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Thermal Properties of Solids and The Size Effect

Thermal Properties of Solids and The Size Effect. Shin Dongwoo 2010-20690. Contents. 5.1 Specific Heat of Solids 5.1.1 Lattice Vibration in Solids : The Phonon gas 5.1.2 The Debye Specific Heat Model 5.1.3 Free Electron Gas in Metals. 5.1.1 Lattice Vibration in Solids : The Phonon gas.

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Thermal Properties of Solids and The Size Effect

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  1. Thermal Properties of Solids and The Size Effect Shin Dongwoo 2010-20690

  2. Contents 5.1 Specific Heat of Solids • 5.1.1 Lattice Vibration in Solids : The Phonon gas • 5.1.2 The Debye Specific Heat Model • 5.1.3 Free Electron Gas in Metals

  3. 5.1.1 Lattice Vibration in Solids : The Phonon gas • Lattice Vibrations • Thermal Energy Storage • Heat Conduction • Free electrons for metals • Electrical transport • Heat conduction Fig. 5.1 The harmonic oscillator model of an atom in a solid.

  4. The Dulong-Petit law ( is universal gas constant) Limits • Cannot predict low-temperature behavior • Overpredictsthe specific heat for diamond, graphite, and boron at room temperature.

  5. Einstein Model is Einstein Temperature Each atom is treated as an independent oscillator and all atoms are assumed to vibrate at the same frequency.

  6. Einstein Model The deriving procedure is similar to the analysis of vibration energies for diatomic gas molecules. Limits • The bonding in a solid prevents independent vibrations.

  7. 5.1.2 The Debye Specific Heat Model Einstein Model(1907) : Each atom as an individual oscillator Debye model (1912): Vibration like standing waves Frequency upper bound : Total number of vibration modes : 3N (N : # of atoms) Phonon : the quanta of lattice waves Energy of phonons : Momentum ( is propagation speed) For elastic vibrations, 1 longitudinal wave + 2 transverse waves in a crystal.

  8. 5.1.2 The Debye Specific Heat Model Determine (Density of states of phonons) The total number of phonon depends on temperature. => NOT conserved. ( at BE statistics) : BE distribution function Using 1 longitudinal + 2 transverse waves is a weighted average

  9. 5.1.2 The Debye Specific Heat Model Determine (Debye temperature) ( :the number density of atoms)

  10. 5.1.2 The Debye Specific Heat Model Determine the Debye Specific Heat ()

  11. 5.1.2 The Debye Specific Heat Model 1. 2. , using

  12. 5.1.2 The Debye Specific Heat Model Summary

  13. 5.1.2 The Debye Specific Heat Model

  14. 5.1.3 Free Electron Gas in Metals The translational motion of free electrons within the solid • Electrical Conductivity • Thermal Conductivity The order of the free electrons number = The order of the number of atoms Electrons obey the Fermi-Dirac distribution

  15. 5.1.3 Free Electron Gas in Metals Fermi energy :

  16. 5.1.3 Free Electron Gas in Metals in a volume V, spherical shell in the velocity space. due to the existence of positive and negative spins. Using ,

  17. 5.1.3 Free Electron Gas in Metals From At

  18. Sommerfeld expansion depends on T “Sommerfeld expansion” Proof. Because , It should follow this condition

  19. 5.1.3 Free Electron Gas in Metals Using Sommerfelt expansion =0

  20. 5.1.3 Free Electron Gas in Metals The specific heat of free electrons Electronic contribution to the specific heat of solids is negligible except at very low temperature. The specific heat of metals at very low temperature

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