Frequency-Dependent Relaxation Time of Electron-Phonon Systems. II. Effects of Finite Lifetime of the Phonons

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Abstract
In a previous paper a theory of the frequency-dependent relaxation time for a simple model of metal was developed, assuming the phonons to have infinite lifetime. In the present paper the problem is investigated from a more sophisticated point of view. The effects of low temperature are reconsidered and the effect of the finite lifetime of the phonons is included. The Green's function technique is used to obtain the response of the electron-phonon system to an oscillating electric field. Our treatment, which employs the time-dependent currents correlation function, does not have the time scale restriction of the usual transition probability approach, and does give a proper description of the time-dependent collective effects.