- Title Pages
- Dedication
- Preface
- Acknowledgments
- 1 Introduction
- 2 The Classical Ideal Gas
- 3 Discrete Probability Theory
- 4 The Classical Ideal Gas: configurational Entropy
- 5 Continuous Random Numbers
- 6 The Classical Ideal Gas:Energy-Dependence of Entropy
- 7 Classical Gases: Ideal and Otherwise
- 8 Temperature, Pressure, Chemical Potential, and All That
- 9 The Postulates and Laws of Thermodynamics
- 10 Perturbations of Thermodynamic State Functions
- 11 Thermodynamic Processes
- 12 Thermodynamic Potentials
- 13 The Consequences of Extensivity
- 14 Thermodynamic Identities
- 15 Extremum Principles
- 16 Stability Conditions
- 17 Phase Transitions
- 18 The Nernst Postulate: the Third Law of Thermodynamics
- 19 Ensembles in Classical Statistical Mechanics
- 20 Classical Ensembles: Grand and Otherwise
- 21 Irreversibility
- 22 Quantum Ensembles
- 23 Quantum Canonical Ensemble
- 24 Black-Body Radiation
- 25 The Harmonic Solid
- 26 Ideal Quantum Gases
- 27 Bose–Einstein Statistics
- 28 Fermi–Dirac Statistics
- 29 Insulators and Semiconductors
- 30 Phase Transitions and the Ising Model
- Appendix: Computer Calculations and VPython
- Index

# The Harmonic Solid

# The Harmonic Solid

- Chapter:
- (p.291) 25 The Harmonic Solid
- Source:
- An Introduction to Statistical Mechanics and Thermodynamics
- Author(s):
### Robert H. Swendsen

- Publisher:
- Oxford University Press

The harmonic solid is a model of vibrations in a crystal. It goes beyond the simpler Einstein model, which was discussed in Chapter 23. The low-temperature behaviour is shown to be extremely similar to that of black-body radiation, which was discussed in Chapter 24. The Debye approximation is introduced, and its advantages and disadvantages are discussed.

*Keywords:*
crystal, black-body radiation, Einstein model, Debye approximation, vibrations

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- Title Pages
- Dedication
- Preface
- Acknowledgments
- 1 Introduction
- 2 The Classical Ideal Gas
- 3 Discrete Probability Theory
- 4 The Classical Ideal Gas: configurational Entropy
- 5 Continuous Random Numbers
- 6 The Classical Ideal Gas:Energy-Dependence of Entropy
- 7 Classical Gases: Ideal and Otherwise
- 8 Temperature, Pressure, Chemical Potential, and All That
- 9 The Postulates and Laws of Thermodynamics
- 10 Perturbations of Thermodynamic State Functions
- 11 Thermodynamic Processes
- 12 Thermodynamic Potentials
- 13 The Consequences of Extensivity
- 14 Thermodynamic Identities
- 15 Extremum Principles
- 16 Stability Conditions
- 17 Phase Transitions
- 18 The Nernst Postulate: the Third Law of Thermodynamics
- 19 Ensembles in Classical Statistical Mechanics
- 20 Classical Ensembles: Grand and Otherwise
- 21 Irreversibility
- 22 Quantum Ensembles
- 23 Quantum Canonical Ensemble
- 24 Black-Body Radiation
- 25 The Harmonic Solid
- 26 Ideal Quantum Gases
- 27 Bose–Einstein Statistics
- 28 Fermi–Dirac Statistics
- 29 Insulators and Semiconductors
- 30 Phase Transitions and the Ising Model
- Appendix: Computer Calculations and VPython
- Index