Lattice Dynamics

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Lattice Dynamics
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Thermal Expansion

• As temperature changes, density changes
• Thermodynamics
• Relates this change to changes in other
  properties
• Cannot tell the magnitude or even the sign!
• Why positive alpha?
• What value vs. P,T,X?
• Macroscopic to Microscopic
• Thermodynamics to Statistical Mechanics

3
Interatomic Forces

• Ambient Structure
• Minimum
• Bulk Modulus
• Curvature
• Thermal Expansivity
• Beyond harmonic
• Molecules
• Solids

4
One Dimensional Lattice
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One Dimensional Lattice

• Periodicity reflects that of the lattice
• Brillouin zone center, k0 ?0
• Brillouin zone edge, k?/a ?maximum
• All information in first Brillouin zone

Frequency, ?
1st Brillouin Zone
2(K/m)1/2
0
Wavevector, k
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One Dimensional Lattice
k?/a2?/? ?2a
k?0 ???
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Acoustic Velocities

• k?0
• ?2(K/m)1/2ka/2
• ?/kdw/dka(K/m)1/2
• Acoustic Velocity
• va(K/m)1/2
• Three dimensions
• Wavevector, ki
• Polarization vector, pi
• For each ki, 3 acoustic branches
• ki?? pi longitudinal (P) wave
• ki?pi transverse (S) waves (2)

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Polyatomic Lattice
a

• Unit cell doubled
• Brillouin Zone Halved
• Acoustic Branches folded
• New, finite frequency mode at k0
• Optic Branch

New Brillouin Zone
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General Lattice

• Number of modes 3N organized into 3Z branches
• Z number of atoms in unit cell
• 3 Acoustic branches
• 3Z-3 optic branches
• Experimental Probes
• Optic zone center
• Raman
• Infrared
• Acoustic near zone center
• Brillouin
• Full phonon spectrum
• Inelastic neutron scattering

Quartz
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MgSiO3 Perovskite Movie
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Internal Energy
Energy

• Sum over all vibrational modes
• Energy of each mode depends on
• Frequency
• Population
• Frequency
• Temperature

n3
n2
n1
n0
Displacement
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Heat Capacity

• or
• CV3R per mol of atoms (Dulong-Petit)

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Thermal Pressure 1

• Compression Increases
• Vibrational frequencies
• Vibrational energy
• ? Thermal pressure

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Thermal Pressure 2

• Thermal Pressure
• Bulk Sound Velocity
• Temperature Dependence of VB

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Interatomic Forces

• Ambient Structure
• Minimum
• Bulk Modulus
• Curvature
• Thermal Expansivity
• Beyond harmonic
• Molecules
• Solids