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We introduce the Planck distributionDistributionPlanck distributionPlanckPlanck distribution functionDistribution functionPlanck distribution functionPlanckPlanck distribution function for phonons and calculate the phonon contributionContributionphonon contribution to the thermodynamic quantities and derivatives (including Gruneisen constantConstantGruneisen constantGruneisenconstant and the coefficient of thermal expansion). We find low- temperature (quantum) and high-temperature (classical) behavior of the specific heat and mean squared atom displacementDisplacementatom displacement in the Debye modelDebyeDebye model. We illustrate important infrared divergencies of the mean squared displacement in crystals of reduced dimensionality.

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Lecture 9. Phonon Distribution Function: Specific Heat of the Crystal

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  • 发表日期：

    2026年

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  • Maxim Kagan

摘要

We introduce the Planck distributionDistributionPlanck distributionPlanckPlanck distribution functionDistribution functionPlanck distribution functionPlanckPlanck distribution function for phonons and calculate the phonon contributionContributionphonon contribution to the thermodynamic quantities and derivatives (including Gruneisen constantConstantGruneisen constantGruneisenconstant and the coefficient of thermal expansion). We find low- temperature (quantum) and high-temperature (classical) behavior of the specific heat and mean squared atom displacementDisplacementatom displacement in the Debye modelDebyeDebye model. We illustrate important infrared divergencies of the mean squared displacement in crystals of reduced dimensionality.