The Multiple-Oscillator Model

A simple application of the multiple-oscillator theory is to fit measured reflectance data for MgO in the Reststrahlen region. In Section 9.1 we considered the electronic excitations of MgO, whereas we now turn our attention to its lattice vibrations. A glance at the far-infrared reflectance spectrum of MgO in Fig. 9.7 shows that it does not completely exhibit one-oscillator behavior there is an additional shoulder on the high-frequency side of the main reflectance peak, which signals a weaker, but still appreciable, second oscillator. The solid curves in Fig. 9.7 show the results of a two-oscillator calculation using (9.25) the reflectance data were taken from Jasperse et al. (1966), who give the following parameters for MgO at 295°K  

Notice the second, weaker oscillator at 640 cm-1 from a quantum viewpoint, this oscillator is interpreted as the excitation of two phonons by a photon. In order for momentum to be conserved, two phonons with almost equal and opposite momenta must be created a photon has negligible momentum compared with phonon momenta. Such two-phonon transitions usually occur to a noticeable extent in ionic crystals, which necessitates a multiple-oscillator correction to the main one-oscillator Reststrahlen band. 

We note that in MgO there is a rather large difference between the 

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