Scherrer shape factor

K is known as the shape factor or Scherrer constant which varies in the range 0.89 < XT < 1, and usually K = 0.9 [H.P. Klug and L.E. Alexander, X-ray diffraction procedures for polycrystalline and amorphous materials, Second edition, John Wiley, NY (1974) p. 656]. L.W. Finger, D.E. Cox, A.P. Jephcoat. A correction for powder diffraction peak asymmetry due to axial divergence, J. Appl. Cryst. 27, 892 (1994). 

The crystallite size was determined by the Scherrer method, and a shape factor of 0.9 was applied (10). A computer program was used to digitize the selected x-ray (slow scan) peaks and determine the peak width. 

While the position and intensity of peaks in a powder pattern are determined by the unit cell size and contents, their shapes and widths are determined by instrumental effects (which can be coiTected for or modeled) and sample properties, such as the sizes and strains of crystallites and stacking faults.The simplest expression for peak broadening due to sample size (the Scherrer formula) predicts that peak width and particle size are related hy fwhm = A,/ /z cos0, where K is a shape factor (often 0.9), fivhm the peak full width at half maximum. and X the wavelength absolute numbers from this expression should be treated with caution. A sample strain leads to a peak width dependence on tanO. In more sophisticated treatments, hkl-dependent peak widths can be used to obtain information on the anisotropies of size and strain in a sample. More details on the interpretation of peak shapes are given elsewhere. ... 

The diffraction pattern of polyamide fibers, as well as those of many other fibers, is characterized by a considerable broadening of the wide-angle diffraction maxima. This is usually explained in terms of the crystallites size responsible for the diffraction. The formal relationship between crystallite size and 6 is given by the Scherrer equation [299] D = K /A(26) cos 6, where D is the diameter of the crystallite, A(26) is the angular diffraction width of the particular crystal reflection, and K is the shape factor, which is a constant with a value close to unity. 

The end-view is the Fourier transform of the projection along/from the c-axis of the rod or string to the equatorial plane. As the electron density of nanocrystals on the projected plane should smear and become uniform, we observe the shape factor of nanocrystals only, and no unoriented peak like a Debye-Scherrer ring in Figure 4.20c. 

To study the structure of NCD hy WAXS method, samples of amorphized cellulose are required. These samples can be prepared by ball-milling, saponification of cellulose acetate in non-water alkali solutions, etc. (Jeffries, 1968). The typical X-ray diffractogram of amorphous cellulose has a wide peak with maximum at 20 = 20-20.5° (Fig. 7.9). Using the modified Scherrer s equation for amorphous pol5miers with the shape factor K = 1.8, an average size of partially ordered mesomorphous clusters of NCD can be estimated = 1.8-1.9 nm. Average Bra s... 

Equation (11) uses strongly varying values for the Scherrer constant D, depending on the Miller index of the peak under consideration and on the particle shape. Table 2 is a list of the correction factors from a literature compilation (Alexander and Klug, 1974). The deviations in values are significant in view of the custom in many literature studies of setting the constant equal to unity. 

In utilizing the Scherrer equation, care must be exercised to properly account for instrumental factors which contribute to the measured peak width at half maximum. This "intrinsic" width must be subtracted from the measured width to yield a value representative of the sample broadening. When the experimental conditions have been properly accounted for, reasonably accurate values for the average crystallite size can be determined. Peak shapes and widths, however, can also provide other information about the catalyst materials being studied. For example, combinations of broad and sharp peaks or asymmetric peak shapes in a pattern can be manifestations of structural disorder, morphology, compositional variations, or impurities. 

To separate the ciystalline peak only, the background and non-ciystalline scattering should be subtracted. The shape of the peak can be corrected using the correction coefficient, K(0), which includes the Lorentz-polarization factor and initial intensity of the X-ray beam. Finally, averaged sizes and standard deviation are calculated. Comparative results of the calculation of the crystallite sizes using Scherrer s and updated equations are shown in Table 7.3. 

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