We are interested in the phonon response in the frustrated magnets SrCr$_{9x}$Ga$_{12-9x}$O$_{19}$ (SCGO) and Ba$_{2}$Sn$_{2}$ZnCr$_{7x}$Ga$_{10-7x}$O$_{22}$ (BSZCGO). The motivation of the study is the recently discovered, phonon-driven, magnetic relaxation in the SCGO compound [Mutka et al. PRL {bf 97} 047203 (2006)] pointing out the importance of a low-energy ($hbaromegasim$7 meV) phonon mode. In neutron scattering experiments on these compounds the phonon signal is partly masked by the magnetic signal from the Cr moments and we have therefore examined in detail the non-magnetic isostructural counterparts SrGa$_{12}$O$_{19}$ (SGO) and Ba$_{2}$Sn$_{2}$ZnGa$_{10}$O$_{22}$ (BSZGO). Our {it ab-initio} lattice dynamics calculations on SGO reveal a peak in the vibrational density of states matching with the neutron observations on SGO and SCGO. A strong contribution in the vibrational density of states comes from the partial contribution of the Ga atoms on the 2b and 12k sites, involving modes at the M--point of the hexagonal system. These modes comprise dynamics of the kagome planes of the pyrochlore slab magnetic sub-lattice, 12k sites, and therefore can drive magnetic relaxation via spin-phonon coupling. Both BSZCGO and BSZGO show a similar low-energy Raman peak but no corresponding peak in the neutron determined density of states of BSZGO is seen. However, a strong non-Debye enhancement of low-energy phonon response is observed. We attribute this particular feature to the Zn/Ga disorder on the 2$d$ -site, already evoked earlier to affect the magnetic properties of BSZCGO. We propose that this disorder-induced phonon response explains the absence of a characteristic energy scale and the much faster magnetic relaxation observed in BSZCGO.
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