  Radioactive radon atoms originating from the long-lived primordial
$^{238}\mathrm{U}$ and $^{232}\mathrm{Th}$ decay chains are constantly emanated
from the surfaces of most materials. The radon atoms and their radioactive
daughter isotopes can significantly contribute to the background of
low-background experiments. The $^{222}\mathrm{Rn}$ progeny
$^{214}\mathrm{Pb}$, for example, dominates the background of current liquid
xenon-based direct dark matter detectors. We report on a new detector system to
directly quantify the $^{222}\mathrm{Rn}$ surface emanation rate of materials.
Using cryogenic physisorption traps, emanated radon atoms are transferred from
an independent emanation vessel and concentrated within the dedicated detection
vessel. The charged daughter isotopes are collected electrostatically on a
silicon PIN photodiode to spectrometrically measure the alpha decays of
$^{218}\mathrm{Po}$ and $^{214}\mathrm{Po}$. The overall detection efficiency
is $\sim 36\,\%$ for both polonium channels. The radon emanation activity of
the emanation vessel was measured to be $(0.16\pm 0.03)\,\mathrm{mBq}$,
resulting in a detection sensitivity of $\sim 0.06\,\mathrm{mBq}$ at
$90\,\%\,\mathrm{C.L.}$.
