A table summarizing the results for phonons is given below. Reading Kittel chapter 4: Crystal vibrations or R. Gross und A. Marx: Thermische Eigenschaften des Kristallgitters. Resources Table summarizing the properties of phonons Phonon software Using complex numbers to describe oscillations International Tables for Crystallography: Phonons Phonon dispersion Java Phonon Applet Program to calculate phonon dispersion of fcc crystals The solid state simulation project Ashcroft and Mermin: Chapters A recent article on phonons: Phonon spectrum and specific heat of silicon nanowires , Y.
Zhang, J. Cao, Y. Xiao, and X. Yan, Journal of Applied Physics , Marx: Thermische Eigenschaften des Kristallgitters For the exam Be able to write down Newton's law for a periodic arrangement of atoms connected by linear springs. Because of the translational symmetry we can show that the normal mode of a crystal with one atom per Bravais lattice point has the form,. What are these eigen functions for a hexagonal close packed lattice? Show that by substituting the general form for the normal mode solutions into this equation, the dispersion relation for a linear chain can be determined.
When there are more atoms in the basis, the phonon normal modes are described by a displacement vector for every atom in the basis. For two atoms in the basis, the displacement vectors are,. What are the equations of motion that describe this system? Show that by substituting the general form for the normal mode solutions into this equation, the dispersion relation can be determined. The dispersion relation in this case is a set of 6 equations with 6 unknowns.
To sketch approximately the phonon dispersion for a crystal, first consider the reciprocal lattice of the Bravais lattice of that crystal. Label horizontal axis with symmetry points of the reciprocal lattice.
Take care that the spacing between those symmetry points equals the distance between the symmetry points in reciprocal space. There is a longitudinal acoustic branch and two transverse acoustic branches. The transverse modes will have the same frequency if the crystal has a symmetry that maps one transverse mode onto the other. There are two atoms in the basis.
The positions of the two carbon atoms given in terms of the fractional coordinates of the conventional unit are,. Sketch the phonon dispersion relation frequency vs. Which of the figures below is the phonon dispersion curve for CsI?
Explain your reasoning. For example GaN. What is the ratio of the integral of the density of states over all acoustic states divided by the integral of the density of states over all of the optical states? Many thermodynamic properties can be calculated from the density of states. Since phonons are bosons, the formula's are:.
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