Alternating-spring chain dispersion relation
= 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 are
$$
\omega_\pm^2(k)=\frac{\lambda}{m}
\left[1+\alpha\pm\sqrt{(1-\alpha)^2+4\alpha\cos^2(ka)}\right].
$$
The minus and plus signs give the <acoustic phonon branch> and <optical phonon branch>, respectively.