Math 6410 • ODE

Consider a linear chain of \(2N\) atoms consisting of two different masses \(m\), \(M\)
\(M>m\), placed alternately. The atoms are equally spaced with lattice spacing \(a\) with
nearest neighbor interactions represented by Hookean springs with spring constant
\(\beta\). Label the light atoms by even integers \(2n\), \(n=0,\ldots,N-1\) and the heavy
atoms by odd integers \(2n-1\), \(n=1,\ldots,N\). Denoting their displacements from
equilibrium by the variables \(U_{2n}\) and \(V_{2n-1}\) respectively, Newton's law of
motion gives
\[
\begin{aligned}
  m\ddot{U}_{2n} &= \beta[V_{2n-1}+V_{2n+1}-2U_{2n}],\\
  M\ddot{V}_{2n-1} &= \beta[U_{2n}+U_{2n-2}-2V_{2n-1}]
\end{aligned}
\]
Assume periodic boundary conditions \(U_0=U_{2N}\) and \(V_1=V_{2N+1}\).
\begin{enumerate}
\item[(a)] Sketch the configuration of atoms and briefly explain how the dynamical
equations arise from Newton's law of motion.
\item[(b)] Assuming a solution of the form
\[
  U_{2n}=\Phi e^{2inka}e^{-iwt},\qquad
  V_{2n+1}=\Psi e^{i(2n+1)ka}e^{-iwt},
\]
derive an eigenvalue equation for the amplitudes \((\Phi,\Psi)\) and determine the
eigenvalues.
\item[(c)] Using part (b), show that there are two branches of solution and determine
the speed \(w/k\) on the two branches for small \(k\).
\end{enumerate}