Blundell Thermal Physics · Problem 24.7

Phonons

Physics — Statistical Mechanics · phonon-dispersion-next-nearest-neighbour

Problem

The treatment of the monatomic linear chain in Section 24.3 included only nearest-neighbour interactions. Show that if a force constant

K_j

links an atom with one

j

atoms away, then the dispersion relation becomes

\omega^2 = \frac{4}{m}\sum_j K_j \sin^2\frac{jqa}{2}. \tag{24.38}

A measurement is made of

\omega(q)

. Show that the force constants can be obtained from

\omega(q)

using

K_j = -\frac{ma}{2\pi}\int_{-\pi/a}^{\pi/a} {\rm d}q\,\omega^2(q)\cos jqa. \tag{24.39}