1. Lattice Dynamics related to movement of atoms
about their equilibrium positions
determined by electronic structure
Physical properties of solids
Sound velocity
Electronic properties the topic coming up after phonons and thermal effects
(have flexible ending because the speed of this lecture varies a lot due to questions)
Thermal properties: -specific heat
-thermal expansion
-thermal conductivity
(for semiconductors)
Hardness of perfect single crystals (without defects)

2. Uniform Solid Material
There is energy associated with the vibrations of atoms.
They are tied together with bonds, so they can't vibrate independently.
The vibrations take the form of collective modes which propagate through the material.
Last time we learned that there is energy associated with the vibrations of atoms. But they are tied together, so they don’t vibrate independently.
Sound velocity, thermal expansion, thermal conductivity, specific heat need lattice dynamics to get correct
(X-1)
(X)
(X+1)

3. Wave-Particle Duality
They don’t like to be seen together 
Phonons are another quasi-particle. What did it mean for something to be a quasi-particle? It acted like a free particle, but could be due to collective interactions including collisions.
We are much more familiar with treating light as a particle, what does a phonon particle look like?
Unlike static lattice model , which deals with average positions of atoms in a crystal, lattice dynamics extends the concept of crystal lattice to an array of atoms with finite masses that are capable of motion.
Just as light is a wave motion that is considered as composed of particles called photons, we can think of the normal modes of vibration in a solid as being particle-like.
Quantum of lattice vibration is called the phonon.

4. Phonon: A Lump of Vibrational Energy
Propagating lattice vibrations can be considered to be sound waves, and their propagation speed is the speed of sound in the material.
Roughly how big is ?
Phonon:
Sound
Wavepackets
So how big is this? (animation and picture from Wikipedia)
If you wanted to probe this oscillation, what wavelength would you want your probe to have? We will come back to this.

5. Reminder to the physics of oscillations and waves (a few slides)
Harmonic oscillator in classical mechanics
Equation of motion:
Example: vertical springs or
Hooke’s law
where
Solution with
where
X=A cos ωt X Kx
Do you remember why we treat waves like harmonic motion? Waves are generated by a oscillation. x

6. Traveling plane waves:
Displacement as a function of time and k Y or X
(Phonon wave vector also often given as q instead of k)
Consider a particular state of oscillation
Y=constant traveling along
In a crystal, there are often traveling waves. Different waves might result in the atom movement, which we’ll come back to later.
To illustrate some points, let’s consider a wave that causes all of the atoms in a region to move together.
You sometimes see q used instead of k for electrons too, but I have seen it more with phonons.
In order for Y to be constant, the argument inside of the cosine must be constant. If it’s constant, it’s derivative is equal to zero; let’s see what that implies.
Also get the same solution if you solve for the wave equation.
solves wave equation

7. Transverse wave Longitudinal wave Standing wave
What if you subtracted the waves?
Standing wave

8. Like diffraction, simple for large wavelengths
Crystal can be viewed as a continuous medium:
good for
Speed of longitudinal wave:
where Bs: bulk modulus
(ignoring anisotropy of the crystal)
Bs determines elastic deformation energy density
(click for details in thermodynamic context)
Relate bulk modulus to our conversation on volume stress and strain
dilation
E.g.: Steel
Bs= N/m2
ρ=7860kg/m3

9. > interatomic spacing continuum approach fails
It is only by considering these quantized phonons that we can finally get an accurate calculation for specific heat (which we will discuss more after a few classes)
In addition:
vibrational modes quantized
phonons

10. Vibrational Modes of a Monatomic Lattice
Linear chain:
Remember: two coupled harmonic oscillators
Symmetric mode
Anti-symmetric mode
Superposition of normal modes
Today 1D, Next time: 3D

11. ? generalization Infinite linear chain
How to derive the equation of motion in the harmonic approximation n
n-2
n-1
n+1
n+2 a C
un-2
un-1
un+1
un+2 un un
un+1
un+2
un-1
un-2
fixed

12. ? Total force driving atom n back to equilibrium n n
equation of motion
Old solution for continuous wave equation was . Use similar? ?
approach for linear chain
Let us try! , ,

13. Continuum limit of acoustic waves:k
Note: here pictures of transversal waves
although calculation for the longitudinal case
Continuum limit of acoustic waves:

14. Technically, only have longitudinal modes in 1D
(but transverse easier to see what’s happening)
x = na un un
Will need to trim out some of this but wait until see how much we get through in prior lecture
(a) Chain of atoms in the absence of vibrations.
(b) Coupled atomic vibrations generate a traveling longitudinal (L) wave along x. Atomic displacements (un) are parallel to x.
(c) A transverse (T) wave traveling along x. Atomic displacements (un) are perpendicular to the x axis.
(b) and (c) are snapshots at one instant.

15. Results in same motion of the atomsk
Results in same motion of the atoms
, here h=1
1-dim. reciprocal
lattice vector
Same as before!
I improved this a bit. Did the class flow better in this section this time?
Region
is called
first Brillouin zone
Means we only have to consider a small range of k values range than all possible wavelengths!

16. Note different convention for a than Kittel. I have a good reason!
Vibrational Spectrum for structures with 2 or more atoms/primitive basis
Linear diatomic chain: 2n
2n-2
2n-1
2n+1
2n+2
Note different convention for a than Kittel. I have a good reason! C a 2a
u2n
u2n+1
u2n+2
u2n-1
u2n-2
Equation of motion for atoms on even positions:
A and B are amplitudes of the motion of the two different atoms. The amplitudes need not be the same, but they could be.
The book actually makes the distance between same atoms a instead of my choice of 2a. Both are common. You will see shortly why I choose the axis this way. By doing this, I’m able to directly compare my result to the monatomic lattice as I make the masses more similar (next slide)
Equation of motion for atoms on odd positions:
Solution with:
and

17. , Click on the picture to start the animation M->m
note wrong axis in the movie
The books solves this with matrices, which is also fine. I’ll show you another way.
Only got here in 75 minute class (slide 18), got to 24 nice time (have flexible ending) 2 2 ,

18. Transverse optical mode for diatomic chain
Amplitudes
of different atoms
A/B=-m/M
Transverse acoustic mode for diatomic chain
Again, in 1D, these should all be longitudinal, but in 3D we have both. (longitudinal is harder to interpret/visualize in my opinion, so that’s why I keep showing these tranverse ones
A/B=1

19. Longitudinal Eigenmodes in 1D
What if the atoms were opposite 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?

20. Summary: What is a phonon?
Consider the regular lattice of atoms in a uniform solid material.
There should be energy associated with the vibrations of these atoms.
But they are tied together with bonds, so they can't vibrate independently.
The vibrations take the form of collective modes which propagate through the material.
Such propagating lattice vibrations can be considered to be sound waves.
And their propagation speed is the speed of sound in the material.

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