The Scherrer equation and the dynamical theory of X-ray diffraction

2016 ◽  
Vol 72 (3) ◽  
pp. 385-390 ◽  
Author(s):  
Francisco Tiago Leitão Muniz ◽  
Marcus Aurélio Ribeiro Miranda ◽  
Cássio Morilla dos Santos ◽  
José Marcos Sasaki
Keyword(s):  
Crystallite Size ◽  
Dynamical Theory ◽  
Polycrystalline Materials ◽  
X Ray Diffraction ◽  
Linear Absorption ◽  
X Ray ◽  
Crystallite Sizes ◽  
Scherrer Equation ◽  
High Degree ◽  
Kinematical Theory

The Scherrer equation is a widely used tool to determine the crystallite size of polycrystalline samples. However, it is not clear if one can apply it to large crystallite sizes because its derivation is based on the kinematical theory of X-ray diffraction. For large and perfect crystals, it is more appropriate to use the dynamical theory of X-ray diffraction. Because of the appearance of polycrystalline materials with a high degree of crystalline perfection and large sizes, it is the authors' belief that it is important to establish the crystallite size limit for which the Scherrer equation can be applied. In this work, the diffraction peak profiles are calculated using the dynamical theory of X-ray diffraction for several Bragg reflections and crystallite sizes for Si, LaB6and CeO2. The full width at half-maximum is then extracted and the crystallite size is computed using the Scherrer equation. It is shown that for crystals with linear absorption coefficients below 2117.3 cm−1the Scherrer equation is valid for crystallites with sizes up to 600 nm. It is also shown that as the size increases only the peaks at higher 2θ angles give good results, and if one uses peaks with 2θ > 60° the limit for use of the Scherrer equation would go up to 1 µm.

X-ray diffraction in superabsorbing crystals: absorption intrinsic width

2019 ◽  
Vol 75 (5) ◽  
pp. 772-776
Author(s):  
A. N. C. Lima ◽  
M. A. R. Miranda ◽  
J. M. Sasaki
Keyword(s):  
Diffraction Theory ◽  
Dynamical Theory ◽  
X Rays ◽  
X Ray Diffraction ◽  
Linear Absorption ◽  
New Techniques ◽  
Diffraction Profile ◽  
X Ray ◽  
Kinematical Theory ◽  
Special Case

The several mathematical formulations of X-ray diffraction theory facilitate its understanding and use as a materials characterization technique, since one can opt for the simplest formulation that adequately describes the case being studied. As synchrotrons advance, new techniques are developed and there is a need for simple formulations to describe them. One of these techniques is soft resonant X-ray diffraction, in which the X-rays suffer large attenuation due to absorption. In this work, an expression is derived for the X-ray diffraction profiles of reflections where the linear absorption is far greater than primary extinction; in other words, the crystal is superabsorbing. The case is considered of a parallel plate crystal, for which the diffraction profile of the superabsorbing crystal is computed as a function of crystal size normal to the diffraction planes. For thin crystals or those with negligible absorption, the diffraction profile of a superabsorbing crystal coincides with the result of the kinematical theory. For thick crystals, the absorption intrinsic profile is obtained, described by a Lorentzian function and characterized by the absorption intrinsic width. This absorption intrinsic width is proportional to the linear absorption coefficient and its expression is similar to that for the Darwin width, while the absorption intrinsic profile is a special case of the Laue dynamical theory, and it is similar to the Ornstein–Zernike Lorentzian. The formulation of X-ray diffraction of superabsorbing crystals is simple and provides new perspectives for the soft resonant X-ray diffraction technique.

A study on the limit of application of kinematical theory of X-ray diffraction

2020 ◽  
Vol 235 (11) ◽  
pp. 523-531
Author(s):  
Diego Felix Dias ◽  
José Marcos Sasaki
Keyword(s):  
Critical Thickness ◽  
Dynamical Theory ◽  
Perfect Crystal ◽  
Bragg Angle ◽  
Crystal Thickness ◽  
X Ray Diffraction ◽  
Linear Absorption ◽  
X Ray ◽  
Integrated Intensities ◽  
Kinematical Theory

AbstractIn this work, the limit of application of the kinematical theory of X-ray diffraction was calculate integrated intensities was evaluated as a function of perfect crystal thickness, when compared with the Ewald–Laue dynamical theory. The percentual difference between the dynamical and kinematical integrated intensities was calculated as a function of unit cell volume, Bragg angle, wavelength, module, and phase of structure factor and linear absorption coefficient. A critical thickness was defined to be the value for which the intensities differ 5%. We show that this critical thickness is 13.7% of the extinction length, which a specific combination of the parameters mentioned before. Also, we find a general expression, for any percentage of the difference between both theories, to determine the validity of the application of the kinematical theory. Finally, we also showed that the linear absorption decreases this critical thickness.

