Lennard-Jones chain sound speed ($v=\sqrt{72ϵ/m}$)

A nearest-neighbour chain with the Lennard-Jones potential $V(x)=4ϵ[(\sigma /x{)}^{12}-(\sigma /x{)}^6]$ has equilibrium spacing $a=2^{1/6}\sigma$ and spring constant $K=V^{\prime \prime }(a)=72ϵ/a^2$. The monatomic-chain dispersion relation gives speed of sound $v=a\sqrt{K/m}=\sqrt{72ϵ/m}$ and maximum angular frequency $2v/a$. The cancellation of $\sigma$ from the sound speed holds in this chain model; it does not remove the spacing from the cutoff frequency.