We report on electron spin resonance (ESR) measurements of phosphorus donors
localized in a 200 square micron area below the inductive wire of a lumped
element superconducting resonator. By combining quantum limited parametric
amplification with a low impedance microwave resonator design we are able to
detect around 20000 spins with a signal-to-noise ratio (SNR) of 1 in a single
shot. The 150 Hz coupling strength between the resonator field and individual
spins is significantly larger than the 1 - 10 Hz coupling rates obtained with
typical coplanar waveguide resonator designs. Due to the larger coupling rate,
we find that spin relaxation is dominated by radiative decay into the resonator
and dependent upon the spin-resonator detuning, as predicted by Purcell
