A comparison of modern data analysis methods for X-ray and neutron specular reflectivity data

Abstract

Data analysis methods for specular X-ray or neutron reflectivity are compared. The methods that have been developed over the years can be classified into different types. The so-called classical methods are based on Parrat's or Abelè s' formalism and rely on minimization using more or less evolved Levenberg-Marquardt or simplex routines. A second class uses the same formalism, but optimization is carried out using simulated annealing or genetic algorithms. A third class uses alternative expressions for the reflectivity, such as the Born approximation or distorted Born approximation. This makes it easier to invert the specular data directly, coupled or not with classical least-squares or iterative methods using over-relaxation or charge-flipping techniques. A fourth class uses mathematical methods founded in scattering theory to determine the phase of the scattered waves, but has to be coupled in certain cases with (magnetic) reference layers. The strengths and weaknesses of a number of these methods are evaluated using simulated and experimental data. It is shown that genetic algorithms are by far superior to traditional and advanced least-squares methods, but that they fail when the layers are less well defined. In the latter case, the methods from the third or fourth class are the better choice, because they permit at least a first estimate of the density profile to be obtained that can be refined using the classical methods of the first class. It is also shown that different analysis programs may calculate different reflectivities for a similar chemical system. One reason for this is that the representation of the layers is either described by chemical composition or by scattering length or electronic densities, between which the conversion of the absorptive part is not straightforward. A second important reason is that routines that describe the convolution with the instrumental resolution function are not identical. research papers J. Appl. Cryst. (2007). 40, 820-833 A. van der Lee et al. Data analysis of specular reflectivity data 821 research papers J. Appl. Cryst. (2007). 40, 820-833 A. van der Lee et al. Data analysis of specular reflectivity data 823

Magnetic reflectometry of heterostructures

“…Keeping all of these details in mind, it becomes obvious that fitting is a difficult task and consequently a lot of research has been performed on this topic [112][113][114].…”

Probing the Local Order of Single Phospholipid Membranes Using Grazing Incidence X-Ray Diffraction

“…1(a). The corresponding electron density profiles from box model fitting procedures [10] are shown in Fig. 1(b).…”

“…Synchrotron x-ray measurements on DPPE bilayers were carried out at beam lines 9-ID and 6-ID at the advanced photon source (APS), Argonne National Laboratory (ANL) (Argonne, IL) at wavelengths of 0.690 and 0.545 Å . By modeling the measured specular XR, detailed information on the average electron density distribution in the direction normal to the interface was determined [10].For the GIXD experiments, Bragg peaks arise from scattered intensity along the scattering vector component parallel to the quartz substrate, q xy or q jj 4= sin2 xy =2, where 2 xy is the angle between the incident and diffracted beam projected onto the quartz plane [11]. The width of the Bragg peaks, corrected for the instrumental resolution [12], gives the finite size of the crystalline domains in the direction of the reciprocal scattering vector q xy (the 2D in-plane crystalline coherence length, L xy ) according to the Scherrer formula [13].…”

“…Synchrotron x-ray measurements on DPPE bilayers were carried out at beam lines 9-ID and 6-ID at the advanced photon source (APS), Argonne National Laboratory (ANL) (Argonne, IL) at wavelengths of 0.690 and 0.545 Å . By modeling the measured specular XR, detailed information on the average electron density distribution in the direction normal to the interface was determined [10].…”

Kramers-Kronig constrained modeling of soft x-ray reflectivity spectra: Obtaining depth resolution of electronic and chemical structure

“…Rather, obtaining the spatial distribution of resonant optical properties directly by simultaneously modeling spectroscopy and structure in fitting experimental data would provide a useful alternative. Fixed-energy determinations of optical constants are common in the x-ray region (generally through fitting of angular resolved reflectivity, R(Q)) [13][14][15][16][17][18][19][20][21] . While this approach has been extended to fitting fixed energy reflectivity data at a number of energies spanning the atomic core level as a means of spectroscopy 22 , full spectroscopic analysis is not generally applied to reflectivity data.…”