1. Lattice Dynamics Plays an important role for anything not at low temperature – like most things on Earth
In past, class ran a little short and is pretty math heavy. Maybe break up with fire eating demo? Recently added second velocity discussion so that may eat up the remaining extra time.

2. Actually longitudinal in 1D (looks like this in 3D though)
Optical Branch
Acoustic Branch
Actually longitudinal in 1D (looks like this in 3D though)
Last time: Compared monatomic and diatomic phonon dispersion curves

3. Objectives By the end of today you should be able to:
Expand 1D 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

4. 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.
Lower left image shows transverse acoustic modes (TA), as opposed to longitudinal acoustic (LA) modes in upper left.
3D model

5. 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 perpendicular displacements have different force (“spring”) constants from the longitudinal force constants, and sometimes transverse are different too.
How many of you have ever experienced an earthquake? After the shock of realizing you were in an earthquake, did you notice if it felt the same way throughout the shaking?
But frequently along directions of high symmetry, the two transverse frequencies will be the same. Often the case for cubic systems. Less common in others.
Example: Earthquake waves

6. Number and Type of Branches
SmMnO3
1D model
3D model
Sometimes transverse will be degenerate
Every 3D crystal has 3 acoustic branches, 1 L and 2 transverse Every additional atom in the basis contributes 3 optical
How many will a perovskite have?
3*5 = 15 branches, 1LA, 2TA, 4LO and 8TO (perovskite example is SmMnO3. I picked this one because you can see all 15 and none are negative, which can happen if not a stable structure.)
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?
How many branches and of what type for a perovskite ABO3?

7. 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?
Substitute on the board
2D Square Lattice
Ul,m-1

8. Plot w vs k for the [10] and [11] directions.
Ul,m+1 K
Plot w vs k for the [10] and [11] directions.
Identify the values of  at k=0 and at the BZ edges. (Hint: draw the BZ first)
Ulm
Ul-1,m
Ul+1,m
Ul,m-1
2D Lattice

9. What are some differences? Why might they be different?
Real Phonon Spectra Might Look Slightly Different
What are some differences?
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?
Equation we obtained is valid only for an isotropic solid, where vibrational frequency does not depend on the direction.

10. Predict for Diamond along one direction (try 001)
Longitudinal Optical
Transversal Optical
degenerate
Longitudinal Acoustic
(0,0,0)
fcc lattice with basis
Transversal Acoustic
degenerate
P=2
2x3=6 branches expected
Which branch is which?
(acoustic, longitudinal, optical, transverse)

11. Can you guess what this is? What makes this look quite different?
How could we measure this?
All measurements before from neutrons.
One of these transverse modes is really different! The out of plane. Why might that happen?
At least something like this is also going to happen for other 2D materials, as well.

12. Diffraction Methods Diffraction Neutron Electron
X-ray
Neutron
Electron
The physical basis for the diffraction of electron and neutron beams is the same as that for the diffraction of X-rays, the only difference is the mechanism of scattering.

13. Advantages and disadvantages of X-rays
X-ray is the cheapest, the most convenient and widely used method.
X-rays are not absorbed very much by air, so the specimen need not be in an evacuated chamber (unlike electron diffraction).
Disadvantages:
They do not interact very strongly with lighter elements.
Not strongly sensitive to magnetism.

14. Neutron Diffraction for Phonons
Neutrons were discovered in 1932 and their wave properties was shown in 1936.
λ ~1A°
Energy E~0.08 eV. On the same order of magnitude as the thermal energy (kBT) at room temperature, eV, and for this reason they are good at probing phonons.
E = p2/2m p = h/λ
E=Energy λ=Wavelength
p=Momentum
mn=Mass of neutron = 1.67x10-27kg
H=6.6x10^-34

15. The energy of other methods
X-Ray
λ = 1A°
E ~ 104 eV
interact with electron
Penetrating
Neutron
E ~ 0.08 eV
interact with nuclei
Highly Penetrating
Electron
λ = 2A°
E ~ 150 eV
Less Penetrating
Note that penetration does not scale with energy. It depends more on the mechanism (and then the energy).
~ Bulk Measurement Bulk Measurement Surface Sensitive if sample is thin (<100s nm)

16. Neutron Diffraction
Unlike the x-ray, which is scattered entirely by electrons, neutrons are scattered by nuclei.
Neutrons are more useful than X-rays for determining the crystal structures of solids containing light (low atomic number) elements.
Although uncharged, neutron has an intrinsic magnetic moment, so it will interact strongly with atoms and ions in the crystal which also have magnetic moments.
Neutron sources in the world are limited so neutron diffraction is a very special tool.

17. Oak Ridge, Tennessee
Start with negatively charged hydrogen ions, produced by an ion source (1 proton and 2 electrons). Ions injected into a linear accelerator.
The ions pass through a foil, which strips off the electrons, converting it to a proton.
The high-energy protons strike a target of liquid mercury, kicking out neutrons (known as spallation).
The spalled neutrons are guided through beam lines.
This is not the only way to create a neutron source. Uranium is a common target, instead of mercury.
Spallation is a process in which fragments of material (spall) are ejected from a body due to impact or stress. In neutron scattering instruments, neutrons are generated by bombarding a target with a stream of atoms. The neutrons that are ejected from the target are known as spall.

