  Long-lived alpha and beta emitters in the $^{222}$Rn decay chain on detector
surfaces may be the limiting background in many experiments attempting to
detect dark matter or neutrinoless double-beta decay. Removal of tens of
microns of material via electropolishing has been shown to be effective at
removing radon daughters implanted into material surfaces. Some applications,
however, require the removal of uniform and significantly smaller thicknesses.
Here, we demonstrate that electropolishing < 1 $\mu$m from stainless-steel
plates reduces the contamination efficiently, by a factor > 100. Examination of
electropolished wires with a scanning electron microscope confirms that the
thickness removed is reproducible and reasonably uniform. Together, these tests
demonstrate the effectiveness of removal of radon daughters for a proposed
low-radiation, multi-wire proportional chamber (the BetaCage), without
compromising the screener's energy resolution. More generally, electropolishing
thin layers of stainless steel may effectively remove radon daughters without
compromising precision-machined parts.
