  We consider the steady state limiting current that can be carried by an
infinite periodic array of thin electron sheets spaced by period p in a planar
diode of gap voltage V and gap separation d. Our primary assumptions are (1)
electron motion is restricted by an infinite magnetic field to the direction
normal to the electrode surfaces, (2) all electrons are emitted from the
cathode with initial kinetic energy Ein, and (3) electron motion is
non-relativistic. The limiting current density, averaged over a period and
normalized to the classical 1D Child-Langmuir (CL) current density (including a
factor that accounts for non-zero Ein), is found to depend only on the two
dimensionless parameters p/d and Ein/eV. This average limiting current density
is computed from the maximum current density for which the iterative solution
of a non-linear integral equation converges. Numerical results and empirical
curve fits for the limiting current are presented, together with an analysis as
p/d and Ein/eV approach zero or infinity, in which cases previously published
results are recovered. Our main finding is that, while the local anode current
density within each electron sheet is infinite in our model (that is, it
exceeds the classical 1D CL value by an 'infinite' factor), the period average
anode current density is in fact still bounded by the classical 1D CL value.
This study therefore provides further evidence that the classical 1D
Child-Langmuir current density is truly a fundamental limit that cannot be
circumvented.