X-ray diffraction Warren–Averbach mullite analysis in whiteware porcelains: influence of kaolin raw material

Clay Minerals ◽  
2018 ◽  
Vol 53 (3) ◽  
pp. 471-485 ◽  
Author(s):  
Angel Sanz ◽  
Joaquín Bastida ◽  
Angel Caballero ◽  
Marek Kojdecki
Keyword(s):  
Crystallite Size ◽  
Firing Temperature ◽  
Microstructural Analysis ◽  
Stoichiometric Composition ◽  
Raw Material ◽  
Size Distributions ◽  
X Ray Diffraction ◽  
X Ray ◽  
Different Temperatures ◽  
Crystallite Sizes

ABSTRACTCompositional and microstructural analysis of mullites in porcelain whitewares obtained by the firing of two blends of identical triaxial composition using a kaolin B consisting of ‘higher-crystallinity’ kaolinite or a finer halloysitic kaolin M of lower crystal order was performed. No significant changes in the average Al2O3 contents (near the stoichiometric composition 3:2) of the mullites were observed. Fast and slow firing at the same temperature using B or M kaolin yielded different mullite contents. The Warren–Averbach method showed increase of the D110 mullite crystallite size and crystallite size distributions with small shifts to greater values with increasing firing temperature for the same type of firing (slow or fast) using the same kaolin, as well as significant differences between fast and slow firing of the same blend at different temperatures for each kaolin. The higher maximum frequency distribution of crystallite size observed at the same firing temperature using blends with M kaolin suggests a clearer crystallite growth of mullite in this blend. The agreement between thickening perpendicular to prism faces and mean crystallite sizes <D110> of mullite were not always observed because the direction perpendicular to 110 planes is not preferred for growth.

Applicability of Routine Methods of Crystallite Size Analysis*

1962 ◽  
Vol 6 ◽  
pp. 191-201
Author(s):  
Robert C. Rau
Keyword(s):  
Crystallite Size ◽  
Beryllium Oxide ◽  
Testing Procedure ◽  
Individual Performance ◽  
Size Analysis ◽  
X Ray Diffraction ◽  
X Ray ◽  
Diffraction Geometry ◽  
Crystallite Sizes ◽  
Performance Results

AbstractSeveral methods for the routine determination of crystallite size by means of X-ray diffraction line-broadening have previously been reported. Although these techniques have proven useful and reliable when utilized with the single X-ray diffractometer and instrumental geometry used to originally develop the methods, it was not known whether other instruments would provide similar reliability. Therefore a study was performed to evaluate the applicability of routine methods of crystallite size analysis to other X-ray diffraction units. A series of six beryllium oxide powder specimens, whose average crystallite sizes ranged stepwise from about 35 to nearly 3000 Å, were used to test a number of X-ray diffractometers. By using a predetermined diffraction geometry for each instrument tested, measured crystallite sizes were found to be quite reproducible and well within the limits of experimental error. The testing procedure, instrumental conditions, and individual performance results are presented in this paper.

A structural study of size-dependent lattice variation: In situ X-ray diffraction of the growth of goethite nanoparticles from 2-line ferrihydrite

2020 ◽  
Vol 105 (5) ◽  
pp. 652-663
Author(s):  
Peter J. Heaney ◽  
Matthew J. Oxman ◽  
Si Athena Chen
Keyword(s):  
Metal Oxides ◽  
Crystallite Size ◽  
Coefficient Of Thermal Expansion ◽  
Particle Growth ◽  
Rietveld Analysis ◽  
Order Kinetic ◽  
X Ray Diffraction ◽  
Time Resolved ◽  
X Ray ◽  
Crystallite Sizes

