Particle size distribution from small-angle X-ray scattering data

Abstract

An improved transform technique has been developed [Gerber (1983). Thesis, Wilhelm‐Pieck‐University Rostock, German Democratic Republic] for calculating the particle size distribution N(R) for spherical particles with radii R from small‐angle X‐ray scattering data. This method permits N(R) to be calculated from analytical expressions that were derived for point collimation and for infinitely long slit collimation. A special procedure has been introduced in order to reduce termination errors. The technique described and those developed by Schmidt [Brill, Weil & Schmidt (1968). J. Colloid Interface Sci.27, 479–492], Vonk [J. Appl. Cryst. (1976), 9, 433–440] and Glatter [J. Appl. Cryst. (1980), 13, 7–11] were used for calculating particle size distributions from theoretical scattering curves and from an experimental scattering curve of suspended SiO2 particles (Ludox). The results obtained by the different techniques were compared, and reasonable results are given by all methods employed. The accuracy of the size distributions calculated by the improved method is somewhat higher than that obtained by Schmidt's transform technique. With Glatter's procedure, the deviations from the exact distributions are comparable to those from this improved transform technique, but the use of Glatter's program requires a large computer, whereas the new method has the advantage of being suitable for a small computer. Vonk's program also requires a large computer, and the deviations obtained are larger than those produced by other methods. The experimental scattering curve of the Ludox sample was also evaluated by assuming a log‐normal distribution for the particles. The parameters μ and σ of this function were determined from a set of small‐angle X‐ray scattering structural parameters. The resulting log‐normal distribution is significantly different from the size distribution calculated by our method.

Formation of magnetic nanocrystals in a glass ceramic studied by small-angle scattering

“…7,9 Correspondingly, the magnitude of the errors of the correlation function C(r) is comparable with the symbol size. The width of the error band of the volume size distributions dw/dD amounts to about 5%.…”

“…Both C(r) and dN(D)/dD can be determined from the measured scattering curve I(s) by series expressions. 7 Since the magnetic properties of the glass ceramic are very strongly connected with the volume fraction, w, of the magnetite phase the differential volume size distribution, dw/dD, is an appropriate structural function. It is calculated from dN(D)/dD by weighting the size distribution with the particle volume D 3 /6 corresponding to the given size grade D. The function dw/dD obtained is normalized to the volume fraction of the particle phase, i.e., the area under dw/dD equals w. 8 Correspondingly, the integral of dN(D)/dD yields the crystallite number density N v in the sample ͓see Eq.…”

Small-angle X-ray scattering study of the growth kinetics of CuCl nanocrystals in NaCl

“…In Fig. 4 the LND function resulting from a fit of equations (2), (3) and (4) to the experimental scattering curve of a CuCl:NaCl sample is shown together with the size distribution of the nanocrystals calculated by the Fourier transform technique (FTT) described by Walter et al (1985), which does not require any a priori assumption about the form of the distribution function.…”

“…The distribution denoted by FTT was calculated using a Fourier transform technique (Walter et al, 1985) without a priori assumptions concerning the form of the distribution function.…”

Distribution Analysis of Nanoparticle Size by Small Angle X-ray Scattering

“…This question has led to significant debates in the drafting of the ISO/TS13762-2001 and other national standards. In the calculations, all current SAXS data processing methods encounter the negative value problem or significant error problems caused by uncertainties in physical modeling as well as in the measurement system (e.g., the Guinier tangent, Indirect Fourier Transformation, Relevant Function, Normal Distribution Function, Γ-distribution function, and Gradual Logarithm methods, as well as the Dividing Distribution Function method presented by Zhang Jinyuan and Liu Chunlan) (Guinier, 1952;Glatter, 1980;Walter, et al, 1985;Sasaki, 2005;Fan et al, 2008;Zhang et al, 1974). Meanwhile, some studies focused on the nano particles with the shape of thin plates (Meng Z.F,1991;.…”