Analysis of the Broadening of Powder Pattern Peaks using Variance, Integral Breadth, and Fourier Coefficients of the Line Profile

1965 ◽  
Vol 9 ◽  
pp. 91-102 ◽  
Author(s):  
N. C. Halder ◽  
C. N. J. Wagner
Keyword(s):  
Particle Size ◽  
Fourier Analysis ◽  
Line Profile ◽  
Room Temperature ◽  
Fourier Coefficients ◽  
Powder Pattern ◽  
Initial Slope ◽  
Particle Sizes ◽  
Line Profiles ◽  
Integral Breadth

AbstractThe broadening of powder pattern peaks has been studied by three methods—Fourier analysis, integral breadth measurements, and variance of the line profiles. The results obtained from the variances are compared with those obtained from the integral breadths and Fourier coefficients.Tungsten filings were prepared at room temperature and their powder pattern peaks were recorded with a Norelco diffractometer using filtered Cu Kα radiation. The variances, integral breadths, and Fourier coefficients were calculated with the IBM 7094 computer. The results indicate that the variance is very sensitive to the range of integration s2 − s1 = (2θ2 − 2θ1) cos θ0/λ. An error of ± 10% in this range due to the difficulty in choosing the correct background changes the values of the variance significantly and the integral breadth to a lesser extent. However, the same error does not affect the values of the Fourier coefficients.Comparing the particle sizes and strains obtained by the three methods, it was found that the strains agreed remarkably well. The particle size calculated from the variance was smaller (DeW = 150Å) than that evaluated from the initial slope of the Fourier coefficients (De – 210Å) and from the integral breadths 2De ≃ D1 = 430Å.

The Structure of Nanocrystalline Iron and Tungsten Powders Prepared by High-Energy Ball Milling

1991 ◽  
Vol 35 (A) ◽  
pp. 585-592 ◽  
Author(s):  
C.N.J. Wagner ◽  
E. Yang ◽  
M.S. Boldrick
Keyword(s):  
Particle Size ◽  
Fourier Analysis ◽  
Domain Size ◽  
Fluorescence Analysis ◽  
High Energy ◽  
Nanocrystalline Powders ◽  
Line Profiles ◽  
Mechanical Working ◽  
Integral Breadth ◽  
X Ray

AbstractNanocrystalline powders of Fe and W were prepared by mechanical working in a highenergy Spex 8000 mixer/mill. The diffraction patterns were recorded with Co Kα radiation and the line profiles were subjected to a Fourier analysis. The size 〈D〉 of the coherently diffracting domains (x-ray particle size) and the root-mean square strains 〈ε2L〉1/2 were determined with the Warren-Averbach method. In addition, the integral breadths were evaluated and corrected for instrumental broadening assuming Cauchy line profiles. In order to separate particle size and strains, the corrected breadths β(s) = βcosθ/λ were plotted as a function of s = 2sinθ/λ, i.e., β(s) =(1/D) + 2ε s, where D = 〈D2/〈D〉 and ε is a strain averaged over the domain size D.X-ray fluorescence analysis indicated that the W powders contained an iron and chromium contamination due to the abrasion of the stainless steel balls reaching a value of 24 at% Fe+Cr after 20h of milling. Since W is elastically isotropic, all available (hkl) reflections can be used in the Warren-Averbach and line breaddi analyses. After 20 h of milling, the W powder exhibited a particle size 〈D〉 = 35 Å and a strain 〈ε2〉1/2 = 0.52% at L = 30 Å. The integral breadths yielded the particle size D1 = 70 Å and the strain ε = 0.38%. in the case of Fe powder, also milled for 20 h, the (110) - (220) pair of reflections was used to calculate the particle size and strains. The Fourier analysis yielded the values 〈D〉 = 105 Å and 〈ε2〉1/2 = 0.59% at L = 30 Å. The corresponding integral breadth values are D1 = 280 Å and ε1 = 0.7%. The sum of the particle size Fourier coefficients is equal to the integral breadth particle size D1 = 125 Å, which is very close to value 〈D〉 = 105 Å indicating that the particle or domain sizes have a very narrow size distribution.

