Monatomic-chain dispersion relation
= 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> $u_j=u_0e^{i(kjd-\omega t)}$ gives
$$
\omega(k)=2\sqrt{K/m}\,|\sin(kd/2)|,\qquad |k|\leq\pi/d.
$$
This single <acoustic phonon branch> is linear at small <wavevector> and has zero <group velocity> at the first <Brillouin zone> edge.