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    "summary": "Debye replaces a single atomic vibration frequency with an elastic-continuum mode spectrum and a finite3N cutoff, yielding the low-temperature cubic lattice heat-capacity law and the classical high-temperature limit within the model.\n\nWhy does a cold solid absorb less heat than classical physics predicts? Einstein had shown that quantized vibrations could explain part of the effect, but his single-frequency model predicted the wrong very-low-temperature trend. Debye instead treats a solid as many collective vibrations. Long-wavelength waves are described by elasticity; their frequency density grows as the square of frequency. A cutoff retains three vibration degrees of freedom per atom. Combining this spectrum with Planck oscillator energies gives a heat-capacity curve that approaches the classical limit at high temperature and a cubic temperature dependence in the macroscopic low-temperature acoustic regime. The characteristic temperature can also be estimated from elastic properties. Debye compares the model with existing measurements of diamond and several metals, while reporting correction uncertainties and mismatches. This is a model of lattice vibrations, not a complete theory of every source of heat capacity.",
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        "title": "P. Debye, Zur Theorie der spezifischen Warmen; complete original1912 Annalen der Physik789-839",
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        "description": "The model uses harmonic elastic-continuum modes and a cutoff matching the number of atomic degrees of freedom. The low-temperature behavior and high-temperature limit depend on these approximations. The sphere calculation and large-mode asymptotics are not an exact finite-atom spectrum. Real materials, anharmonicity, electronic heat capacity and every experimental fit are outside this scoped model.",
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      "statement": "# Debye: a vibration spectrum explains the heat capacity of solids\n\nDebye replaces a single atomic vibration frequency with an elastic-continuum mode spectrum and a finite3N cutoff, yielding the low-temperature cubic lattice heat-capacity law and the classical high-temperature limit within the model.\n\nWhy does a cold solid absorb less heat than classical physics predicts? Einstein had shown that quantized vibrations could explain part of the effect, but his single-frequency model predicted the wrong very-low-temperature trend. Debye instead treats a solid as many collective vibrations. Long-wavelength waves are described by elasticity; their frequency density grows as the square of frequency. A cutoff retains three vibration degrees of freedom per atom. Combining this spectrum with Planck oscillator energies gives a heat-capacity curve that approaches the classical limit at high temperature and a cubic temperature dependence in the macroscopic low-temperature acoustic regime. The characteristic temperature can also be estimated from elastic properties. Debye compares the model with existing measurements of diamond and several metals, while reporting correction uncertainties and mismatches. This is a model of lattice vibrations, not a complete theory of every source of heat capacity.\n\n## Scientific scope\n\nThe model uses harmonic elastic-continuum modes and a cutoff matching the number of atomic degrees of freedom. The low-temperature behavior and high-temperature limit depend on these approximations. The sphere calculation and large-mode asymptotics are not an exact finite-atom spectrum. Real materials, anharmonicity, electronic heat capacity and every experimental fit are outside this scoped model.\n\n## People and systems\n\nP. Debye\n\n\n\n## Original sources\n\n- [P. Debye, Zur Theorie der spezifischen Warmen; complete original1912 Annalen der Physik789-839](https://gilles.montambaux.com/files/histoire-physique/Debye-1912.pdf)\n- [Publisher-deposited DOI10.1002/andp.19123441404 identity and references](https://api.crossref.org/works/10.1002/andp.19123441404)\n",
      "summary": "Debye replaces a single atomic vibration frequency with an elastic-continuum mode spectrum and a finite3N cutoff, yielding the low-temperature cubic lattice heat-capacity law and the classical high-temperature limit within the model.\n\nWhy does a cold solid absorb less heat than classical physics predicts? Einstein had shown that quantized vibrations could explain part of the effect, but his single-frequency model predicted the wrong very-low-temperature trend. Debye instead treats a solid as many collective vibrations. Long-wavelength waves are described by elasticity; their frequency density grows as the square of frequency. A cutoff retains three vibration degrees of freedom per atom. Combining this spectrum with Planck oscillator energies gives a heat-capacity curve that approaches the classical limit at high temperature and a cubic temperature dependence in the macroscopic low-temperature acoustic regime. The characteristic temperature can also be estimated from elastic properties. Debye compares the model with existing measurements of diamond and several metals, while reporting correction uncertainties and mismatches. This is a model of lattice vibrations, not a complete theory of every source of heat capacity.",
      "title": "Debye: a vibration spectrum explains the heat capacity of solids"
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