Size Broadening

As described in Chapter 1, for a perfect, infinite crystal the reciprocal lattice is made of points, each representing a set of planes with Miller indices hkl). The diffraction condition in reciprocal space is then defined in terms of a geometrical relation diffraction takes place when incident and diffracted beam are such that the scattering vector = (v — Vq) connects the origin with an hkl) point  

When the crystalline domains (or crystallites, i.e. coherently scattering regions) have a finite extension, the diffracted intensity is no longer confined to a point, but spreads over a region whose size and shape are related to the crystallite size and shape. 

The expression is different for other (hkl) and crystallite shapes, thus providing intensity profiles with different width and shape. 

The integral breadth, defined as the ratio between integrated intensity (peak area) and peak maximum, is frequently considered as a measure of the peak width. For the case of (00/) reflections from cubic crystallites of edge D = Na, the IB of the PD peak profile is readily obtained using the properties of the profile function of Equation (2)  

This is the well-known Scherrer equation, relating the peak width with the crystallite size, in this case IB in reciprocal space and cube edge, respectively. The inverse proportionality between IB and domain size is valid whatever the crystal shape (and symmetry of the lattice).In a more general form. Equation (3) can be written as  

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Scherrer equation to estimate the size of organized regions Imperfections in the crystal, such as particle size, strains, faults, etc, affect the X-ray diffraction pattern. The effect of particle size on the diffraction pattern is one of the simplest cases and the first treatment of particle size broadening was made by Scherrer in 1918 [16]. A more exact derivation by Warren showed that. 

Why is it possible to separate crystal size from lattice distortion — Limited crystal size broadens every reflection by the same amount20. On the other hand, the higher the order of a reflection is, the higher is the smearing effect caused by lattice distortions. 

Figure 1. Observed x-ray (CuKa) powder diffraction pattern for LaPOt showing crystallite-size broadening...

Fourier transform method. The method used most widely for the separation of size and distortion in peak profiles from metals and inorganic materials is the Fourier analysis method introduced by Warren and Averbach (21). The peak profile is considered as a convolution of the size-broadening profile fg and the distortion broadening profile fj), so that the resolved and corrected profile f(x) is given by... 

We determined Tc for films of various thickness by analyzing the T dependence of both the out-of-plane lattice parameter [4] and the satellite intensity [5], The latter method was more accurate for ultrathin films, where the finite-size broadening of the Bragg peaks becomes severe. The values of Tr (d) obtained are shown in Figure 8.7 (b). For thicker films we observe that Tc approaches the predicted value of 752°C for coherently strained PbTiC>3 on SrTiC>3 as d — oo. As <7 decreases, Tc is gradually suppressed by hundreds of degrees below... 

The factors that are included when calculating the intensity of a powder diffraction peak in a Bragg-Brentano geometry for a pure sample, composed of three-dimensional crystallites with a parallelepiped form, are the structure factor Fhkl 2=l/ TS )l2, the multiplicity factor, mm, the Lorentz polarization factor, LP(0), the absorption factor, A, the temperature factor, D(0), and the particle-size broadening factor, Bp(0). Then, the line intensity of a powder x-ray diffraction pattern is given by [20-22,24-26]... 

In Equation 4.1, the factor fiF(0) is included, which is the peak profile function, that describes particle size broadening and other sources of peak broadening. The XRD method can be used as well for the measurement of the crystallite size of powders by applying the Scherrer-Williamson-Hall methodology [4,35], In this methodology, the FWHM of a diffraction peak, p, is affected by two types of defects, that is, the dislocations, which are related to the stress of the sample, and the grain size. It is possible to write [35]... 

In addition, diffraction line breadth contains information on lattice strain, lattice defects, and thermal vibrations of the crystal structure. The chief problem to determine crystallite size from line breadth is the determination of /3(20) from the diffraction profile, because broadening can also be caused by the instrument. To correct for the instrumental broadening on the pattern of the sample, it is convenient to run a standard peak from a sample in which the crystallite size is large enough to eliminate all crystallite size broadening. By use of a convolu-... 

Plus unit cell, grain size broadening (A), peak asymmetry (a) and preferred orientation (PO) along [001] 13.3 16.6 18.7 16.1... 

An improved modelling of the average structure of ice Ic includes linear combination of stacking-probability driven structure models and anisotropic size broadening. It allows for a quantitative modelling of neutron diffraction data of different ice Ic samples. It will serve as well for the description of ice Ih samples with stacking faults. 

Layer-layer correlation probabilities. The analytical approach described by Berliner and Wemer is worth to be pursued further. From the four independent stacking probabilities corresponding to an interaction range of s=4, layer-layer correlation probabilities can be readily computed by a numerical approach for any finite crystallite size. It is then less trivial to compute the precise diffraction pattern corresponding to a correlation probability, and then to refine the parameters - the stacking probabilities - to fit an observed pattern. In principle, even the anisotropic size broadening can be taken into account in this approach as well as deviations from the layer positions. 

Vol. III. Physical and Chemical Tables (1962). Includes data on characteristic wavelengths, absorption coefficients, atomic scattering factors, Compton scattering, etc. Also treatments of intensity measurements, texture determination, particle size broadening, small angle scattering, and radiation hazards. 

GRE 85] GREAVES C., Rietveld analysis of powder neutron diffraction data displaying anisotropic crystallite size broadening , J. Appl. Cryst, vol. 18, p. 48-50,1985. 

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