Single Phonon Approximation

The single phonon approximation is obtained by expanding the exponential expression for the influence functional F into series and keeping two terms only. The fist term describes the elastic scattering while the second one corresponds to single phonon scattering and can be splitted into two first parts, corresponding to the creation and annihilation of a phonon ... 

A very much simplified lattice-dynamical model is that of Debye. In the Debye approximation, discussed in the following section, a single phonon branch is assumed, with frequencies proportional to the magnitude of the wavevector q. 

Fig. 17. A comparision of the temperature dependence of the line-shape function (G) of the transition probability for the multimode case (solid line) as against a single mode approximation (dashed line). Here the phonon frequency spectrum (A) is assumed to be of Gaussian form, A a>) = 2 2) 1,2exp [—(to — cu0)2/2

Although in principle detailed knowledge of the actual phonon phase space is necessary in order to predict accurately the temperature dependence of the line width, a simple approximation can suffice. We simply assume a variation of the Einstein model for the specific heat dependence of solids, namely, that there exists a single phonon mode of energy It is then straightforward to show that the fwhm cr(r) of the band must vary with temperature as... 

For single phonon-vibrational interactions, the field density of states Pg (ro) can be approximated with the elassieal version of the Debye vibrational spectral density of a solid (Section LB) shown in the second row of Scheme 5 ... 

An accurate calculation of the heat conductivity requires solving a kinetic equation for the phonons coupled with the multilevel systems, which would account for thermal saturation effects and so on. We encountered one example of such saturation in the expression (21) for the scattering strength by a two-level system, where the factor of tanh((3co/2) reflected the difference between thermal populations of the two states. Neglecting these effects should lead to an error on the order of unity for the thermal frequencies. Within this single relaxation time approximation for each phonon frequency, the Fermi golden rule yields, for the scattering rate of a phonon with Ha kgT,... 

In Fig. 14 we have plotted also the expected ratio Tc/TF(BCS) for the critical temperature in the single o band of MgB2 calculated by the standard BCS approximations using the McMillan formula, the density of states and electron phonon coupling obtained by band structure theory that gives for MgB2 Tc(5CS)=20K [148],... 

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