  Coherent elastic neutrino-nucleus scattering (CE$\nu$NS), a process recently
measured for the first time at ORNL's Spallation Neutron Source, is directly
sensitive to the weak form factor of the nucleus. The European Spallation
Source (ESS), presently under construction, will generate the most intense
pulsed neutrino flux suitable for the detection of CE$\nu$NS. In this paper we
quantify its potential to determine the root mean square radius of the
point-neutron distribution, for a variety of target nuclei and a suite of
detectors. To put our results in context we also derive, for the first time, a
constraint on this parameter from the analysis of the energy and timing data of
the CsI detector at the COHERENT experiment.
