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From Ashcroft's "Solid State Physics", for one-dimensional monatomic Bravais lattice, the equations of motion of ions are:

\begin{equation} M\ddot u(na)=-K[2u(na)-u([n-1]a)-u([n+1]a)] \end{equation}

And we seek solutions of the form: \begin{equation} u(na,t)\propto e^{i(kna-\omega t)} \end{equation}

My question is:
Why do we know the solutions to above equations can be expressed in this form?

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    $\begingroup$ There are two questions here. Please post one question at a time. You will get much better answers if you do this. $\endgroup$
    – DanielSank
    Dec 27 '14 at 13:54
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    $\begingroup$ The title should be definitely improved, a suggestion: "planewave Ansatz for modelling phonon dispersion in crystals", something along those lines. $\endgroup$
    – Ellie
    Dec 27 '14 at 15:07
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This is the classical treatment to model vibrations in solids, using the analogy with vibrations of a one-or-two dimensional monatomic or diatomic chains. Which basically boils down to writing Newton's equation of motion to find out the force on each mass when the whole system constitutes of masses attached by Hookean springs, i.e. for our purposes the potential holding the atoms together is taken as quadratic.

The first differential equation you wrote corresponds to the monatomic 1D case, which in a more compact notation would be (only considering its 2 closest neighbours and $u_n$ being the relative displacement of atom $n$): $$M\ddot{u_n}=F_n=K(u_{\rm n+1}+u_{\rm n-1}-2u_n)$$

As an attempt to find a solution to Newton's equations here one usually makes use of the planewave Ansatz in order to describe the normal modes of vibration as waves: $$u=Ae^{ikna-i\omega t}$$

Where $A$ is just an amplitude of oscillation and $k$ and $\omega$ are the wavenumber and frequency of the proposed wave. Now bear in mind that all we're trying to reach at the end is the phonon dispersion relation $\omega(k)$ in a given crystal. The educated guess of planewaves here stems from the fact that we're dealing with perfectly periodic systems, in which the normal vibrations, or in other words phonons are modelled as infinitely extended plane waves. Of course for a real crystal, filled with imperfection, things will not be so simple anymore. This is why we need models like PCM: Phonon Confinement Model, where due to imperfections, the perfect periodicity is perturbed and the waves describing phonons become localized, i.e. not simple planewaves anymore. But that's another story.

For a more general understanding on planewave Ansatz, you will find more useful information on this post. At the end of the day, the planewave choice is nothing but an educated guess which works out perfectly for our purposes.

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