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School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
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
Design Lab
3
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
School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
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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.
School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
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.
School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
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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.
School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
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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.
School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
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
School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
9
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
Design Lab
10
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
Design Lab
11
5.1.2 The Debye Specific Heat Model
1.
2. , using
School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
5.1.2 The Debye Specific Heat Model
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Summary
𝑫 (𝝂 )=𝟏𝟐𝝅𝝂𝟐
𝒗𝒂𝟑
School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
13
5.1.2 The Debye Specific Heat Model
School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
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
School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
15
5.1.3 Free Electron Gas in Metals
• Fermi energy :
School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
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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 ,
School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
17
5.1.3 Free Electron Gas in Metals
• From
• At
School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
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Sommerfeld expansion
• depends on T “Sommerfeld expansion”
Proof.
Because ,
It should follow this condition
School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
19
5.1.3 Free Electron Gas in Metals
• Using Sommerfelt expansion
=0
School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
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