Line broadening faulting

Fig. 4. Hexagonal MoS2 with stacking faults sharp lines for h—k = 3 n, lines broadened if h — k 3 n...

The determination of stacking faults, which can lead to diffraction line broadening (i.e. twinning) and peak shifts as well. Recent examples using X-rays are the examination of stacking faults in Ni(OH)2, which can be related to the electrochemical behavior of this material and neutron diffraction work by Berliner and Werner on the 9R structure of lithium below 20 K. 

Line broadening due to APBs is somewhat similar to faulting, even if the FT in this case is real and there is no peak shift. According to Wilson, Afi f L) can be written as an exponential function of L. For the case of APBs in CU3AU (Section 13.2.3) ... 

Many of the line profiles in the x-ray diffraction pattern of ZSM-3 are asymmetric. There is also some line broadening, some of it caused by crystallite size, and a shifting of some lines. This makes the structure determination of these polytype materials extremely difficult, especially from powder data. The large number of possible stacking arrangements and the existence of random stacking faults make it difficult to arrive at the correct structure by trial and error methods. 

In the preparation in the presence of bentyl alcohol many autiiors report the formation of platelets with stacking faults (deduced from the preferential line broadening of tire (001) reflection) attributed to the trapping of the alcohol between the layms of the precursor and its release during activation (6,46,49). 

Alan D. Franklin ARPA) X-ray line-broadening arises from several sources, strain and stacking faults in particular. Have you taken these into account in your treatment Detailed analyses have been worked out in the past by Bertaut, by Warren and his students, and recently by Bienenstock. 

Microstructural imperfections (lattice distortions, stacking faults) and the small size of crystallites (i.e. domains over which diffraction is coherent) are usually extracted from the integral breadth or a Fourier analysis of individual diffraction line profiles. Lattice distortion (microstrain) represents departure of atom position from an ideal structure. Crystallite sizes covered in line-broadening analysis are in the approximate range 20-1000 A. Stacking faults may occur in close-packed or layer structures, e.g. hexagonal Co and ZnO. The effect on line breadths is similar to that due to crystallite size, but there is usually a marked / fe/-dependence. Fourier coefficients for a reflection of order /, C( ,/), corrected from the instrumental contribution, are expressed as the product of real, order-independent, size coefficients A n) and complex, order-dependent, distortion coefficients C (n,l) [=A n,l)+iB n,l)]. Considering only the cosine coefficients A(n,l) [=A ( ).AD( ,/)] and a series expansion oiAP(n,l), A (n) and the microstrain e (n)) can be readily separated, if at least two orders of a reflection are available, e.g. from the equation... 

Figure 6 Example of a Williamson-Hall plot for a sample of nanocrystalline ZnO powder exhibiting size and stacking-feult diffraction line broadening according to the hW values. (O) reflections unaffected by mistakes hkD and hWwith /even, h-k= 3n), (A) first reflection set affected by stacking faults (/rWwith / odd, h- k=3n 1), ( ) second reflection set affected by stacking faults (/r/c/with / even, h-k=3n ).

Instrumentation for studies of this nature are usually variations of the normal powder X-ray diffractometer. Except for faulting and strain in single crystals, which are better treated as defects, the very nature of the material limits studies to powders or aggregates. X-ray powder patterns of simple metals can be analyzed to yield information on particle size, deformation fault probability, mean-square strain, and twinning. The theory and techniques used to study diffraction line broadening, peak shifts, and line profile asymmetry have been derived and applied by Warren, " and Warren and Averbach. To assess faulting probability, certain drastic assumptions are necessary, reducing the detectability limit to approximately one faulted layer in 200. 

In 1949, however, Warren pointed out that there was important information about the state of a cold-worked metal in the shape of its diffraction lines, and that to base conclusions only on line width was to use only part of the experimental evidence. If the observed line profiles, corrected for instrumental broadening, are expressed as Fourier series, then an analysis of the Fourier coefficients discloses both particle size and strain, without the necessity for any prior assumption as to the existence of either [9,3, G.30, G.39]. Warren and Averbach [9.4] made the first measurements of this kind, on brass filings, and many similar studies followed [9.5]. Somewhat later, Paterson [9.6] showed that the Fourier coefficients of the line profile could also disclose the presence of stacking faults caused by cold work. (In FCC metals and alloys, for example, slip on 111 planes can here and there alter the normal stacking sequence ABCABC... of these planes to the faulted... 

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