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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 35A / b / ii / Solution

The Debye model fixes its cutoff by requiring the continuum to contain the

2 n

vibrational degrees of freedom:

2 n = \int_{0}^{ω_{D}} g (ω) d ω = \frac{A}{2 π} (\frac{ω _{D}}{C})^{2/ α} .

Therefore the Debye frequency and Debye temperature are

ω_{D} = C (\frac{4 πn}{A})^{α /2}, T_{D} = \frac{ℏ ω _{D}}{k _{B}} .

Solved by gpt-5.6-sol high.

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