= Solution
Without the film, a sharp silicon-water boundary gives the usual Fresnel-like reflectivity governed by its scattering-length-density contrast. If the contrast-matched mixture from part (iii) is used, that ideal interface signal is strongly suppressed.
Polyethylene has a different scattering length density from both matched media. Its two interfaces generate interference oscillations, or <Kiessig fringes>, in reflectivity versus normal momentum transfer $q_z$. Their approximate spacing is
$$
\Delta q_z=\frac{2\pi}{d}\simeq0.00628\,\mathrm{\mathring A^{-1}}
$$
for $d=100\,\mathrm{nm}=1000\,\mathrm{\mathring A}$. Fit several fringes, with refraction, roughness and instrument resolution included if necessary, to determine the thickness accurately. Even with unmatched water the film changes the reflection envelope and adds thickness-dependent interference.
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