Lattice dynamics and Raman spectra of isotopically mixed diamond

We present coherent-potential-approximation (CPA) calculations and first- and second-order Raman spectra for diamonds with varying concentrations of ¹²C and ¹³C. The calculations are based on the valence-force model of Tubino, Piseri, and Zerbi [J. Chem. Phys. 56, 1022 (1972)]. Contrary to previous claims, we find that this model does not give a sharp peak in the density of states (DOS) near the Raman mode. Alternative dispersion curves that do give such a peak are discussed. Raman results are reported for high-quality, single-crystal synthetic diamonds with isotopic compositions ranging from nearly pure ¹²C to nearly pure ¹³C. A measurable deviation in the Raman frequency away from a simple M^-1/2 (``virtual-crystal'') dependence and an observable broadening of the first- and second-order spectra are qualitatively consistent with CPA predictions for the effects of isotopic disorder. Quantitative agreement between theory and experiment is achieved only if the reference DOS contains the above-mentioned peak. This supports the interpretation of the controversial second-order Raman peak in naturally abundant diamond (1.1 at. % ¹³C) at 2667 cm^-1 as a DOS effect. At all compositions, the effects of isotopic disorder are relatively weak because of the small mass difference between ¹²C and ¹³C. For 1.1 at. % ¹³C, the maximum broadening predicted in the CPA is less than 1 cm^-1, nearly two orders of magnitude smaller than a previous estimate. For the lowest-frequency modes most relevant to the thermal conductivity, the CPA scattering rate reduces to the usual ω⁴ dependence first derived by Klemens for phonon-isotope scattering. Using Callaway's theory, we show that this term can easily account for the recently observed 50% enhancement in room-temperature thermal conductivity upon elimination of ¹³C impurities, provided that sufficient normal scattering also occurs.