Monatomic-chain dispersion relation

For equal masses $m$ at spacing $d$ joined by nearest-neighbour springs of spring constant $K$, Newton's second law gives $m{\ddot{u}}_j=K(u_{j+1}+u_{j-1}-2u_j)$. A normal mode $u_j=u_0{e}^{i(kjd-\omega t)}$ givesThis single acoustic phonon branch is linear at small wavevector and has zero group velocity at the first Brillouin zone edge.

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.

With real wave number and time convention $e^{-i\omega t}$, an on-site friction force $-\Gamma {\dot{u}}_j$ shifts the frequencies as displayed. Underdamped oscillations have amplitude envelope $e^{-\Gamma t/(2m)}$, while overdamped modes have two nonoscillatory rates. At small wave number a stable acoustic chain has slow decay $\omega \simeq -im{c}_s^2{k}^2/\Gamma$. Exactly uniform displacement remains a zero-restoring-force mode, so a constant displacement is not forced to vanish.

A periodic chain of $N$ sites and length $Na$ has allowed signed wave numbers separated by $2\pi /(Na)$. The continuum state density on the signed axis is $Na/(2\pi )$. Combining opposite propagation directions onto $0<|q|<\pi /a$ doubles it to $Na/\pi$. This counts the same $N$ modes, not two independent acoustic phonon branches.