linewidth and lineshape information behind line shape contains
linewidth and lineshape
information behind line shape contains information on the realstructure of the phase - What does line shape include? - width - tails (Gauss, Lorentz) - asymmetry - convolution of instrumental and physical profile only interested in physical profile Which information can be extracted from the line shape?
information behind line shape linewidths - conversions definition b … integral breadth FWHM … full width at half maximum mutual conversion Lorentz b … is more sensitive to peak tails Gauss
information behind line shape crystallite size - Scherrer - crystallite size broadening interference: structure factor/amplitude
information behind line shape crystallite size – Scherrer - crystallite size broadening the more scattering centres the sharper the peaks Scherrer equation: viz: moving objects into rciprocal space K ≈ 0. 9
information behind line shape line width increases with increasing diffraction angle -> results from the Scherrer equation
information behind line shape crystallite size - Scherrer - crystallite size broadening - maximal size of crystallites which can be determined through diffraction depends on: - instrumental resolution - Darwin limit (coherence length of the radiation) longitudinal coherence: - distance LL across which two waves of the same source are totally out of phase - emission with l and l-Dl typical values: l = 1. 54 Å Dl = 10 -4 LL ≈ 1 µm
information behind line shape crystallite size - Scherrer What is the crystallite size determined by XRD? - XRD: size of coherently scattering domains What destroys coherency? - grain boundaries (tilting of the lattice) - dislocation cell walls (sublattice tilting) - TEM: - hard contrasts only at true grain boundaries - dislocations produce a contrast due to their strain fields - contrasts of dislocations are typically blurred XRD crystallite sizes are typically smaller than crystallite sizes determined by other methods
information behind line shape microstrain - isotropic - residual stresses, II. and III. kind - are homogeneous across small areas, but for XRD inhomogeneous (integral method) - local change of interplanar spacings d c) - vacancies interstitials dislocations planar defects
information behind line shape linewidth - for standard line shapes, linewidths are additive (Stokes+Wilson, 1944) isotrop anisotrop broadening through small crystallites and microstrain can be separated
information behind line shape Williamson-Hall plot - linear regresson of sin q vs. linewidths (FWHM, b) e represent the rms elongation
information behind line shape Williamson-Hall plot - conditions for the Williamson-Hall analysis - Lorentzian peaks - Gaussian strain distribution - isotropic elastic properties - size distribution and strain distribution are not correlated - no preferrd orientation - …
information behind line shape Warren-Averbach analysis - more fundamental approach to the linewidth analysis with less strict assumptions as the Williamson-Hall method - peaks are expressed as Fourier sums (sin terms are neglected) deconvolution of peaks each order of (h, k, l) is treated separately size effects are independent on (h, k, l), strain effects depend on (h, k, l) - relevant values are hidden in the Fourier coefficients - area weighted, average crystallite size - crystallite size distribution - average strain for each crystallographic direction - is exact when strain distribution is Gaussian - good approximation for small distortion of crystallites requires mutually separated peaks!
information behind line shape anisotropic line broadening – Stephens model in general: - strain field of the microstructure defect is subject to the crystal symmetry (secondorder elastic constants of single crystals) A … F: parameters, accounting for the deviation from the isotropic case s 2 … Varianz
information behind line shape anisotropic line broadening Sn
information behind line shape defects and their strain fields - e. g. edge dislocations xx Dd/d < 0 : compression, Dd/d > 0 : dilatation zz yy xy
information behind line shape defects - dislocations slip systems for hcp metals
information behind line shape dislocations – dislocation density chkl … dislocation contrast factor M … Wilkens factor b … modulus of the Burgers vector r … dislocation density - contrast factors - depend on shape of the strain field, resp. the crystallographic anisotropy - contain dislocation type and their slip system
information behind line shape dislocations – dislocation density hexagonal boron nitride: {001}<110> edge dislocations, r = 1013 m-2
information behind line shape dislocations – dislocation density
information behind line shape
information behind line shape l = 2 n+1 hexagonal BN: stacking faults on basal planes, asf = 0. 015
information behind line shape turbostratic disorder - known from graphite, hexagonal BN and clay minerals
information behind line shape turbostratic disorder - known from graphite, hexagonal BN and clay minerals
information behind line shape turbostratic disorder - known from graphite, hexagonal BN and clay minerals - relatively well ordered structure along the c axis - heavily distorted along the a and b axis - basal planes are rotated around the c axis and shifted perpendicular to it - results in: - broadening of all lines of type hk 0 and hkl with l ≠ 0 and h or k ≠ 0 - lines of type 00 l unaffected (broadening only due to small crystallites)
information behind line shape turbostratic disorder hexagonal BN: turbostratic disorder, g = 0. 07 alternative description via supercells…
information behind line shape crystallite size - Scherrer - partial coherence - observed in nanocrystalline materials - reciprocal lattice points partially overlap for slightly mutually rotated crystallites - separation of crystallite size and cluster size misorientation angle crystallite size cluster size
information behind line shape peak asymmetry - modelled through split peak functions - left/right part have their own width and shape parameter - results from: instrumental function or physical effects (microstructure) peak asymmetry due to microstructure effects - means an asymmetric distribution of d values - chemical inhomogeneities (concentration gradients, core-shell structures, doping, solid solutions, …) affects all peaks in the same way - symmetry reduction (i. e. tetragonal distortion) affects only selected peaks - peak overlap due to multiple phases - stacking faults/twins - turbostratic diorder
information behind line shape peak asymmetry - Sn. O 2 (tetragonal), ground by hand
information behind line shape peak asymmetry - kaolinite, turbostratic
information behind line shape peak asymmetry 31
information behind line shape concentration gradient – Vegard‘s rule (1921) - linear dependence of the lattice parameter of a (binary) substitutional solid solution on the percentage of the two components - both components have the same structure type - both components have similar atomic or ionic radii - empirical approximation - deviations from Vegard‘s rule are known Pt-Au
information behind line shape summary: effect of microstructure defects on the peak shape
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