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A4 HANDWRITTEN THEN BOX THE FINAL ANSWERS (DO NOT USE AI) IV. PHONON HEAT CAPACITY [20 pts] Consider a three-dimensional monatomic crystal at thermal equilibrium.

A4 HANDWRITTEN THEN BOX THE FINAL ANSWERS (DO NOT USE AI)

IV. PHONON HEAT CAPACITY [20 pts] Consider a three-dimensional monatomic crystal at thermal equilibrium. Use the Debye model to show that the phonon contribution to the low temperature heat ca- pacity of the crystal behaves as T3

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