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A Kiessig fringe is an oscillation in reflected intensity caused by wave interference between contributions from the two sides of a thin film. For specular neutron scattering at grazing angle $\theta$ and wavelength $\lambda$, the normal wave vector transfer is $q_z=4\pi \sin \theta /\lambda$. Well above the critical edge, a film of thickness $d$ contributes a relative wave phase approximately $q_{z}d$, giving fringe spacingThus closely spaced fringes indicate a thicker film. As a calculated example, spacing $0.0628\mathrm{n}{\mathrm{m}}^{-1}$ corresponds to about $100\mathrm{nm}$. Interface surface roughness, contrast, additional layers and instrumental resolution affect the fringe pattern; thickness is inferred with an appropriate reflectivity model, especially near the critical edge.^[ref]

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 isfor $d=100\mathrm{nm}=1000\mathring{\mathrm{A}}$. 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.