Past exam of the condensed matter physics course of the University of Oxford / 2013 / 3 / c / Solution

For a viscously damped monatomic-chain phonon, the extra force gives $m{\ddot{u}}_j+\Gamma {\dot{u}}_j$ on the left of the preceding equation. Keep a real wave number $k$ and permit complex frequency in the convention $u_j\propto e^{i(kja-\omega t)}$. The dispersion equation isWriting $\gamma =\Gamma /(2m)$, its two roots areThe negative imaginary part corresponds to decay with the chosen time convention. Here $\Gamma$ has units of mass per time, while $\gamma$ is a decay rate. A driven experiment at specified real frequency would instead solve for a generally complex wave number; it is a different way of specifying the response, not a contradictory sign choice.