18. Table top alternatives
Very expensive and involved experiments (construction of Oak Ridge neutron source was ~ $1 billion)
Table top alternatives ?
Yes, infrared absorption and
inelastic light scattering (Raman and Brillouin)
However only k~0 accessible (atomic scattering factors decrease as k2 for light)
Atomic scattering factors decrease as Q squared for light methods
Brillouin when photon absorbed is an acoustic phonon, Raman when its an optical phonon. Absorbed or emitted is anti-Stokes vs Stokes.
Lonely scientist in the reactor hall

19. Conditions for: elastic scattering
Phonon spectroscopy =
Conditions for: elastic scattering in
Constraints:
Conservation laws of
Momentum
Energy
Maybe draw Feynman diagram for phonon absorption (and maybe creation too)
In all interactions involving phonons, energy must be conserved and crystal momentum must be conserved to within a reciprocal lattice vector.

20. Magnetic Applications of Neutron Diffraction
Establishing magnetic structure of materials
Neutrons carry magnetic moments and their scattering is affected by magnetic moments of atoms
Example:
Magnetism of MnO
Mn2+ ions
face-centered cube
O2- ions
in the centers of all edges
and in the body center
What crystal structure? Mn O
Fcc with how many atoms in the basis? 2
This particular basis gives the rocksalt structure (same for NaCl)
What happens to the unit cell when you add the magnetic structure?

21. Simple Cubic Magnetic Super-Structure showing only magnetic elements
Ignoring this effect what peaks do we expect for MnO in diffraction? c c a a
Anti-Ferromagnetic=FM in one plane but antiparallel in next plane, M=0
Magnetic and charge have the different unit cells. Magnetic unit cell doubles nuclear unit cell.
Ferromagnetic (FM)=has a spontaneous magnetization whose direction can be changed with a magnetic field
Magnetic and charge have the same unit cell

22. Chemical unit cell (FCC)
Magnetism of MnO
(from Kittel)
Chemical unit cell (FCC)
(all odd or
even h,k,l)
Magnetic unit cell
Next two small peaks are 331 and 511, a=8.85 Angstroms as opposed to 4.43 in other example
Could also discuss magnetic reflectometry
How do we expect this to change below the magnetic ordering temperature?

23. Group: Consider Neutron Diffraction
Qualitatively discuss the atomic scattering factor (e.g., as a function of scattering angle) for neutron diffraction (compared to x-ray) by a crystalline solid.
For x-rays, we saw that f is related to Z and has a strong angular component. For neutrons?
The same concept applies, but since the neutron scatters off a tiny nucleus, scattering is more point-like, and f is independent of .
3.9
Only at 2=0 does f=Z

24. Neutrons are sensitive to both the nuclear and magnetic structure.
But not constant for neutrons studying magnetism. Why?
As the magnetism arises from unpaired electrons in outer shells and not the nucleus there is a dependence on intensity, similar to the sin(q) / l used for x-rays
Got here (slide 17) in 75 minute class
Neutrons are sensitive to both the nuclear and magnetic structure.

25. Like Telescope Time, Beamtime is Competitive
Ranking scheme:
Must do
High Priority
Medium Priority
Low priority
Don’t do
3 pages to describe why your work is important, what you expect to learn, how you will do it and why you can’t do it anywhere else!
Write 3 pages on why your work is important
Due ~ 1 year before the work is done!
Cutoff

26. You might get three days to do all of your experiments for the year!
If You Do Get Time
Magnetism experts
XAS expert
I = Installation/Maintanence me
You might get three days to do all of your experiments for the year!

27. SUMMARY: NEUTRON DIFFRACTION
Major differences as compared to X-rays
Neutrons are scattered by atomic nuclei
The scattering intensity does not depend on sin, don’t interact with electrons
The energy is ideal for probing phonon dispersion
Scattering by light atoms might be as intense as scattering by heavy atoms
Isotopes of the same element might have different scattering power
Scattering depends on magnetic moments of atoms
Much larger beam and sample are typically required (due to lower beam intensity). Neutron sources (like high brightness synchrotrons) in the world are limited so neutron diffraction is a very special tool.

28. Applications of Neutron Diffraction Scattering length (10-12 cm)
Distinguishing between neighboring elements
Neighboring elements have similar number of electrons and are difficult to distinguish using X-ray methods
Some of the neighboring element pairs exhibit substantially different neutron scattering
Element
Atomic #
Scattering length (10-12 cm)
Titanium (Ti) 22
–0.38
Vanadium (V) 23
–0.05
Chromium (Cr) 24
0.35
Manganese (Mn) 25
–0.36
Iron (Fe) 26
0.96
Cobalt (Co) 27
0.25
Nickel (Ni) 28
1.03
Copper (Cu) 29
0.79
Zinc (Zn) 30
0.59
Negative scatter (example: hydrogen) means that neutrons deflected from hydrogen are 180° out of phase relative to those deflected by the other elements.
The technique of contrast variation (or contrast matching) relies on the differential scatter of elements or isotopes, such as hydrogen (H) vs. deuterium (D). H D
0.6671

29. Examples where phonons are important
Energy loss is problem for solar cells
I moved this because we haven’t talked about band diagrams yet.
One way to handle this is with multilayers, but this increases the production cost

30. Indirect semiconductors
What temperature do you want to do these measurements at? What would happen if you measured at low temperature?
You also see that the band gap changes a little with temperature
Transition can only be excited if there are phonons present