Solution to the phase problem for specular x-ray or neutron reflectivity from thin films on liquid surfaces

Abstract

The phase problem for specular x-ray and neutron reflectivity from liquid surfaces and thin films on liquid surfaces can be solved in the distorted-wave Born approximation. The gradient of the scattering-length density ͑SLD͒ profile normal to the plane of the surface is bounded in these cases. This provides a powerful constraint allowing the phase problem to be solved with no a priori assumptions via an iterative Fourier refinement procedure applied to the Fresnel-normalized reflectivity. The critical boundary condition can be determined experimentally from the autocorrelation of the gradient profile obtained via an inverse Fourier transform of the Fresnel-normalized reflectivity without phase information. The phase solution and the resulting gradient SLD profile can be shown to be unique, and therefore unambiguously determined, when all of phase space is systematically explored for particular cases, especially for thin films on liquid surfaces. This gradient SLD profile can then be integrated either numerically, or better, analytically to provide the scattering-length density profile itself on an absolute scale.

Probing surface and interface morphology with Grazing Incidence Small Angle X-Ray Scattering

“…In the inversion procedure of the density profile n 2 (z), the main drawback is of course the loss of phase. Various approaches have been developed to bypass the problem using several measurements across an absorption edge [113] like the heavy atoms method in crystallography or relying on physical constraints on the density profile [114,115]. Other approximations have been developed to handle the non-linear relationship between the reflectivity and the interface profile; the widest discussion of approximate methods has been given by Lekner [94,116].…”

Specular neutron reflectivity and the structure of artificial protein maquettes vectorially oriented at interfaces

“…However, a simple scaling of the data sets to each other was achieved by noting that the inverse Fourier transform of the Fresnel-normalized reflectivity data in the first or second Born approximation is the autocorrelation of the gradient of the neutron scatteringlength density (SLD) profile [10]. Consideration of this autocorrelation at z = 0 Å readily demonstrates that the integral of the Fresnel-normalized reflectivity data is then equal to the integral of the square of the gradient SLD.…”

“…Neutron scattering-length density (SLD) profiles can now be derived unambiguously from single monolayers of vectorially oriented peptides at liquid-vapor, liquid-liquid, solid-vapor, or solid-liquid interfaces. In the absence of a solid, recently developed, model-independent refinement methods may be used to derive these profiles with no a priori assumptions for monolayers at liquid-vapor or liquid-liquid interfaces [10]. When a solid is present, well-developed interferometric methods may be used [11].…”

Continuous distribution model for the investigation of complex molecular architectures near interfaces with scattering techniques

“…These approaches were refined to parameterize molecular components and their configurations within a sample, an approach termed composition-space refinement. 10,11 In reflectometry, even if scattering length density (SLD) profiles can be worked out from first principles or can be determined by direct inversion, 15,16 composition-space refinement has advantages in that it allows intuitive models to be implemented that constrain the full range of SLD profiles by retaining only those relevant for plausible chemical structures. This is particularly important in neutron scattering when using isotopic variation of selected chemical species, typically by 1 H= 2 H substitution in biological samples.…”

“…Also widely used are models employing Gaussian functions, 29 or direct inversion techniques. 16,30 In this paper, we implement a general modeling strategy for NR or XR, which we will call "continuous distribution" (CD) modeling, that takes advantage of the self-consistency intrinsic to the composition-space refinement approach. The CD model addresses efficiently the requirement of complete space-filling within the sample.…”