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 two-atom spring figure on the Scattering from lattice vibrations page, and the basic definition are given in this equation 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 plane wave form type 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 this equation from the Scattering from lattice vibrations page.