We have developed a laboratory exercise using an ultrasonic test system to measure elastic constants in copper single crystals, as well as moduli in polycrystalline samples. This laboratory is coordinated with the Introduction to Physics of Materials class taught to junior-level Materials Science and Engineering majors. Students in the class have usually taken a course in Mechanical Behavior of Solids, and are familiar with macroscopic (and destructive) testing of elastic moduli. This exercise provides an excellent basis for a discussion of the propagation of phonons in solids, the relationship between microscopic and macroscopic properties, destructive vs. non-destructive test methods, and an application of matrix algebra. We have recently added a computational component to the lab: use of density functional theory (using the program Plane Wave Self-Consistent Field, PWSCF) to calculate the elastic constants and compare to the measured values. The free software is part of the Quantum-ESPRESSO package. The use of DFT also allows the demonstration of the nature of the approximations that must be made to solve the many-body Schrodinger equation for electrons in a condensed phase. This follows discussions in the first half of the course about the hydrogen atom and how extending such calculations require approximations due to the many-body problem.
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