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We are interested in the phonon response in the frustrated magnetsSrCr$_{9x}$Ga$_{12-9x}$O$_{19}$ (SCGO) andBa$_{2}$Sn$_{2}$ZnCr$_{7x}$Ga$_{10-7x}$O$_{22}$ (BSZCGO). The motivation of thestudy is the recently discovered, phonon-driven, magnetic relaxation in theSCGO compound [Mutka et al. PRL {\bf 97} 047203 (2006)] pointing out theimportance of a low-energy ($\hbar\omega\sim$7 meV) phonon mode. In neutronscattering experiments on these compounds the phonon signal is partly masked bythe magnetic signal from the Cr moments and we have therefore examined indetail the non-magnetic isostructural counterparts SrGa$_{12}$O$_{19}$ (SGO)and Ba$_{2}$Sn$_{2}$ZnGa$_{10}$O$_{22}$ (BSZGO). Our {\it ab-initio} latticedynamics calculations on SGO reveal a peak in the vibrational density of statesmatching with the neutron observations on SGO and SCGO. A strong contributionin the vibrational density of states comes from the partial contribution of theGa atoms on the 2b and 12k sites, involving modes at the M--point of thehexagonal system. These modes comprise dynamics of the kagom\'e planes of thepyrochlore slab magnetic sub-lattice, 12k sites, and therefore can drivemagnetic relaxation via spin-phonon coupling. Both BSZCGO and BSZGO show asimilar low-energy Raman peak but no corresponding peak in the neutrondetermined density of states of BSZGO is seen. However, a strong non-Debyeenhancement of low-energy phonon response is observed. We attribute thisparticular feature to the Zn/Ga disorder on the 2$d$ -site, already evokedearlier to affect the magnetic properties of BSZCGO. We propose that thisdisorder-induced phonon response explains the absence of a characteristicenergy scale and the much faster magnetic relaxation observed in BSZCGO.

Phonon Control of Magnetic Relaxation in the Pyrochlore Slab Compounds SCGO and BSZCGO