  We present neutron scattering measurements on single crystals of lightly
doped $La_{2-x}Ba_{x}CuO_{4}$, with $0 \leq x \leq? 0.035$. These reveal the
evolution of the magnetism in this prototypical doped Mott insulator from a
three dimensional (3D) commensurate (C) antiferromagnetic ground state, which
orders at a relatively high TN, to a two dimensional (2D) incommensurate (IC)
ground state with finite ranged static correlations, which appear below a
relatively low effective TN. At low temperatures, the 2D IC magnetism co-exists
with the 3D C magnetism for doping concentrations as low as ? 0.0125. We find
no signal of a 3D C magnetic ground state by x $\sim$? 0.025, consistent with
the upper limit of x $\sim$? 0.02 observed in the sister family of doped Mott
insulators, $La_{2-x}Sr_{x}CuO_{4}$. The 2D IC ground states observed for
$0.0125 \leq x \leq 0.035$ are diagonal, and are rotated by 45 degrees within
the orthorhombic basal plane compared with those previously reported for
samples with superconducting ground states: $La_{2-x}Ba_{x}CuO_{4}$, with $0.05
\leq? x \leq? 0.095. We construct a phase diagram based solely on magnetic
order parameter measurements, which displays much of the complexity of standard
high temperature superconductivity phase diagrams discussed in the literature.
Analysis of high energy-resolution inelastic neutron scattering at moderately
low temperatures shows a progressive depletion of the very low energy dynamic
magnetic susceptibility as x increases from 0.0125 to 0.035. This low energy,
dynamic susceptibility falls off? with increasing temperature on a scale much
higher than the effective 2D IC TN appropriate to these materials. Appreciable
dynamic 2D IC magnetic fluctuations inhabit much of the "pseudogap" regime of
the phase diagram.
