Molecular Vibrations

• Molecules vibrate due to the stretching, bending, and twisting of bonds. Typical vibrational frequencies range from 10¹² to 10¹³ Hz (in the infrared region of the electromagnetic spectrum).

Lattice Vibrations in Solids (Phonons)

• In crystalline materials, atoms vibrate in periodic patterns called phonons. Frequencies range from 10⁹ to 10¹³ Hz, depending on the material and type of vibration (acoustic vs. optical phonons).
• Room temperature vibrations typically involve thermal phonons with energies comparable to kBTk_BTkB​T (kBk_BkB​ is Boltzmann constant, TTT is temperature).

3. Quantum Considerations

• Vibrational modes are quantized, with energy levels given by: Ev=(v+12)hνE_v = \left(v + \frac{1}{2}\right)h\nuEv​=(v+21​)hνwhere vvv is the vibrational quantum number, hhh is Planck’s constant, and ν\nuν is the vibrational frequency.
• At room temperature, transitions to higher energy states are minimal unless the vibrational frequency is low.

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