Debye T³ law

The Debye T³ law states that a solid’s lattice heat capacity is proportional to the cube of absolute temperature when temperature is well below its Debye temperature.

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Debye model: A model of a crystal’s vibrations that treats them as acoustic waves with a continuous spectrum of frequencies. Its low-frequency phonon spectrum produces the cubic heat-capacity dependence.
Specific heat capacity: The heat required to raise the temperature of a unit mass of a substance by one degree. The law is often expressed through the low-temperature specific heat of a solid.
Peter Debye: A Dutch physicist and physical chemist whose work included a quantum model of specific heat in solids. His 1912 model yielded the low-temperature cubic law.
Low-temperature calorimetry: The measurement of heat capacity and related thermal properties near absolute zero. It can test whether a solid’s lattice contribution follows a cubic temperature dependence.
Two-dimensional Debye model: A Debye-type description of vibrations in a two-dimensional solid, with a low-frequency density of states proportional to frequency. Its low-temperature lattice heat capacity scales as T² rather than T³.
Phonon: A quantized mode of vibration in a solid, carrying energy and crystal momentum. The thermal population of low-energy phonons determines the lattice heat capacity.
Equipartition theorem: A classical statistical-mechanics result assigning equal average energy to each quadratic degree of freedom at thermal equilibrium. Its classical prediction differs from the quantum low-temperature behavior captured by the T³ law.
Einstein model of a solid: A quantum model of a solid that treats its atoms as oscillators with one common vibration frequency. It improved on classical theory but predicts an exponential low-temperature heat capacity, not a T³ power law.
Electronic specific heat: The heat-capacity contribution from thermally excited electrons in a material. In metals it often adds a term proportional to temperature, distinguishable from the lattice T³ term.
One-dimensional Debye model: A Debye-type description of vibrations in a one-dimensional system with acoustic modes. Its ideal low-temperature heat capacity scales as T rather than T³.