  Transverse 2D phase space distribution of a 2.1 MeV, 5 mA H$^-$ beam is
measured at the PIPIT test accelerator at Fermilab with an Allison scanner. The
paper describes the design, calibration, and performance of the scanner as well
as the main results of the beam measurements. Analyses of the recorded phase
portraits are performed primarily in action-phase coordinates; the stability of
the action under linear optics makes it easier to compare measurements taken
with different beamline conditions, e.g. in various locations. The intensity of
a single measured point (\pixel") is proportional to the phase density in the
corresponding portion of the beam. When the Twiss parameters are calculated
using only the high-phase density part of the beam, the pixel intensity in the
beam core is found to be decreasing exponentially with action and to be
phase-independent. Outside of the core, the intensities decrease with action at
a significantly slower rate than in the core. This `tail' comprises 10-30% of
the beam, with 0.1% of the total measured intensity extending beyond the action
10-20 times larger than the rms emittance. The transition from the core to the
tail is accompanied by the appearance of a strong phase dependence, which is
characterized in action-phase coordinates by two `branches' extending beyond
the core. A set of selected measurements shows, in part, that there is no
measurable emittance dilution along the beam line in the main portion of the
beam; the beam parameters are practically constant over a 0.5 ms pulse; and
scraping in various parts of the beam line is an effective way to decrease the
transverse tails by removing the branches.
