Using broadband ferromagnetic resonance, we measure the damping parameter of [Co(5 r{A})/Pt(3 r{A})]${times 6}$ multilayers whose growth was optimized to maximize the perpendicular anisotropy. Structural characterizations indicate abrupt interfaces essentially free of intermixing despite the miscible character of Co and Pt. Gilbert damping parameters as low as 0.021 can be obtained despite a magneto-crystalline anisotropy as large as $10^6~textrm{J/m}^3$. The inhomogeneous broadening accounts for part of the ferromagnetic resonance linewidth, indicating some structural disorder leading to a equivalent 20 mT of inhomogenity of the effective field. The unexpectedly relatively low damping factor indicates that the presence of the Pt heavy metal within the multilayer may not be detrimental to the damping provided that intermixing is avoided at the Co/Pt interfaces.
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