  Prompted by plans to conduct a new neutron oscillation experiment at the
European Spallation Source (ESS), we consider issues associated with the
magnetic field that must be present, some of which are potentially exacerbated
by the significantly larger length $l$ contemplated for the neutron propagation
region. To this end, we introduce a stochastic model of the residual magnetic
field within the propagation region which draws on features of magnetic
profiles measured during the last free neutron oscillation experiment
[conducted at the Institut Laue-Langevin (ILL) in the 1990's]. We average over
both fluctuations in the magnetic field sampled by neutrons, and representative
spectra of neutron speeds. We find that deviations from the quasi-free result
for the antineutron probability do not depend quadratically on $l$ (as a naive
perturbative estimate would suggest) but increase only linearly with $l$. As
regards the large spikes in the magnetic field which can be expected at, for
example, joints in the magnetic shielding of the propagation region (despite
compensating currents and magnetic idealization of the shield), we demonstrate
that their effect scales as $l/D^{3/2}$, where $D$ is the diameter of the
cylindrical magnetic shielding. Our arguments suggest that, provided the
dimensions of the propagation region are such that the ratio $l/D^{3/2}$ does
not exceed the value pertinent to the ILL experiment, and these spikes occur
close to either end of the propagation region, they can be neglected. We also
establish that any large magnetic field encountered after the propagation
region is exited will not diminish the probability for antineutron detection.
For the range of values of $l$ of most interest to the ESS experiment, it
should suffice to improve on the level of magnetic suppression achieved at the
ILL by a factor of two.
