1. Transverse optical mode for diatomic chain
A real 1D system only has longitudinal, but harder to visualize
Amplitudes
of different atoms
A/B=-m/M
Transverse acoustic mode for diatomic chain
A/B=1

2. Amplitude of vibration is strongly exaggerated!
Analogy with classical mechanical pendulums attached by spring
Amplitude of vibration is strongly exaggerated!

3. Longitudinal Eigenmodes in 1D
What if the atoms were oppositely charged?
Optical Mode: These atoms, if oppositely charged, would form an oscillating dipole which would couple to optical fields with λ~a
For example, an ionic crystal, where we know an electron transfers from one atom to the other. What would be different between these two?

4. Objectives By the end of today you should be able to:
Expand this model into 2 and 3 dimensions
Analyze the dispersion curves for real crystals
Understand why neutron scattering is sensitive to the phonon dispersion curve
Next time: use to understand experimental specific heat

5. Phonon Dispersion in 3D
1D model
The 1D model can be extended to 3D if the variables u refer not to displacements of atoms but planes of atoms.
Need to include motions that are perpendicular to the wave vector.
These are called transverse acoustic modes (TA), as opposed to longitudinal acoustic modes (LA).
3D model

6. Example: Earthquake waves
3D Dispersion curves
Every 3D crystal has 3 acoustic branches, 1 longitudinal and 2 transverse Are the branches degenerate?
“Primary wave” = faster wave LA
“Secondary wave” = slower wave TA
The 2011 Virginia earthquake occurred on August 23 at 1:51:04 p.m. Could also ask how many have felt an earthquake?
Could draw a rectangular lattice on the board to discuss why the two transverse directions would be different (wave would be going into or out of board to show this)
But frequently allow directions of high symmetry, the two transverse frequencies will be the same
Why longitudinal faster? A little challenging to explain. Think about springs. Linking very strong along the wave motion. We will see how this breaks down when talk about very different kinds of bonds.
No, the perpendicular displacements will have different force (“spring”) constants from the longitudinal force constants.
Example: Earthquake waves

7. Number and Type of Branches
Every crystal has 3 acoustic branches, 1 longitudinal and 2 transverse
Every additional atom in the primitive basis contributes 3 further optical branches (again 2 transverse and 1 longitudinal)
Sometimes transverse will be degenerate
How would you know which
branch is longitudinal?
How many will a perovskite have?
3*5 = 15 branches, 1LA, 2TA, 4LO and 8TO
Question for me to look up sometime: What happens when perovskite is no longer cubic and unit cell is larger (such as rhombohedral). The above logic would suggest more bands. Perhaps some bands split into slightly different bands? What effect does that have on material properties?
p atoms/primitive unit cell ( primitive basis of p atoms):
3 acoustic branches
+ 3(p-1) optical branches
= 3p branches
1LA +2TA
(p-1)LO +2(p-1)TO
How many branches and of what type for a perovskite ABO3?

8. 3 translational degrees of freedom xz y
Intuitive picture:
1atom
3 translational degrees of freedom x
3+3=6 degrees of freedom=3 translations+2rotations
+1vibraton
# atoms
in primitive
basis
# of primitive
unit cells
But each unit cell should have the same possible vibrations as the next one, since they are unique
3D Solid: p N atoms
3p N vibrations
no translations, no rotations

9. 2D Lattice Write down the equation(s) of motion and guess solution
Ul,m+1 K
Ulm
Ul-1,m
Ul+1,m
How will ṻ relate to u?
2D Lattice
Ul,m-1

10. Ul,m+1 K
What should it be for a cubic lattice?
Similar to electronic energy bands, plot w vs k for the [10] and [11] directions.
Identify the values of  at k=0 and at the BZ edges. (Might be helpful to draw the BZ)
Ulm
Ul-1,m
Ul+1,m
Got here in 50 minute class
Ul,m-1
2D Lattice

11. Discuss similarities & differences. Why? Why might they be different?
Real Phonon Spectra Might Look Slightly Different
What are some differences?
Discuss similarities & differences. Why?
Why might they be different?
What is this?
Remember we made several simplifications:
Interactions beyond nearest neighbors are not included
Assumed harmonic potential
Ignored electron-phonon coupling
Differences: More than one branch, not always largest at BZ edge
First hint is the labeling. Show the structure of BZ. Don’t worry if you can’t easily visualize that the BZ is BCC (thus real is FCC). Takes practice.
Similar to: Neon, monatomic FCC lattice
Not much difference between gammaX and gammaL lines. Why? If you only knew what direction you were measuring and you saw this similarity, this would be one way to identify the crystal type (however, XRD is a much easier and cheaper experiment, so it wouldn’t typically make a lot of sense to do this for that purpose). What do transverse not look as similar?