Long-range harmonic-chain dispersion (${\omega }^2(k)=4M^{-1}{\sum }_{n>0}{C}_n{\sin }^2(nka/2)$)

For a translation-invariant monatomic chain with symmetric pair force constants, $M{\ddot{u}}_p={\sum }_{n>0}{C}_n(u_{p+n}+u_{p-n}-2u_p)$. Substituting a traveling normal mode gives the displayed even, reciprocal-periodic phonon dispersion relation. If $\sum n^2{C}_n$ converges and the chain is stable, its long-wavelength speed is $a\sqrt{\sum n^2{C}_n/M}$. A truly long-range interaction need not have a finite second moment or a finite linear acoustic slope.

Let $\alpha =C_2/C_1>0$ and $x={\sin }^2(ka/2)$. The squared frequency in units $4C_1/M$ is $(1+4\alpha )x-4\alpha x^2$. For $\alpha \le 1/4$ its maximum on $0\le x\le 1$ is at the zone edge and equals one. For $\alpha >1/4$ it moves inside the zone to $x=(1+4\alpha )/(8\alpha )$, with the displayed frequency ratio. The small-wavevector speed instead increases by $\sqrt{1+4\alpha }$; maximizing the frequency and finding the sound speed are different operations.