School of Mechanical and Aerospace Engineering

Seoul National University Computer Aided Thermal

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Thermal Properties of Solids and The Size Effect

Shin Dongwoo2010-20690

School of Mechanical and Aerospace Engineering

Seoul National University Computer Aided Thermal

Design Lab

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

School of Mechanical and Aerospace Engineering

Seoul National University Computer Aided Thermal

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5.1.1 Lattice Vibration in Solids : The Phonon gas

Fig. 5.1 The harmonic oscil-lator model of an atomin a solid.

1.Lattice VibrationsI. Thermal Energy Stor-

ageII. Heat Conduction

2.Free electrons for metalsI. Electrical transportII. Heat conduction

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The Dulong-Petit law

• ( is universal gas constant)

• Limits– Cannot predict low-temperature behavior– Overpredicts the specific heat for diamond,

graphite, and boron at room temperature.

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Einstein Model

• is Einstein Temperature• Each atom is treated as an independent oscillator

and all atoms are assumed to vibrate at the same frequency.

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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.

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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.

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Seoul National University Computer Aided Thermal

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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

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5.1.2 The Debye Specific Heat Model

• Determine (Debye temperature)

• ( :the number density of atoms)

School of Mechanical and Aerospace Engineering

Seoul National University Computer Aided Thermal

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5.1.2 The Debye Specific Heat Model

• Determine the Debye Specific Heat

()

School of Mechanical and Aerospace Engineering

Seoul National University Computer Aided Thermal

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5.1.2 The Debye Specific Heat Model

1.

2. , using

School of Mechanical and Aerospace Engineering

Seoul National University Computer Aided Thermal

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5.1.2 The Debye Specific Heat Model

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Summary

𝑫 (𝝂 )=𝟏𝟐𝝅𝝂𝟐

𝒗𝒂𝟑

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5.1.2 The Debye Specific Heat Model

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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

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5.1.3 Free Electron Gas in Metals

• Fermi energy :

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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 ,

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5.1.3 Free Electron Gas in Metals

• From

• At

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Sommerfeld expansion

• depends on T “Sommerfeld expansion”

Proof.

Because ,

It should follow this condition

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5.1.3 Free Electron Gas in Metals

• Using Sommerfelt expansion

=0

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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