1. CIDER/ITP Short Course
Lattice Dynamics
11/23/2018
CIDER/ITP Short Course

2. 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 Bulk Modulus Thermal Expansivity
Minimum
Bulk Modulus
Curvature
Thermal Expansivity
Beyond harmonic
Molecules
Solids

4. One Dimensional Lattice

5. One Dimensional Lattice
Periodicity reflects that of the lattice
Brillouin zone center, k=0: =0
Brillouin zone edge, k=/a: =maximum
All information in first Brillouin zone
Frequency, 
1st Brillouin
Zone
2(K/m)1/2
Wavevector, k

6. One Dimensional Lattice
k=/a=2/; =2a
k0; 

9. Acoustic Velocities Acoustic Velocity Three dimensions k0
=2(K/m)1/2ka/2
/k=dw/dk=a(K/m)1/2
Acoustic Velocity
v=a(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)

10. Polyatomic Lattice Unit cell doubled Brillouin Zone Halved
Acoustic Branches folded
New, finite frequency mode at k=0
Optic Branch
New Brillouin
Zone

11. 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

12. MgSiO3 Perovskite Movie

13. Internal Energy Sum over all vibrational modes
Energy of each mode depends on
Frequency
Population
Temperature
n=3
n=2
n=1
n=0
Displacement

14. Heat Capacity or
CV=3R per mol of atoms (Dulong-Petit)

15. Thermal Pressure 1 Compression Increases  Thermal pressure
Vibrational frequencies
Vibrational energy
 Thermal pressure

16. Thermal Pressure 2 Thermal Pressure Bulk Sound Velocity
Temperature Dependence of VB

17. Interatomic Forces Ambient Structure Bulk Modulus Thermal Expansivity
Minimum
Bulk Modulus
Curvature
Thermal Expansivity
Beyond harmonic
Molecules
Solids