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PHY.K02UF Molecular and Solid State Physics

## Student projects

• Over the years there have been many contributions to the course from students and now we need to correct errors and remove unclear contributions before too much more is added. If you find something that is wrong or unclear and can improve it, that would be a suitable project.
• Many students have provided projects in pdf format but this is an awkward format to use. Most of these projects contain small errors or omissions and it is usually not possible to edit them. It is much easier to use material that is in html like the web page. The equations are rendered by mathjax which uses the same syntax as latex math mode. You can embed links, images, javascript simulations and videos into html. It is a much better format than pdf.
• Make a 2 page summary of one of the sections like this one for electron bands. The format should be html. Some material can probably be borrowed from the exercise questions. The summary should include the topics given in the "For the exam" sections.
• There are some examples discussions of the atomic orbitals and how to calculate matrix elements with them: 1s orbital, 2p orbital. This discussion could be expanded to more atomic orbitals. Code in various languages such as Fortran, c, Java, JavaScript, Matlab, and Mathematica would be useful to have.
• Four important quantum mechanical problems that can be solved analytically are the infinite quare well, the finite square well, the harmonic oscillator, and the hydrogen atom. For this course it is assumed that you have seen each these problems before. Nevertheless, it would be useful to have pages summarizing the quantum descriptions of each of these systems. These pages should include dynamic plots and the code needed to address these problems similar to the page for the atomic orbitals. There are already some pages that could incorporated into this such as the page on Hermite polynomials or the page on the finite potential well.
• Calculate the molecular orbitals of ethylene, and butadiene similar to the calculation for benzene. Since these molecules are chains of carbon atoms, refer to the discussion of molecular chains. Separate projects would be the numerical calculation of $H_{11}$, $H_{12}$, and $S_{12}$ for ethylene, butadiene, or benzene. The matrix elements can be calculated with the code on the 2pz orbital page.
• Some students have calculated the molecular orbitals of CO. It would be useful to plot the molecular orbitals.
• Make similar calculations to the molecular orbitals of CO for other simple molecules.
• Two students calculated the molecular orbitals of H2O but they got different results for the Hamiltonian matrix. The calculation is described here. Because of the shape of the molecule, the calculation would be improved by adding 3d orbitals.
• In the course outline there is a long list of chemical bonds. Make a brief description of those bonds.
• Upload a video to YouTube (<10 minutes) that explains some topic in the course outline.
• Upload a video to YouTube that explains how to use a piece of laboratory equipment in the physics building related to this course.
• x-ray diffractometer
• atomic force microscope
• Low energy electron diffraction
• etc.
• The page that draws the Wigner-Seitz cell often has numerical errors. The routine that calculates where the courners of the WS cell are could be replaced by the routine used in the Brillouin zone calculation.
• Make a page like the ones for diamond, hcp, or NaCl for one of the nine crystal structures listed here. The JSmol part that rotates the images is already written (see: 'Examples of crystal structures' in the outline).
• Make a web page that specifies the symmetry points and lines of a Brillouin zone like the one for fcc. The shapes of some of the Brillouin zones depends on the lattice paramters. It would be good to add more plots for different lattice parameters.
• Triclinic
• Simple Monoclinic
• Base centered Monoclinic
• Base centered Orthorhombic
• Face centered Orthorhombic
• Trigonal
• Construct the empty lattice approximation for photons for one of the Bravais lattices not already calculated. See: Empty lattice approximation for photons.
• Write a program that will plot the phonon dispersion curves in any direction in k-space. This can be done quickly for crystals with one unit atom per unit cell because the matrix that has to be diagonalized is 3×3 and the roots of the determinant can be found with Cardano's formula. This program for bcc phonons has implemented Cardano's formula but only allows you to plot along a high symmetry axis.
• Add a column to the table of phonon properties. One case that would be interesting is the 2D square lattice. There already exists a discussion of this case here.
• Digitize the phonon density of states for some material (make a table of data from a plot) so we can make plots of the phonon density of states like we have for the electron density of states. This will allow us to make buttons to calculate the phonon properties of some materials like the buttons at the bottom of this page.
• Add a column to the table of band structures calculated by tight binding http://lampx.tugraz.at/~hadley/ss1/bands/tbtable/tbtable.html.
• Compare the electron dispersion relation for a one-dimensional chain of finite potential wells in the tight binding model with the exact results from the Kronig-Penney model.
• Calculate the dispersion relation and density of states for a one dimensional chain of atoms. To do this take the limit $N\rightarrow\infty$ in the formula for the energy levels of a ring of atoms from the molecules section of the course.
• Calculate the electron density of states in the tight binding model for one of the cases where the dispersion relation has been calculated. See: http://lampx.tugraz.at/~hadley/ss1/bands/tbtable/tbtable.html. You should provide a page that can be used to plot and tabulate the density of states like this one: http://lampx.tugraz.at/~hadley/ss1/bands/tbtable/simple_cubic_dos.html.
• Calculate the effective density of states in the conduction band $N_c$ of silicon from the dispersion relation near the band edges. A similar discussion for $N_v$ can be found here.