# planewave Ansatz for modelling phonon dispersion in crystals

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?

• There are two questions here. Please post one question at a time. You will get much better answers if you do this. Dec 27 '14 at 13:54
• The title should be definitely improved, a suggestion: "planewave Ansatz for modelling phonon dispersion in crystals", something along those lines. Dec 27 '14 at 15:07

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.