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A program for SAXS data processing and analysis

Zhi-hong Li

arXiv:1307.0358v1physics.data-ancond-mat.mtrl-sci

TL;DR

SAXS data processing and analysis rely on computer programs, motivating development of S.exe in Intel Visual Fortran. The program introduces theory and functions for processing, calibration, and structural analysis, and is useful for standard-sample calibration when the beam is not perfectly stable.

  • Problem

    Computer programs play a key role in SAXS data analysis, motivating a dedicated program for SAXS data processing and analysis.

  • Method

    S.exe processes SAXS data, calibrates scattering angle or vector using wavelength and sample-to-detector distance, and analyzes structural information.

  • Results

    S.exe provides structural information and is particularly useful for standard-sample calibration when the beam is not perfectly stable.

  • Takeaways & Limitations

    The program supports SAXS calibration and structural analysis, including distinguishing whether scatterers form volume or surface fractals.

Abstract

from arXiv · show

A computer program for small angle X-ray scattering (SAXS) data processing and analysis named S.exe written in Intel Visual Fortran has been developed. This paper briefly introduces its main theory and function.

I. Introduction · II. Main theory · 1 Primary data processing

The paper introduces S.exe, an Intel Visual Fortran program for SAXS data processing and analysis, covering its main theory and functions. The discussed workflow addresses calibration, normalization, background subtraction, and optional desmearing for point-collimation and slit-smeared data.

  • I. Introduction: SAXS determines nanoscale structure by detecting X-rays scattered by samples at very low angles.The method applies to systems exhibiting electron-density fluctuations, including biopolymers, nanoparticles, and nanopores.
  • I. Introduction: S.exe is an Intel Visual Fortran program developed for SAXS data processing and analysis.The paper briefly introduces the program’s main theory and function.
  • II. Main theory: The main theory discussion uses point-collimation scattering data, while S.exe also supports slit-smeared data.Slit-smeared data can be desmeared or processed directly with special formulae.
  • 1 Primary data processing: Primary data processing includes scattering-angle or scattering-vector calibration, incident-beam normalization, and background subtraction.These operations define the principal preprocessing workflow described for SAXS data.
  • 1 Primary data processing: Calibration uses the incident X-ray wavelength and sample-to-detector distance or diffraction data from a standard sample.Before calibration, detector data are preprocessed to check the direct-beam centre, determine radial distance, and select useful data.
  • 1 Primary data processing: Normalization and background subtraction use sample and background intensities together with transmitted-beam intensities through the sample and background.The resulting I(q) is the background-subtracted scattering intensity, while Is(q) and Ib(q) are the raw sample and background intensities.
  • 1 Primary data processing: The transmitted-beam intensities Ks and Kb correspond to the sample and background, respectively, and slit-collimation desmearing is optional.These quantities complete the definitions used in the normalization and background-subtraction expression.

2 Data analysis · 1) Porod analysis · (1) No deviation

The paper introduces Porod analysis in SAXS, using Porod’s law to describe high-q intensity behavior and characterize ideal two-phase systems. It also presents the Porod method for calculating specific surface from the Porod constant, phase fraction, and invariant, with extrapolations supplying unmeasured q-ranges.

  • 1) Porod analysis: Porod’s law describes the asymptotic behavior of SAXS intensity as a function of scattering vector for different sample structures.The corresponding scattering characteristics are illustrated in a Porod plot with point-collimation.
  • (1) No deviation: Only an ideal two-phase system with a sharp boundary and constant, different electron densities in each phase obeys Porod’s law exactly.
  • (1) No deviation: At high q-values, ln[q4I(q)] plotted against q2 tends to a constant equal to the Porod constant K.Here q is the scattering vector, q=4πsinθ/λ, 2θ is the scattering angle, λ is the incident X-ray wavelength, and I(q) is the pinhole scattering intensity.
  • (1) No deviation: The Porod method calculates the sample’s specific surface, defined as total surface per unit volume, from K, the dispersed-phase volume fraction P, and invariant Q.For porous systems, P is termed porosity and is usually derived from sample densities or the primary beam’s absolute intensity.
  • (1) No deviation: The invariant Q is obtained from the scattering intensity through integration over q.The invariant is represented by the integral involving I(q) and q2.
  • (1) No deviation: Because experimental data do not extend to zero or infinitely large scattering angles, Q must be calculated by extrapolating the intensity into both unmeasured regions.
  • (1) No deviation: The low-q extrapolation uses the Guinier approximation, whereas the high-q extrapolation uses Porod’s q-4 dependence.

(2) Negative deviation · (3) Positive deviation · (4) Correction of deviation

The paper models negative and positive deviations from Porod’s law as effects of diffuse interfaces and electron-density micro-fluctuations, respectively. It then corrects each deviation using the parameter obtained from its corresponding analysis.

