Ab initiotest of the Warren–Averbach analysis on model palladium nanocrystals

Abstract

Model powder diffraction patterns were calculated via the Debye formula from atom positions of a range of energy‐relaxed closed‐shell cubooctahedral clusters. The energy relaxation employed the Sutton–Chen potential scheme with parameters for palladium. The assumed cluster size distribution followed lognormal distribution of a crystallite volume centred with the diameter of 5 nm, as well as two bimodal lognormal distributions centred around 4 nm and 7 nm. These models allowed an in‐depth analysis of the Warren–Averbach method of separating strain and size effects in a peak shape Fourier analysis. The atom‐displacement distribution in the relaxed clusters could be directly computed, as well as the strain Fourier coefficients. The results showed that in the case of the unimodal size distribution, the method can still be successfully used for obtaining the column length distribution. However, the strain Fourier coefficients obtained from three reflections (002, 004 and 008) cannot be reliably estimated with the Warren–Averbach method. The primary cause is a non‐Gaussian strain distribution and a shift of the diffraction maximum, inherent to the nanoparticles, differing for every constituent cluster in the size distribution. For the bimodal size distributions, the obtained column length distributions tend to be shifted towards the centres of the modes and are less sensitive to the larger size mode.

X-ray scattering characterisation of nanoparticles

“…Erroneous size and strain values will also be obtained for small particles if their size polydispersity is too large. [86] Shape anisotropy. For sufficiently large nanoparticles where crystallite size broadening applies according to the Scherrer equation, particle shape anisotropy can be manifested as peak broadening that is dependent on the hkl indices of the diffraction peaks.…”

The real background and peak asymmetry in diffraction on nanocrystalline metals

“…The powder diffraction peak shape can be represented by the Fourier series (Warren & Averbach, 1950;Warren, 1959;Kaszkur et al, 2005) Pð2Þ…”

“…For small defined molecular fragments it is then accepted to use the Debye summation formula to calculate the pattern. Diffraction peaks calculated this way can still be analyzed using the Warren-Averbach method, provided the crystallites contributing to the model have the same lattice constant (Kaszkur et al, 2005). Problems with Fourier analysis arise when the lattice constant's dependency on the crystal size makes the peak profile from the CSD asymmetric.…”

X-ray diffraction analysis of ultradisperse systems: The Debye formula

“…In [22], powder diffraction was simulated using the Debye formula for the models where positions of atoms were also corrected for energy reasons. The number of atoms in palladium clusters with cuboctahedral packing ranged from 561 to S157 14993.…”