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Exchange interactions with itinerant electrons are known to act as arelaxation mechanism for individual local spins. The same exchange interactionsare also known to induce the so called RKKY indirect exchange interactionbetween two otherwise decoupled local spins. Here we show that both the spinrelaxation and the RKKY coupling can be seen as the dissipative and reactiveresponse to the coupling of the local spins with the itinerant electrons. Wethereby predict that the spin relaxation rates of magnetic nanostructures ofexchanged coupled local spins, such as as nanoengineered spin chains, have anoscillatory dependence on $k_F d$ , where $k_F$ is the Fermi wave length and$d$ is the inter-spin distance, very much like the celebrated oscillations inthe RKKY interaction. We demonstrate that both $T_1$ and $T_2$ can be enhancedor suppressed, compared to the single spin limit, depending on the interplaybetween the Fermi surface and the nanostructure geometrical arrangement. Ourresults open a route to engineer spin relaxation and decoherence in atomicallydesigned spin structures.

RKKY oscillations in the spin relaxation rates of atomic scale nanomagnets