Hydrothermal Synthesis of CoFe2O4 Nanoparticles and their Magnetic Properties

2013 ◽  
Vol 821-822 ◽  
pp. 1358-1361 ◽  
Author(s):  
Fan Zhang ◽  
Rui Liang Su ◽  
Li Zhi Shi ◽  
Yang Liu ◽  
Yan Na Chen ◽  
...  
Keyword(s):  
Particle Size ◽  
Magnetic Properties ◽  
Room Temperature ◽  
Magnetic Measurements ◽  
Spinel Ferrite ◽  
Synthesis Method ◽  
Ferrite Nanoparticles ◽  
Particle Sizes ◽  
Naoh Solution ◽  
20 Nm

CoFe2O4 (CFO) nanoparticles was synthesized by a simple hydrothermal method using NaOH solution as a mineralizer at 200 °C for 4 h. It was found that CFO particle sizes decreased firstly and then increased with the increasing of NaOH concentration, and had a minumum value about 10-20 nm when selected 4 mol/L NaOH solution, indicating the NaOH concentration played an important role in controlling the particle size of CFO powders. The room temperature magnetic measurements showed that the saturation magnetization value was 48 emu/g, which is less than the bulk value. The synthesis method is possible to be a general approach for the preparation of other spinel ferrite nanoparticles.

scholarly journals Diffraction line profiles of spherical hollow nanocrystals

2019 ◽  
Vol 9 ◽  
pp. 184798041983238
Author(s):  
Emiliano Burresi ◽  
Leander Tapfer
Keyword(s):  
Particle Size ◽  
Explicit Formula ◽  
Shell Thickness ◽  
Diffraction Line ◽  
Crystallographic Structure ◽  
Line Profiles ◽  
Integral Breadth ◽  
Analytical Expression ◽  
Inner Radius ◽  
Spherical Hollow

An analytical expression of diffraction line profiles of spherical hollow nanocrystals (NCs) is derived. The particular features of the profile lines, enhanced peak tail intensity, are analyzed and discussed as a function of the NC size parameters (outer and inner radius, shell thickness). The explicit formula for the integral breadth, the Fourier particle size, and the Scherrer constants are also obtained and discussed in detail. The diffraction line profiles of hollow CdS NCs of zincblende and wurtzite crystallographic structure are calculated and compared with Debye scattering profiles. The diffraction profiles of both approaches exhibit an enhanced peak tail intensity that can be considered as a fingerprint of the hollow NC structure.

An evaluation of methods of diffraction-line broadening analysis applied to ball-milled molybdenum

2004 ◽  
Vol 37 (2) ◽  
pp. 300-311 ◽  
Author(s):  
I. Lucks ◽  
P. Lamparter ◽  
E. J. Mittemeijer
Keyword(s):  
Line Profile ◽  
Strain Field ◽  
Fourier Coefficients ◽  
Field Method ◽  
Diffraction Line ◽  
Line Profile Analysis ◽  
Cauchy Type ◽  
Line Broadening ◽  
Integral Breadth ◽  
Crystallite Sizes

A comparison has been carried out of different methods of X-ray diffraction-line profile analysis for the determination of crystallite sizes and microstrains, namely the integral breadth method and three methods based on Fourier analysis of diffraction lines, namely the Warren–Averbach method, an `alternative method' and a profile synthesis strain field method. The analyses have been applied to Mo powder ball milled in two types of mills: an attritor and a planetary mill. Using the Williamson–Hall integral breadth method, the line broadening at moderate deformation is attributed solely to microstrain,i.e.practically no size broadening is detected. The three methods based on the Fourier coefficients of diffraction lines yield comparable values for crystallite sizes and microstrains. With the profile synthesis strain field method, if a size effect is included, it is possible to fit the experimental Fourier coefficients over the entire range of the relevant scale of correlation distances. The line profile shape due to microstrains, as derived with the strain field method, exhibits a systematic dependence on the integral breadth. With increasing breadth, the shape changes from a Cauchy type to a Gaussian type, suggesting a change of the dislocation arrangement with increasing plastic deformation of molybdenum powders.

