Beam-helicity asymmetries have been measured at the MAMI accelerator in Mainz for the photoproduction of neutral pion pairs in the reactions $vec{gamma}prightarrow ppi^0pi^0$ and $vec{gamma}drightarrow (n)ppi^0pi^0$, $vec{gamma}drightarrow (p)npi^0pi^0$ off free protons and off quasi-free nucleons bound in the deuteron for incident photon energies up to 1.4 GeV. Circularly polarized photons were produced from bremsstrahlung of longitudinally polarized electrons and tagged with the Glasgow magnetic spectrometer. Decay photons from the $pi^0$ mesons, recoil protons, and recoil neutrons were detected in the 4$pi$ covering electromagnetic calorimeter composed of the Crystal Ball and TAPS detectors. After kinematic reconstruction of the final state, excellent agreement was found between the results for free and quasi-free protons. This demonstrates that the free-nucleon behavior of such observables can be extracted from measurements with quasi-free nucleons, which is the only possibility for the neutron. Contrary to expectations, the measured asymmetries are very similar for reactions off protons and neutrons. The results are compared to the predictions from the Two-Pion-MAID reaction model and (for the proton) also to the Bonn-Gatchina coupled channel analysis.
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