Alternating-spring chain dispersion relation

For equal masses $m$ separated by $a$ and alternating spring constants $\lambda$ and $\alpha \lambda$, the primitive-cell length is $2a$ and the two branches areThe minus and plus signs give the acoustic phonon branch and optical phonon branch, respectively.

Take equal masses $m$, atomic spacing $a$, and alternating springs $K\gg \kappa >0$. The primitive unit cell contains a strong-bonded pair, so its length is $2a$ and the Brillouin zone ends at $\pm \pi /(2a)$. Expanding the exact alternating-spring chain dispersion relation givesThe low-frequency acoustic phonon branch moves almost rigid dimers of mass $2m$ coupled by weak springs; its speed of sound tends to $a\sqrt{2\kappa /m}$. The high-frequency optical phonon branch describes internal dimer stretch. At the zone edge the exact frequencies are $\sqrt{2\kappa /m}$ and $\sqrt{2K/m}$.