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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?

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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