  The bubble nucleation efficiency of low-energy nuclear recoils in superheated
liquids plays a crucial role in interpreting results from direct searches for
weakly interacting massive particle (WIMP) dark matter. The PICO Collaboration
presents the results of the efficiencies for bubble nucleation from carbon and
fluorine recoils in superheated C$_3$F$_8$ from calibration data taken with 5
distinct neutron spectra at various thermodynamic thresholds ranging from 2.1
keV to 3.9 keV. Instead of assuming any particular functional forms for the
nuclear recoil efficiency, a generalized piecewise linear model is proposed
with systematic errors included as nuisance parameters to minimize
model-introduced uncertainties. A Markov-Chain Monte-Carlo (MCMC) routine is
applied to sample the nuclear recoil efficiency for fluorine and carbon at 2.45
keV and 3.29 keV thermodynamic thresholds simultaneously. The nucleation
efficiency for fluorine was found to be $\geq 50\, \%$ for nuclear recoils of
3.3 keV (3.7 keV) at a thermodynamic Seitz threshold of 2.45 keV (3.29 keV),
and for carbon the efficiency was found to be $\geq 50\, \%$ for recoils of
10.6 keV (11.1 keV) at a threshold of 2.45 keV (3.29 keV). Simulated data sets
are used to calculate a p-value for the fit, confirming that the model used is
compatible with the data. The fit paradigm is also assessed for potential
systematic biases, which although small, are corrected for. Additional steps
are performed to calculate the expected interaction rates of WIMPs in the
PICO-60 detector, a requirement for calculating WIMP exclusion limits.