  • (2) Negative deviation: A diffuse interface or transition zone between two phases reduces scattering, especially at high angles, producing a negative deviation from Porod’s law.The interface is represented with a Gaussian-smoothing factor exp(-s^2q^2), where s relates to interfacial-layer thickness.
  • (2) Negative deviation: Fitting the negative-deviation expression yields K and s, after which the average interface-layer thickness E is derived.The paper presents the negative-deviation relation in both direct and logarithmic forms.
  • (3) Positive deviation: Sharply defined phase boundaries with electron-density micro-fluctuations within a phase add scattering, producing a positive deviation from Porod’s law.The paper notes that the actual relationship between the deviation and micro-fluctuation regions remains to be further studied.
  • (3) Positive deviation: The positive deviation is represented with a factor like exp(bq^2), where b is related to the size of regions containing electron-density micro-fluctuations.The corresponding expression is given in direct and logarithmic forms.
  • (4) Correction of deviation: Once s is obtained from negative-deviation analysis, the corrected intensity is calculated as I′(q) = I(q) exp(s^2q^2).This correction is applied to remove the negative deviation from Porod’s law.
  • (4) Correction of deviation: Once b is obtained from positive-deviation analysis, the corrected intensity is calculated as I′(q) = I(q) exp(-bq^2).This correction is applied to remove the positive deviation from Porod’s law.

2) Debye analysis · (1) No deviation

The Debye analysis uses the correlation function g(r), the Fourier transform of SAXS intensity, to characterize electron-density correlations and derive structural parameters. For an ideal two-phase system with a sharp boundary, g(r) is exponential, yielding a linear I(q)^-1/2 versus q^2 plot and estimates of phase, pore, and skeleton dimensions.

  • 2) Debye analysis: The correlation function g(r) is the Fourier transform of SAXS intensity I(q) and describes electron-density correlations between points separated by distance r.It also determines the distribution of scattered intensities and can provide the specific surface of a two-phase system.
  • 2) Debye analysis: The Debye correlation function can be analyzed using an ideal lamellar morphology to obtain structural parameters.The method relates SAXS correlations to structural characteristics of the sample.
  • (1) No deviation: For an ideal two-phase system with a sharp boundary, g(r) has the exponential form γ(r) = A_c exp(-r/A_c).A_c is the correlation distance and measures phase size or long-range order.
  • (1) No deviation: Debye scattering theory predicts that the plot of I(q)^-1/2 versus q^2 is linear across the entire measured scattering-vector range.The relation includes a constant C and the correlation distance A_c.
  • (1) No deviation: The Debye method also provides a route for calculating the specific surface.The passage identifies this as an additional application of the method for two-phase systems.
  • (1) No deviation: For porous systems, the Debye method estimates the average pore size L_P and average skeleton thickness L_S.The corresponding expressions are L_P = 1/A_c and L_S = A_c.

(2) Negative deviation · (3) Positive deviation · (4) Correction of deviation

The paper distinguishes negative and positive deviations from Debye’s theory by their interfacial or intraphase electron-density features. It then describes fitting and correction procedures for these deviations.

  • (2) Negative deviation: A diffuse phase boundary or transition zone of average thickness E between two phases produces a negative deviation from Debye’s theory.This quasi two-phase system is illustrated in Figure 2.
  • (2) Negative deviation: Equation (17) models the negative deviation with an exp(-s^2q^2) term and parameters C, s, and Ac.The exponential term has the same physical meaning as in Equation (5).
  • (2) Negative deviation: Fitting Equation (17) yields C, s, and Ac, after which the interface-layer thickness E is determined from s using Equation (6).The fitting procedure is specifically applied to the negative deviation from Debye’s theory.
  • (3) Positive deviation: Sharp phase boundaries with micro-fluctuations of electron density within either phase produce an overall positive deviation from Debye’s theory.The positive deviation is shown in Figure 2.
  • (3) Positive deviation: Equation (18) represents the positive deviation using an exp(bq^2) term and parameters C, b, and Ac.The exponential term has the same physical meaning as in Equation (7).
  • (4) Correction of deviation: After Debye analysis derives s and b, Equation (8) corrects the negative deviation from Debye’s theory.The correction is based on the parameter associated with the negative-deviation model.
  • (4) Correction of deviation: After Debye analysis derives s and b, Equation (9) corrects the positive deviation from Debye’s theory.The positive-deviation correction is applied separately from the negative-deviation correction.

3) Guinier approximation

The Guinier approximation applies at small angles regardless of scatterer shape and relates scattering intensity to the radius of gyration and forward intensity. Its validity requires specific particle and measurement conditions, while the radius of gyration is the only precise parameter obtainable without supplementary hypotheses.

  • At small angles, the Guinier formula holds regardless of scatterer shape.
  • The Guinier relation expresses ln I(q) as ln I(0) minus a term involving q^2 and the radius of gyration R_G.R_G is the radius of gyration, and I(0) is the forward scattering intensity.
  • The approximation assumes monodisperse, identical, dilute, randomly oriented, and isotropic scatterers, although anisotropic forms can be accommodated.Diluteness makes interference between particles negligible, and matrix or solvent scattering must be removed.
  • The applicable range is qR_G ≤ 1.3 for spherical particles and qR_G ≤ 0.7 for rod-like particles.
  • The radius of gyration is the only precise parameter SAXS can determine without supplementary hypotheses.It is defined as the mean square distance of scatterers from the centre of their distribution.

