The angular frequency of vibrations in a one-dimensional monatomic crystal are given by where w(k) = 4K m sin k = ka 2 2π X is the wavenumber of the vibration pattern in terms of its wavelength >, the spring constant between neighbouring atoms, and m the mass of each atom. (i) Show that the frequency does not change if one shifts 2π k→ k +-. a (ii) Briefly explain the physical reason for this periodicity.

The angular frequency of vibrations in a one-dimensional monatomic crystal are given by where w(k) = 4K m sin k = ka 2 2π X is the wavenumber of the vibration pattern in terms of its wavelength >, the spring constant between neighbouring atoms, and m the mass of each atom. (i) Show that the frequency does not change if one shifts 2π k→ k +-. a (ii) Briefly explain the physical reason for this periodicity.