View source for Problem:Classical vibrations with a two-atom unit cell

We here consider the one-dimensional chain with two atoms per unit cell, of masses \(m\) and \(M\), with one spring constant, \(K\), and with distances between atoms of \(d/2\). This system is illustrated in the [[Scattering from lattice vibrations#label-fig:springs2|two-atom spring figure]]<!--Figure~\ref{fig:springs2}--> on the [[Scattering from lattice vibrations]] page, and the basic definition are given in [[Scattering from lattice vibrations#label-eq:twoatoms|this equation]]<!--(\ref{eq:twoatoms})--> on the [[Scattering from lattice vibrations]] page.

=====Question 1=====
Write down the equations of motion for each of the two atoms.

=====Question 2=====
Write down the trial solutions for the two atoms of the [[Scattering from lattice vibrations#label-eq:fourier|plane wave form]] type <!--(\ref{eq:fourier})--> introduced on the [[Scattering from lattice vibrations]] page. Remember that there may be a phase shift, \(\phi(q)\) between the two types of atoms.

=====Question 3=====
Show that there are only two allowed values of \(\phi(q)\), which differ by \(\pi\).

=====Question 4=====
Show that the vibration frequencies can be written by [[Scattering from lattice vibrations#label-eq:spring2d|this equation]]<!--(\ref{eq:spring2d})--> from the [[Scattering from lattice vibrations]] page.