Past exam of the general relativity and cosmology and condensed matter physics paper of Part B of the physics course of the University of Oxford / 2011 / Section VI / 6

Let $a$ be a stable equilibrium separation: $V^{\prime }(a)=0$, $K=V^{\prime \prime }(a)>0$. With bond extension $\delta =x-a$, the Taylor expansion isThe harmonic approximation applies when the typical thermal and zero-point bond fluctuations satisfy $|\delta |\ll a$ and the higher terms are negligible, for example $|V^{\prime \prime \prime }(a)\delta |\ll 3K$ and $|V^{\prime \prime \prime \prime }(a){\delta }^2|\ll 12K$. The relevant small quantity is the relative displacement of neighbouring atoms. Small bond excursions about a stable minimum give an amplitude-independent spring constant $K=V^{\prime \prime }(a)$. Large thermal excursions, bond breaking or appreciable anharmonicity invalidate the approximation.

With atomic displacement $u_j$, the harmonic potential energy is $\frac{1}{2}K{\sum }_j(u_{j+1}-u_j{)}^2$. Newton's second law givesFor a periodic chain of $N$ atoms, its normal modes are $u_j=U{e}^{i(kja-\omega t)}$, with $k=2\pi s/(Na)$ and one representative per Brillouin zone. Substitution yields the monatomic-chain dispersion relationA real general displacement is a superposition of these normal modes and their complex conjugates. The $k=0$ mode is a rigid translation. At small $|k|$, the acoustic phonon branch is linear, with speed of soundFor $0<k<\pi /a$, the group velocity is $a\sqrt{K/m}\cos (ka/2)$ and vanishes at the zone boundary. The dashed initial tangent in the sketch has slope $v$.

For the Lennard-Jones potential,Its minimum and harmonic spring constant areUsing the monatomic-chain dispersion relation,The maximum is at $|k|=\pi /a$. Taking the argon atomic mass as $m\simeq 39.95\mathrm{u}=6.63\times 10^{-26}\mathrm{kg}$ and $ϵ=1.602\times 10^{-21}\mathrm{J}$ givesThese are the stipulated one-dimensional nearest-neighbour model estimates; a three-dimensional argon crystal has additional neighbours and phonon polarizations.