch's Theorem](https://lightmysky.com/learn/science/crystal-lattices-the-reciprocal-lattice-and-blochs-theorem-mt_nbNnHpGnZI) - [Coupled Oscillators and Normal Modes](https://lightmysky.com/learn/science/coupled-oscillators-and-normal-modes-mt_vnTGOIqZzP) - [Ideal Quantum Gases and Bose-Einstein Condensation](https://lightmysky.com/learn/science/ideal-quantum-gases-and-bose-einstein-condensation-mt_wR2YnShTwJ) ## Opens up - [Superconductivity: the Meissner Effect and Cooper Pairs](https://lightmysky.com/learn/science/superconductivity-the-meissner-effect-and-cooper-pairs-mt_2H9bAiAAUV) - [Band Structure: Metals, Insulators and the Gap](https://lightmysky.com/learn/science/band-structure-metals-insulators-and-the-gap-mt_9f74Vu6qnW)