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Brillouin light scattering spectroscopy was used to probe theroom-temperature elasticity of crystalline high-$T_c$ superconductorBi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$. A complete set of best-estimate elasticconstants was obtained using established relationships between acoustic phononvelocities and elastic constants along with a simple expression relatingcrystal elastic constant $C_{22}$ to the corresponding constants of theconstituent incommensurate sublattices. This latter relationship, which wasderived and validated in the present work, has important implications for thosestudying incommensurate systems as it appears that it may be applied in itsgeneral form to any composite incommensurate crystal. The results obtained inthis work are also consistent with sublattice assignments of Bi$_2$Sr$_2$O$_4$and CaCu$_2$O$_4$ reported in a previous Brillouin scattering study.

Influence of incommensurate structure on the elastic constants of  crystalline Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ by Brillouin light scattering  spectroscopy