Temperature limits on Debye's Calculationp

[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?

 

[Vagn]

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.

 

[Lord Jestocost]

The heat capacity of a single harmonic oscillator is always k_B in the high temperature limit. In case you have a solid composed of N atoms, you will thus always get 3Nk_B 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