Membrane structure characterization using variable-period x-ray standing waves.
The variable-period x-ray standing wave (XSW) technique is emerging as a powerful tool for studying membrane structure. However, two significant problems arise when the method is used to characterize membranes of thickness dL < 100 A. First, the surface roughness, sigma(r), of the supporting reflecting mirror convolutes with the intrinsic half-width of the marker atom distribution in the membrane, sigma(in), and contributes to an apparent half-width, sigma, which is measured in the XSW experiment. Here we show how the latter terms are related quantitatively [sigma(in) = (sigma2 - sigma(r)2)(1/2)], such that rough mirrors give rise to larger marker atom distribution widths, sigma, and how the required quantity sigma(in) can be determined in the XSW measurement. Second, when the mean position of the marker atom layer, (z), is close to one or both boundaries of the membrane, its distribution function is truncated at the boundary. In such cases, we show why marker atom distribution should be expressed in terms of its first and second moments. We also demonstrate by numerical simulations of realistic samples how the physical parameters, sigma(r), sigma, (z), and dL, affect x-ray reflectivity and fluorescence yield profiles as an aid in their interpretation.