Abstract Unlike most native metals, the unit cells of metal oxides tend to expand when crystallite sizes approach the nanoscale. Here we review different models that account for this behavior, and we present structural analyses for goethite (α-FeOOH) crystallites from ~10 to ~30 nm. The goethite was investigated during continuous particle growth via the hydrothermal transformation of 2-line ferrihydrite at pH 13.6 at 80, 90, and 100 °C using time-resolved, angle-dispersive synchrotron X-ray diffraction. Ferrihydrite gels were injected into polyimide capillaries with low background scattering, increasing the sensitivity for detecting diffraction from goethite nanocrystals that nucleated upon heating. Rietveld analysis enabled high-resolution extraction of crystallographic and kinetic data. Crystallite sizes for goethite increased with time at similar rates for all temperatures. With increasing crystallite size, goethite unit-cell volumes decreased, primarily as a result of contraction along the c-axis, the direction of closest-packing (space group Pnma). We introduce the coefficient of nanoscale contraction (CNC) as an analog to the coefficient of thermal expansion (CTE) to compare the dependence of lattice strain on crystallite size for goethite and other metal oxides, and we argue that nanoscale-induced crystallographic expansion is quantitatively similar to that produced when goethite is heated. In addition, our first-order kinetic model based on the Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation yielded an activation energy for the transformation of ferrihydrite to goethite of 72.74 ± 0.2 kJ/mol, below reported values for hematite nucleation and growth.

X-ray Analysis of Ni/Ti Multilayers

1993 ◽  
Vol 308 ◽  
Author(s):  
J. Chaudhuri ◽  
S.M. Alyan ◽  
A.F. Jankowski
Keyword(s):  
Diffraction Pattern ◽  
Atomic Layer ◽  
Sine Wave ◽  
Dynamical Theory ◽  
X Ray Diffraction ◽  
Rectangular Wave ◽  
X Ray ◽  
Composition Modulation ◽  
Complete Relaxation ◽  
Kinematical Theory

ABSTRACTThe structure, composition and strain in Ni/Ti multilayers are analyzed using x-ray diffraction theories. The repeat period of the multilayers used in this study ranges from 1.3 to 12.8 nm. The composition modulation is obtained by using a kinematical theory of x-ray diffraction. A sine wave for the shorter repeat period and a rectangular wave for the longer repeat period are predicted for the composition modulation. The strain within each atomic layer is found by iteratively fitting the experimental x-ray diffraction pattern with the simulated one from a dynamical theory of x-ray diffraction. The strain at the interface is tensile in Ni and compressive in Ti with a complete relaxation of the strain at a distance away from the interface.

Synthesis and Characterisation of Titania Nanotubes: Effect of Phase and Crystallite Size on Nanotube Formation

2007 ◽  
Vol 29-30 ◽  
pp. 211-214 ◽  
Author(s):  
D.L. Morgan ◽  
E.R. Waclawik ◽  
R.L Frost
Keyword(s):  
Raman Spectroscopy ◽  
Crystallite Size ◽  
Electron Spectroscopy ◽  
Driving Force ◽  
Titania Nanotubes ◽  
X Ray Diffraction ◽  
X Ray ◽  
Material Type ◽  
Transmission Electron ◽  
Crystallite Sizes

Nanotubes were produced from commercial and self-prepared anatase and rutile which were treated with 7.5 M NaOH over a temperature range of 100 – 200°C in 20°C increments. The formation of nanotubes was examined as a function of starting material type and size. Products were characterised by X-Ray Diffraction (XRD), Transmission Electron Spectroscopy (TEM), and Raman Spectroscopy. The results indicated that both phase and crystallite size affected the nanotube formation. Rutile was observed to require a greater driving force than anatase to form nanotubes, and increases in crystallite sizes appeared to impede formation slightly.