4) Shape evaluation

Shape evaluation in SAXS uses form factors for modeled scatterer geometries and refines calculated data against experimental curves. The distance distribution function p(r) provides qualitative shape information and can reveal conformational changes in monodisperse macromolecules.

  • Form factors can be calculated for basic shapes including spheres, cylinders, disks, rods, micelles, lamellas, and ellipsoids.
  • SAXS shape interpretation models the scatterer, calculates expected data, and refines the model to best match the experimental curve.
  • Shape determination often requires supplementary knowledge from crystallography, nuclear magnetic resonance, or electron microscopy.
  • The p(r) function starts at zero at r=0 and returns to zero at the scatterer’s largest diameter r=rmax, while its shape indicates qualitative symmetry and spatial extent.A very symmetric p(r) suggests a highly symmetric scatterer such as a sphere.
  • p(r) is obtained by Fourier transforming measured scattered intensity I(q) and can visibly detect conformational changes within a monodisperse macromolecule.

5) Scatterers size distribution determination

The program determines scatterer size distributions from SAXS intensity for polydisperse, non-interacting particles or pores of common shape and electron density. It supports several distribution-recovery methods and computes the scatterers’ mean size.

  • Theory: For scatterers differing in size but sharing shape and electron density, SAXS intensity I(q) is expressed as an integral over the volume-weighted size distribution D_V(r).The formulation uses a constant C and the normalized scattering intensity I_0(q,r) for a radially symmetric scatterer of size r.
  • Theory: The normalized single-size intensity I_0(q,r) is also referred to as the scatterer form factor P(q).Its forward scattering is normalized to unity.
  • Methods: The implemented options for recovering D_V(r) include cascade tangent, Maxwellian, log-normal, and maximum entropy methods.These methods are adopted from approaches reported in the literature for solving the size-distribution determination equation.
  • Mean size: The scatterers’ mean size is computed from the volume-weighted distribution using the program’s mean-size equation.The equation sums the product of each size r_i and its corresponding distribution value D_V,i.

6) Fractal

The section describes SAXS scattering from fractal objects using a power-law intensity and derives fractal dimensions from the exponent’s value. The exponent also distinguishes volume from surface fractals, while real objects exhibit fractal behavior only over a limited scale range.

  • Fractal: Fractal objects produce SAXS intensity with a simple power-law form, I(q) = I0q^-α, where I0 and α are constants.The exponent α is obtained from the slope of linear regions in logI(q) versus logq plots.
  • Fractal: The exponent α determines mass (Dm), pore (Dp), and surface (Ds) fractal dimensions.For volume fractals, Dm or Dp = α; for surface fractals, Ds = 6 − α.
  • Fractal: The different α values for volume and surface fractals allow the scatterer structure to be classified as a volume or surface fractal.The stated ranges are 1 < α < 3 for volume or pore fractals and 3 < α < 4 for surface fractals.
  • Fractal: Most real objects reveal fractal properties only over a limited range, because physical self-similarity is statistical.The fractal range is measured by the ratio ξ/α, and ξ and α correspond to the scale of the radius of gyration.

III. Program structure and function

S.exe is organized into two modules: module I processes SAXS data from acquisition through correction and normalization, while module II analyzes sample structure. The program supports standard-sample calibration, independent or automated operation, and separately saved ASCII outputs.

  • Program structure: S.exe comprises two modules for SAXS data processing and analysis.Its basic structure and functions are schematically displayed in Figure 3.
  • Module I: primary data processing: Module I performs primary processing, including beam-center correction, radial-distance calculation, data selection, scattering-vector calibration, collimation correction, normalization, and background subtraction.It outputs background-subtracted scattering data containing scattering vector, intensity, and error.
  • Module I: primary data processing: Module I can automatically correct the direct-beam center when the beamstop shadow and scattering profile meet the specified condition, otherwise using standard-sample diffraction peaks for calibration.The standard-sample route creates a calibration file named “calibration.dat” for later use.
  • Module II: data analysis: Module II computes structural parameters such as scatterer shape, size distribution, specific surface, porosity, interface-layer thickness, correlation distance, and fractal dimension.Its functions include Porod, Debye, Guinier, shape, size-distribution, and fractal analyses, using desmeared or slit-smeared data.
  • Outputs and availability: Module II results containing sample structural information are saved in ASCII format, and the program is freely available, registered, used, and updated.The result of each module is saved separately for viewing and reuse.
  • Program advantages: The program is particularly useful for standard-sample calibration when the beam is not perfectly stable and does not require accurate wavelength and sample-detector-distance knowledge.It can run fully automatically or semi-automatically, and each module or subprogram can be called independently.
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