i was looking at my retake from 2024. there is a question about a monoatomic chain with a returnong forse with sprink constant kappa’. Doing the calculations you arrive at a dispertion relation of w(k) = sqrt(4kappasin^2(ka/2)+kappa’)/m^(-1/2). Now, the question is for what temperatures is this comparable to: einstein model, debeye model, law of dulong-petit? i really dont understand where the T dependency comes in or how i can look at it differently.
Let’s first identify the main features of each model:
- Einstein model: all modes that are thermally excited have a comparable energy.
- Debye model: all modes that are thermally excited have a roughly linear dispersion.
- The law of Dulong-Petit: all modes are excited.
To proceed, we then need to identify what are the modes that contribute significantly to heat capacitance at different temperatures.
Does that help?
I am still confused. I see that when i take kappa’ = 0 i get debye, when i take kappa = 0 i get einstein. but i still dont see how this relates to temperature?
You have ignored the key question: which phonon modes contribute significantly to heat capacitance. The contribution of modes with \hbar\omega \ll k_bT is exponentially small. The contribution of modes with \hbar\omega \gtrsim k_bT is C_V\approx k_b. Does that help to make progress?
Its clear that high temp limit is dulong petit, the confusion lies in the distiction between debye and einstein. and i still dont understand