# Vibrations and Harmonic Oscillators

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i) Show that the thermodynamic properties of an assembly of N three-dimensional harmonic oscillators are the same as those of an assembly of 3N of on-dimensional oscillators.

ii) In the Einstein model, each atom in a solid acts as a three-dimensional simple harmonic oscillator. An atom in a crystal of copper has a typical vibrational frequency of v = 5.2 x 10^12 Hz. What is its specific heat at room temperature expressed as a percentage of its maximum specific heat?

iii) The vibrational frequency of the carbon atoms in diamond is of order v = 3.12 x 10^13 Hz. What would the Einstein model predict for its heat capacity at room temperature?

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#### Solution Summary

This solution discusses thermodynamic properties and heat capacity of harmonic oscillators,

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