Coupled Oscillators and the Monoatomic Chain (P.S. Lab 1)
Highlights of the activity
- Students are placed into small groups and will use the “One dimensional Oscillator Chain” program to simulate various monatomic chains.
- Each group will experimentally find the frequency of oscillation for several coupled oscillators and compare it to the theoretical period of oscillation.
- For a five coupled oscillator system, students investigate the behavior of different normal modes by recording the initial positions of each atom, the frequency of oscillation, and the wave number at each mode.
- Students will plot the dispersion relation for the five coupled oscillator system and compare the dispersion relation for when $k > \frac{\pi}{a}\,m^{-1}$, where $a$ is the repeat distance between atoms.
Reasons to spend class time on the activity
This lab is an introduction to using the “Waves and Optics Simulation” for the emulation of one-dimensional oscillators. Students will work in small groups to first analyze a single oscillator and a two coupled oscillator to determine that the wave simulator's experimental data matches with the values theoretically predicted.
Then, each group will move to a five coupled oscillator. Students will visually see that at the normal modes for a coupled oscillator, all particles in the system oscillate with the same frequency. After taking data for the wave vector and frequency of each normal mode, the dispersion relation for the given five coupled oscillator can be plotted. From these plots, the class will find that the dispersion relation is non-linear and is symmetric about the wave number $k = \frac{\pi}{a}\, m^{-1}$.
At the end of the experiment, students will also explore a system with many coupled oscillators. The groups are asked to predict the behavior of several normal modes and experimentally check their predictions. This activity will also transition well into discussing the analysis of an infinite chain of 1-D atoms through exploration of the periodic boundary condition.