We have performed measurements of sodium nuclear recoils in NaI:Tl crystals, following scattering by neutrons produced in a $^{7}$Li(p,n)$^{7}$Be reaction. Understanding the light output from such recoils, which is reduced relative to electrons of equivalent energy by the quenching factor, is critical to interpret dark matter experiments that search for nuclear scattering interactions. We have developed a spectrum-fitting methodology to extract the quenching factor from our measurements, and report quenching factors for nuclear recoil energies between 36 and 401 keV. Our results agree with other recent quenching factor measurements that use quasi-monoenergetic neutron sources. The new method will be applied in the future to the NaI:Tl crystals used in the SABRE experiment.
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