  The amount of light produced by nuclear recoils in scintillating targets is
strongly quenched compared to that produced by electrons. A precise
understanding of the quenching factor is particularly interesting for WIMP
searches and CE{\nu}NS measurements since both rely on nuclear recoils, whereas
energy calibrations are more readily accessible from electron recoils. There is
a wide variation among the current measurements of the quenching factor in
sodium iodide (NaI) crystals, especially below 10 keV, the energy region of
interest for dark matter and CE{\nu}NS studies. A better understanding of the
quenching factor in NaI(Tl) is of particular interest for resolving the
decades-old puzzle in the field of dark matter between the null results of most
WIMP searches and the claim for dark matter detection by the DAMA/LIBRA
collaboration. In this work, we measured sodium and iodine quenching factors
for five small NaI(Tl) crystals grown with similar thallium concentrations and
growth procedures. Unlike previous experiments, multiple crystals were tested,
with measurements made in the same experimental setup to control systematic
effects. The quenching factors agree in all crystals we investigated, and both
sodium and iodine quenching factors are smaller than those reported by
DAMA/LIBRA. The dominant systematic effect was due to the electron equivalent
energy calibration originating from the non-proportional behavior of the
NaI(Tl) light yield at lower energies, potentially the cause for the
discrepancies among the previous measurements.
