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Before getting deeper into the phonons. One of the conditions is that because of their quantization comes from the condition that system is invariant to translation.

And length of the systems I can calculate by: $$ \text { L = Na } $$

Born - Van Karman condition is: $u_{n+N}=u_{n}$, which looks like I put my chain in a ring, circular shape. As I got it it is a mathematical trick to get around with boundaries of a system?

But what bothers me the most is, in most of the books in literature they just interpret and conclude:

$$e^{i k N a}=1$$ that leads to: $$ k=\frac{2 \pi}{N a} \cdot m, m \in \mathbb{Z} $$

where m is number of possible states:

So, last two equation I can not interpret, why this $e^{i k N a}$ equals one, because it is periodic or what? And then this extraction of k that equals: $ k=\frac{2 \pi}{N a} \cdot m$. Am I missing some basic math here or whtat?

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\begin{align} \Psi(x+L)&=\Psi(x) \\ e^{ik(x+L)} &= e^{ikx} \\ e^{ikNa} &= 1 \end{align} Use Euler's identity $ e^{i\pi}=-1$ so $e^{i 2\pi} =1$. Hence, the above expression gives the condition $k=\frac{2\pi}{Na}m$ with $m\in \mathbb{Z}$.

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