Approximating the N-th Normal Mode Frequency for an N-chain Oscillator (10 minutes)

• If an N-chain oscillator is oscillating at the N-th normal mode in the first Burilloun zone (that is, when the wavelength of the envelope function is $k=\frac{\pi}{a}$), the frequency of oscillation can be reasonably approximated using the equation of motion for a particle in the system.

• The key approximation for this calculation is that the displacement of each molecule in the oscillator is equal in magnitude. Using this approximation, the equation of motion for a particular particle becomes

$$m\ddot{x}=-2\kappa x \; - \; 2\kappa x \; \; .$$

Assuming that the equation describing the particle's motion has the form

$$x(t)=Ae^{i \omega t} \; \; ,$$

this equation can be inserted into the equation of motion to find that

$$\omega=\sqrt{\frac{4\kappa}{m}} \; \; .$$

Have the students test this approximation using the “One Dimensional Oscillator Chain” program. This exercise works quite well as an extension in concluding the Coupled Oscillators and the Monatomic Chain Lab.

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