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If we start with the general derivation of a dispersion relation for a 1D system, with atoms coupled by springs, one gets the following relation

$$\omega = 2\sqrt{\frac{k}{m}} |\sin \big( \frac{ka}{2} \big)|$$

whereby it is possible to show that the function is periodic with ##\omega (k + 2\pi/a) = \omega (k)##.

However if we consider Debye's model, which comes as a result of taking the long wavelength (and small k) limit, we get

$$\omega = v_s k$$

where ##v_s## is the speed of sound.

##\textbf{Question}##: Does the concept of a periodic ##k## still apply here? Also, is the number of allowed modes for a chain of N masses still equal to N?

For example if I were to calculate energy contributions by individual frequencies ##\omega (k)## using the following equation

$$\langle E \rangle _k= \hbar \omega(k) \big( n_b (\beta \hbar \omega (k)) + 1/2 \big)$$

clearly energy for the ##k=0## mode differs from that of the ##k = 2\pi /a## mode. Should I pick ##k = 0## for my calculations or ##k = 2 \pi /a##?