The mean crystallite size within a hydrogenated nanocrystalline silicon based photovoltaic solar cell and its role in determining the corresponding crystalline volume fraction

2014 ◽  
Vol 92 (7/8) ◽  
pp. 857-861 ◽  
Author(s):  
K.J. Schmidt ◽  
Y. Lin ◽  
M. Beaudoin ◽  
G. Xia ◽  
S.K. O’Leary ◽  
...  
Keyword(s):  
Crystallite Size ◽  
Volume Fraction ◽  
Nanocrystalline Silicon ◽  
Crystalline Volume Fraction ◽  
X Ray Diffraction ◽  
X Ray ◽  
Photovoltaic Solar Cell ◽  
Crystallite Sizes ◽  
The Mean ◽  
Silicon Based

We examine the dependence of the crystalline volume fraction on the mean crystallite size for hydrogenated nanocrystalline silicon based photovoltaic solar cells; this work builds upon an earlier study by Schmidt et al. (Mater. Res. Soc. Symp. Proc. 1536 (2013)). For each photovoltaic solar cell considered, the X-ray diffraction and Raman spectra are measured. Through the application of Scherrer’s equation, the X-ray diffraction results are used to determine the corresponding mean crystallite sizes. Through peak decomposition, the Raman results are used to estimate the corresponding crystalline volume fraction. Plotting the crystalline volume fraction as a function of the mean crystallite size, it is found that larger mean crystallite sizes tend to favor reduced crystalline volume fractions. The ability to randomly pack smaller crystallites with a greater packing fraction than their larger counterparts was suggested as a possible explanation for this observation.

Routine Crystallite-Size Determination by X-Ray Diffraction Line Broadening

1961 ◽  
Vol 5 ◽  
pp. 104-116 ◽  
Author(s):  
R. C. Rau
Keyword(s):  
Crystallite Size ◽  
Ceramic Materials ◽  
Diffraction Line ◽  
Line Broadening ◽  
X Ray Diffraction ◽  
X Ray ◽  
Routine Basis ◽  
Crystallite Sizes ◽  
Standard Set ◽  
Sintering Characteristics

AbstractIncreasing interest in the sintering characteristics of various ceramic materials has resulted in the need for a knowledge of the crystallite sizes of many constituent ceramic powders. Standard X-ray diffraction line-broadening techniques have been utilized to determine these crystallite sizes. This paper presents a general review of the theory of line broadening as a means of measuring crystallite size and gives the methods and modifications used to perform this type of analysis rapidly and on a routine basis.Four modifications have been used in the determination of crystallite size routinely by X-ray line broadening. These methods are (1) a graded set of powder photographs, (2) a computer program to calculate sizes from diffractometer data, (3) a set of crystallite-size curves for a given material for use with diffractometer data, and (4) a standard set of curves to use with diffractometer data for any strain-free materials. The preparation, use, and limitations of each of these methods is presented.

Investigation of precision, accuracy and confidence of X-ray diffraction for determining crystallite size in nanopowders

2021 ◽  
Vol 54 (3) ◽  
Author(s):  
Alexander P. Moore ◽  
Martin B. Nemer ◽  
Mark A. Rodriguez ◽  
Christine C. Roberts ◽  
Patrick F. Fleig ◽  
...  
Keyword(s):  
Crystallite Size ◽  
Weighted Least Squares ◽  
Bulk Sample ◽  
Quantitative Information ◽  
Particle Identification ◽  
Inversion Method ◽  
X Ray Diffraction ◽  
X Ray ◽  
Peak Broadening ◽  
Crystallite Sizes

X-ray diffraction (XRD) is often utilized as a method of determining bulk sample crystallite size in powder characterization. While it is generally accepted that XRD peak broadening allows for qualitative crystallite size comparisons, its use for quantitative information is still debated. This study investigates the quantitative capability of XRD for determining the crystallite sizes of magnesium oxide nanocrystals by examining the precision, accuracy and uncertainty using the whole pattern (WP) weighted least-squares and Williamson–Hall (WH) methods. The precision of the methods was investigated by re-preparing, re-running and re-analysing identical samples. Both methods were found to be precise within 2 nm. The accuracy of the methods was investigated by comparing them against independent crystallite size analyses using visual particle identification from scanning electron microscopy micrographs and from indirect calculations using Brunauer–Emmett–Teller (BET) adsorption-determined surface areas. The WP method was found to be more accurate than the WH method, which consistently underpredicted the crystallite size. Finally, the confidence of the methods was investigated using a Bayesian inference statistical inversion method. The WP method was found to have a narrower confidence distribution in its crystallite size determination than the WH method. The broad WH confidence indicates that reliable quantitative single-measurement crystallite size determinations are not feasible using the WH technique. However, the WP method demonstrated precision, accuracy and confidence, allowing quantitative crystallite size determinations to be made.

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