Exam 2019 question 3d

Physics Q&A
1

The answer to question 3d in the exam of 2019 is the following :

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It states that the temperature dependence of phonons at low temperatures is quadratic. In the lecture on Debye model I remember the following :

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Is the answer model wrong or am I missing something?
Thanks in advance!

2

Take into account that in that exercise you are dealing with a 2-dimensional system and things might (will) change!

3

In a 3D system we get T^3, in a 2D system we apparently get T^2.
Am I correct in assuming that a phonon system scales with heat in terms of T^D(dimension).

I just found this link, 1D scales linearly, 2D scales with T^2, and 3D scales with T^3 (near T=0)

4

Indeed, though the point is not so much that you memorize that, but that you should be able to compute this in the low temperature limit for any dimensionality.