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The acoustic phonons of cyclotrimethylene trinitramine (RDX) have been studied using Brillouin scattering. The analysis of the acoustic-phonon velocities allowed determination of the complete stiffness tensor for this energetic material. The results are compared to other recent experimental and theoretical determinations of the RDX elastic constants, bulk moduli, and shear moduli. The observed ordering of elastic constants, C11>C22>C33, is qualitatively associated with a (001) cleavage plane and molecular packing. This interpretation is further corroborated by the linear compressibilities plotted in three crystallographic planes, and a comparison to recent theoretical and experimental hydrostatic compression studies on RDX. Finally, the elasticity of RDX is compared to a recently published report on the beta polymorph of cyclotetramethylene tetranitramine’s elasticity, and is related to several proposed mechanisms for detonation initiation.