We have formed a resonant cavity for a massive particle, the neutron,
in an attractive potential between a surface segregated polymer film a
nd its substrate. Standing waves occur when incident waves interfere w
ith the waves reflected from the substrate. Multiple internal reflecti
ons lead to an enhanced intensity at resonant wave vectors below the w
ave vector for total reflection. At these resonant wave vectors the en
hanced neutron intensity incoherently scatters from the hydrogen conta
ining polymer of the potential well causing a marked decrease in the r
eflectivity. The position of the resonant dips provides an aid to fitt
ing the reflectivity data by restricting the possible scattering lengt
h density profiles. This phenomena may find application in the study o
f in-plane correlations by enhancing off-specular scattering.