One of the goals of this class is to help students understand the material by including demonstrations and activities within the course. This page will be constantly updated and added to as the demonstrations are fleshed out and more demonstrations are added to the curriculum.

Bonding and Structures (Topics 1 and 2)

 

Topic 1: Local coordination in ionic, covalent and hydrogen bonded structures

Required videos: Topic 1

Download and install Vesta. Vesta is a free crystal structure viewing software. Download these structure files: NaCl, diamond-C, graphite, heroin, sapphire, iron, GaAs

Load the structures into Vesta; consider the local coordination of each atom. Can you correlate the electronegativity contrast with the local coordination number? Why or why not?

 

Topic 2: Lattice and basis in common crystals; symmetry; slices

Associated videos: Topic 2.1, 2.2, 2.3

Structures: NaCl, GaAs, GaN, CsCl, BN: zip

2.A Basis: Load your structure into Vesta. Shrink the boundary to 0 - 0.99 in each direction. What is the Bravais lattice? How many atoms are in the basis? How many atoms are in the unit cell?

2.B Slices: Draw your structure in slices. After drawing slices for one unit cell, repeat the drawing with 2a and 2b widths (4 unit cells).

2.C Interpenetrating lattices: Open up the bonds menu, select the bond and delete it. Back to the main screen, on the upper left, select the "Objects" tab and turn off the cations. What is the anion sub-lattice structure? Repeat for the opposite atoms. (for BN, just consider this in-plane)

2.D Asymmetric basis: Open up the Edit Structure (ctrl-E) and click on the 'Structure parameters' tab. How many unique atoms are specified? Mark them on your slices drawing. If you have an answer that is different than you got for 2.A, what could explain this? Hint: Under the 'Unit cell' tab, the unit cell and space group are listed.

2.E: Symmetry removal: Go to 'Edit Structure-Unit cell" and set the crystal system to triclinic and the space group to P1. What, if anything, is different? What symmetry elements are now missing?

 

Topic 2: Miller indices

Associated videos: Topic 2.4

2.F Cubic system Draw (100), (111) and (102) planes for a cubic system. Check with your neighbors that you agree.

2G Hexagonal system: Draw (100), (111) and (102) planes for a hexagonal system. Check with your neighbors that you agree.

2H Epitaxy example: During my PhD, I grew some ZnMn2O4 single crystal films on MgAl2O4 single crystal substrates. With some work, I figured out the orientation between the films, as shown below. You can see the cations coordinated by 4 oxygen in a tetrahedra are shown in blue, octahedral sites are shown in orange. By inspection, you may be able to the blue tetrahedral sites of the ZnMn2O4 could replace the blue tetrahedral sites of the MgAl2O4. Per the drawing below, what two planes of the two structures are parallel?

The c axes are the same length and the base diagonal of the tetragonal cell is equal in length to the cubic cell.

 

Topic 2 solutions

 

Topic 3: Plane waves, Fourier series and transforms

Associated videos: Topic 3.0 File: 3.1.nb

3.A Plane wave propagation - Develop an undertanding of how plane waves propagate as a function of k direction and magnitude.

3.B - Amplitude and intensity - How does the intensity vary across a plane wave? Does the intensity depend on the wt term in the exponential?

3.C - Fourier series - Develop an understanding of how Fourier series can be used to approximate other periodic functions. In particular, consider the role of the Fourier coefficient.

3.D - Fourier transforms -(i) What is the connection between a Fourier series and a Fourier transform? (ii) What accounts for the inverse relationship between the real space Gausian and its transform? (iii) Establish general trends for the finite array of Gaussians and the associated transform. See if you can articulate the source of these trends.

Topic 5: Phonons

Associated videos: 5.1, 5.2 Files: PhononDispersion.nb

5.A Einstein model - How would you mathematically describe a crystal where the atoms were confined in a potential but not interacting? This is called an Einstein solid and and is perhaps the simplest solid one could imagine.

5.B Harmonic lattice, single atom basis - Write out the forces present in a one dimensional ball and spring chain if we invoke only nearest neighbor coupling. Visualize these forces with the handy demo. Watch the time evolution as this potential energy is converted into kinetic energy.

5.C Assumptions in the model - Consider an ionic solid. What terms are missing from the expression in 5.A? How would you fix this?

5.D Equations of motion - We're going to invoke an ansatz that the vibrations in the crystal have a traveling wave form. Solve for the relationship between wavevector (q for phonons) and the frequency for a crystal that has periodic boundary conditions. We call this w(q) relationship the dispersion relation.

5.E Visualizing the dispersion relation - Plot the dispersion relation and consider the wavelengths of the associated waves.

5.F Limits to the phonon dispersion - Why does the frequency have a maximum upper value? What happens as we continue to put more thermal energy into a solid? What are the ranges of phonon wavelengths that have a physical significance?

5.G Two atom basis - What's different when the basis increases to two atoms? Write out the forces and equations of motion. What normal modes do you think populate this system? Ansatz time!

5.H Phonon dispersion - How would you solve for the phonon dispersion in this case?

5.I Dispersion's dependence on masses and spring constant - How does the dispersion periodicity compare to the one atom basis? Play with the Manipulate in Mathematica to understand how the dispersion depends on M1, M2 and C. Can you rationalize this behavior? What does the gap in frequency suggest? Recall the E=hbar omega. Physically, what do you expect the highest and lowest energy vibrations to correspond to?

5.J Group and phase velocity - We can treat the sum of two Cos waves as the product of two Cos waves using the derivation provided in the videos. Play with the time axis to see the velocity of the envelope function and the inner phase velocity. How does this vary as you change the dw/dk ratio? What happens when the dw/dk ratio is unity?

5.K Wave packets and group velocity- Here, we build a wave packet out of a bunch of Cos waves that have similar wave vectors. The concept is the same as 5.J, but now we can see the amplitude displacement is more localized.

 

Topic 6: Heat capacity

Associated videos: 5.3, 5.4, 6.1, 6.2 Files: Bose_Einstein_Heat_Capacity.nb

We'll walk through the Mathematica together. The goal is to understand how modes are populated with phonons and the ultimate impact on the system energy and heat capacity.