  We present avalanche gain and associated resolution measurements recorded
with a $^4$He:CO$_2$ (70:30) gas mixture and pure SF$_6$, a Negative Ion (NI)
gas. SF$_6$ is of particular interest to the directional dark matter detection
community, as its low thermal diffusion helps to retain recoil ionization track
features over long drift lengths. With the aid of a general form of the reduced
first Townsend coefficient (RFTC), multiple GEM-based detector data sets are
used to study the high-gain behavior of the $^4$He:CO$_2$ gas mixture. The
high-gain data is well described purely in terms of the reduced electric field
strength and the number of GEMs, and the robust relationship between the RFTC
and the average, reduced, electric field strength across the GEMs is
emphasized. The associated (pulse-height) resolution measurements are used to
discuss the variance of the avalanche distribution and to describe and estimate
the lower limits of energy resolution one should expect to measure using a
simple relationship with the RFTC. In the end, a description of avalanche gain,
its effect on energy resolution, and the contributing experimental parameters
in GEM-based detectors is developed over a broad parameter space for further
use.
