  We present new [CI](1-0) and 12CO(4-3) Plateau de Bure Interferometer
observations of five Sub-Millimeter Galaxies (SMGs) and combine these with all
available [CI](1-0) literature detections in SMGs to probe the gas distribution
within a sample of 14 systems. We explore the [CI](1-0) properties of the SMG
population, particularly investigating the ratio of the [CI](1-0) luminosity to
various 12CO transition and far-infrared luminosities. We find that the SMGs
with new observations extend the spread of L[CI](1-0)/LFIR to much higher
values than found before, with our complete sample providing a good
representation of the diverse z>2 SMG population. We compare the line ratios to
the outputs of photodissociation region (PDR) models to constrain the average
density (n) and radiation field (in terms of the local field value, G0) of the
SMGs. We find the SMGs are most comparable to local ULIRGs in G0 and n, however
a significant tail of 5 of the 14 SMGs are likely best compared to less
compact, local starburst galaxies, providing new evidence that many SMGs have
extended star formation distributions and are therefore not simply scaled up
versions of local ULIRGs. We explore the limitations of using simple PDR models
to understand [CI], which may be concomitant with the bulk H2 mass rather than
PDR-distributed. We therefore also assess [CI] as a tracer of H2, finding that
for our sample SMGs, the H2 masses derived from [CI] are often consistent with
those determined from low excitation 12CO. We conclude that [CI] observations
provide a useful tool to probe the bulk gas and gas processes occurring within
merging SMGs, however more detailed, resolved observations are required to
fully exploit [CI] as a diagnostic.
