  We explore the role of the background plasma ion motion in self-modulated
plasma wakefield accelerators. We employ J. Dawson's plasma sheet model to
derive expressions for the transverse plasma electric field and ponderomotive
force in the narrow bunch limit. We use these results to determine the on-set
of the ion dynamics, and demonstrate that the ion motion could occur in
self-modulated plasma wakefield accelerators. Simulations show the motion of
the plasma ions can lead to the early suppression of the self-modulation
instability and of the accelerating fields. The background plasma ion motion
can nevertheless be fully mitigated by using plasmas with heavier plasmas.
