thermal properties of solids and the size effect
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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
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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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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)
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5.1.2 The Debye Specific Heat Model
• Determine the Debye Specific Heat
()
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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
12
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