Quantization of Energy in Mechanical Oscillators (15 minutes)

  • Before beginning this lecture, ask the students to Draw Some Bound States.
  • The most common bound state is the single mass on a spring, as seen below.

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  • We can also extend this bound state premise to a system with multiple masses and springs as well. Looking at the first normal mode for a five-mass system (if students have already performed the Monatomic Chain Lab, refer to this), if we keep the envelope function the same shape but increase the amplitude, the potential energy for the system changes with very similar behavior to the single mass system.

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In essence, each normal mode for a multi-particle oscillator can be thought of as a collective mass bound by an effective potential.

  • How do we determine the energy for the multi-particle oscillator? Since only certain normal modes are allowed due to the boundary conditions of the system, we can expect that the energies will be quantized as well. The total energy stored in a normal mode is

$$U_{mode} \, = \, \left(n_{phonon}\, + \, \frac{1}{2}\right)\hbar \omega_{mode} \, \, , $$

where $\hbar \omega_{mode}$ is the phonon energy and $n_{phonon}$ is called the phonon number. $n_{phonon}$ increases with temperature and in whole steps (i.e. $n_{phonon}=0,1,2…$).

  • Unfortunately, use of the equipartition theorem for $U_{mode}$ breaks down if $n_{phonon} \sim 1$. It turns out that we do not observe “fractional phonon numbers”; we find that a normal mode “freezes out” when $k_{B}T=\hbar \omega_{mode}$ for a system. This behavior is seen in the graph below.

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Notice the abrupt drop off at the freeze out point. It was later found that this “freeze out” range of temperatures can actually be modeled using the Real Bose-Einstein function

$$f(\omega)=\frac{1}{e^{\frac{\hbar \omega}{kT}}-1} \; \; . $$

  • So, the big question now is this: how do we determine the energy stored in a lattice once the temperature is in this “freeze out” zone?

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