Analysis of neutron and X-ray reflectivity data. II. Constrained least-squares methods

Abstract

Two methods for the determination of scatteringlength density profiles from specular reflectivity data are described. Both kinematical and dynamical theory can be used for calculating the reflectivity. In the first method, the scattering density is parameterized using cubic splines. The coefficients in the series are determined by constrained nonlinear least-squares methods, in which the smoothest solution that agrees with the data is chosen. The method is a further development of the two-step approach of Pedersen [J. Appl. Cryst. (1992), 25, 129-145-1. The second approach is based on a method introduced by Singh, Tirrell & Bates [J. Appl. Cryst. (1993), 26, 650-659] for analyzing reflectivity data from periodic profiles. In this approach, the profile is expressed as a series of sine and cosine terms. Several new features have been introduced in the method, of which the most important is the inclusion of a smoothness constraint, which reduces the coefficients of the higher harmonics in the Fourier series. This makes it possible to apply the method also to aperiodic profiles. For the analysis of neutron reflectivity data, the instrumental smearing of the model reflectivity is important and a method for fast calculation of smeared reflectivity curves is described. The two methods of analyzing reflectivity data have been applied to sets of simulated data based on examples from the literature, including an amphiphilic monolayer and block copolymer thin films. The two methods work equally well in most situations and are able to recover the original profiles. In general, the method using splines as the basis functions is better suited to aperiodic than to periodic structures, whereas the sine/cosine basis is well suited to periodic and nearly periodic structures.

Overcoming rapid inactivation of lung surfactant: Analogies between competitive adsorption and colloid stability

“…Fig. 9 shows the electron density profiles of adsorbed Survanta films, normalized by the subphase electron density ( ρ ( Z )/ ρ sub ) [158–160]. For LS films, the electron density begins at 0 on the air side of the interface (Z = 0), rises sharply through the tail region, and reaches a maximum at the headgroup region at Z~30 Å before quickly decaying to the subphase electron density ( ρ ( Z )/ ρ sub = 1) at Z ~ 42 Å.…”

Small-Angle Neutron Scattering of Precipitates in Ni-Rich Ni–Ti Alloys. II. Methods for Analyzing Anisotropic Scattering Data

“…However, some approximations have to be used to get a manageable expression. In the local monodisperse approximation (Pedersen, 1994) it is assumed that all precipitates are surrounded by precipitates of identical size. This approach is probably a good approximation in the presence of large lattice strains; it would be energetically unfavourable to place precipitates of very different sizes close to one another.…”

“…However, at larger scattering vectors polydispersity influences the data and accurate modelling requires the application of a polydisperse hard-sphere model (see e.g. Pedersen, 1993Pedersen, , 1994.…”

“…The model is based on some of the concepts of the paracrystal model, but uses clusters of finite size. A formfree size distribution is used and the effects of the polydispersity on the scattering intensity are taken into account in the local monodisperse approximation (Pedersen, 1994). Expressions for the decoupling approximation (Kotlarchyk & Chen, 1983) are also given.…”

Numerical analysis of x-ray reflectivity data from organic thin films at interfaces

“…2, 3 To reduce the calculation effort, number of fit parameters, and fit ambiguity various fit procedures and approximations have been developed. [4][5][6] The aim of all fit procedures is always to obtain the best fit, respectively, minimum mismatch between model function and measured curve. However, because the model describes only an idealized film structure it is not definitely clear which set of parameter values really represents the optimum fit.…”