  Material screening for identifying low-energy electron emitters and
alpha-decaying isotopes is now a prerequisite for rare-event searches (e.g.,
dark-matter direct detection and neutrinoless double-beta decay) for which
surface radiocontamination has become an increasingly important background. The
BetaCage, a gaseous neon time-projection chamber, is a proposed ultra-sensitive
(and nondestructive) screener for alpha- and beta-emitting surface contaminants
to which existing screening facilities are insufficiently sensitive.
Sensitivity goals are 0.1 betas per keV-m$^2$-day and 0.1 alphas per m$^2$-day,
with the former limited by Compton scattering of photons in the screening
samples and (thanks to tracking) the latter expected to be signal-limited;
radioassays and simulations indicate backgrounds from detector materials and
radon daughters should be subdominant. We report on details of the background
simulations and detector design that provide the discrimination, shielding, and
radiopurity necessary to reach our sensitivity goals for a chamber with a
95$\times$95 cm$^2$ sample area positioned below a 40 cm drift region and
monitored by crisscrossed anode and cathode planes consisting of 151 wires
each.
