# Temperature limits on Debye's Calculationp

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• cozycoz
A pedagogical note, which shows nicely how you recover the Dulong-Petit limit by summing over all possible modes.In summary, Debye assumed a sound wave dispersion relation for phonons (ω=vK) which corresponds to acoustic modes in the low frequency limit. This explains the low temperature heat capacity well. However, this same assumption cannot fully explain the high temperature limit (C=3k_B per atom). Debye did consider a cut-off frequency to address this, but the overall calculation is still based on the low temperature dispersion relation. This may not hold true at high temperatures (k_BT>>ħω). The cut-off frequency represents the maximum of the dispersion relation, corresponding to a wavelength equal to twicef

#### cozycoz

Debye assumed sound wave dispersion relation for phonons(##ω=vK##) and this corresponds to acoustic modes in low frequency limits. That's why it explains low temperature heat capacity fairly well.

But how could this also explain high temperature limit(##C=3k_B## per atom)? I know Debye considered cutoff frequency to make this result, but anyway the whole calculation rooted from low temperature dispersion relation, and generally the relation would be absolute value of sine function! I think it should fail at ##k_BT>>ħω##.

could you explain this to me?

and plus, is it okay to think that the cutoff frequency represents the maximum of the dispersion relation?

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Debye assumed sound wave dispersion relation for phonons(##ω=vK##) and this corresponds to acoustic modes in low frequency limits. That's why it explains low temperature heat capacity fairly well.

But how could this also explain high temperature limit(##C=3k_B## per atom)? I know Debye considered cutoff frequency to make this result, but anyway the whole calculation rooted from low temperature dispersion relation, and generally the relation would be absolute value of sine function! I think it should fail at ##k_BT>>ħω##.

could you explain this to me?

and plus, is it okay to think that the cutoff frequency represents the maximum of the dispersion relation?
The cut-off frequency corresponds to a wavelength equal to twice the atomic spacing, i.e. the minimum wavelength that the atomic chain can support.

The heat capacity of a single harmonic oscillator is always kB in the high temperature limit. In case you have a solid composed of N atoms, you will thus always get 3NkB in the high temperature limit, even if you assume a certain distribution of 3N normal modes of vibration where each has its own frequency. Have a look at [PDF]Einstein and Debye heat capacities of solids