Analysis of the Broadening of Powder Pattern Peaks from Cold-Worked Face-Centered and Body-Centered Cubic Metals

1963 ◽  
Vol 7 ◽  
pp. 46-65 ◽  
Author(s):  
C. N. J. Wagner ◽  
E. N. Aqua
Keyword(s):  
Particle Size ◽  
Crystallite Size ◽  
Powder Pattern ◽  
Initial Slope ◽  
Maximum Deviation ◽  
Indium Alloys ◽  
Diffraction Geometry ◽  
Cold Worked ◽  
Face Centered ◽  
Definition Of

AbstractThe broadening of powder pattern peaks of metals is normally caused by small crystallite size and by distortions within the crystallites as a consequence of dislocation configurations. In addition, the experimental diffraction geometry contributes to the line broadening. Using Cu Kα radiation, we recorded the peak profiles of the cold-worked and annealed filings of tungsten, niobium, aluminum, and silver-indium alloys with a standard focusing diffractometer. The Rachinger method was applied to separate the Kα1 and Kα2 peaks, and the Kα1 peaks were subjected to a Fourier analysis. Elimination of the instrumental broadening was carried out by the Stokes method, which does not require any assumptions concerning the mathematical description of the diffraction peak profile.The Fourier coefficients, AL, were separated into the fraction produced by particle size, ALFF, and by strains, ALe, by the Warren-Averbach method and plotted as a function of the distance L = ndhkl, normal to the reflecting planes (hkl) of spacing dhkl. The negative, reciprocal initial slope of the ALPF vs. L curve is equal to the effective particle size Deff, which contains the effects of the true crystallite size and faulting. The coefficients ALβ = AL/ALPF are used to calculate the root mean square strain <∊L2>1/2. The sum of the ALPF, which is proportional to a reciprocal integral breadth, leads to a different particle size, DWAPF, which is also dependent upon the crystallite size and faulting. The sum of the AL6 values is a measure of the strain ƐWA.The integral breadths bst, corrected by the Stokes method, i.e.,are separated into a particle size term bstPF and distortion term bSts using the relations . It is found that only the first relation leads to particle sizes DG,StPF and strains ∊G,st similar to those obtained by the summation of the ALPF and AL6, i.e., DWAPF and ∊wA, respectively. The integral breadths of the cold-worked peaks Bew and of the annealed standards bA. were also calculated, using the Lauc definition of B equal to area per peak maximum. A comparison of the values of bst/Bcw with bA/Bew for bA/Bew < 0.6 shows that all data follow the parabolic relation b/Bew = 1 − bA2/Bew2. The maximum deviation of bst/Bew from the parabola is less than 10%.

Effect of Cooling Rate and Particle Size on Compressive Strength of Macroporous Hydroxyapatite

2006 ◽  
Vol 309-311 ◽  
pp. 1047-1050
Author(s):  
Yeon Ung Kim ◽  
Byung Hyun Lee ◽  
Min Chul Kim ◽  
Kyoung Nam Kim ◽  
Kwang Mahn Kim ◽  
...  
Keyword(s):  
Particle Size ◽  
Compressive Strength ◽  
Cooling Rate ◽  
Room Temperature ◽  
Sintering Temperature ◽  
High Strength ◽  
Distilled Water ◽  
Particle Sizes ◽  
Organic Additives ◽  
Polyurethane Sponge

The objective of this study was to produce a macroporous hydroxyapatite(HA) scaffold with high strength by controlling the size of HA particles as well as cooling rate from the sintering temperature. Macroporous polyurethane sponge was employed as template to manufacture the macroporous HA scaffolds. Particle sizes of HA powders selected in this study were 4 µm and 7 µm. They were dispersed in distilled water with organic additives and infiltrated into polyurethane sponge. After drying and sintering at 1300oC, cooled down to room temperature slowly to prevent microcracking either 1oC/min or 3oC/min. Density, porosity and compressive strength were measured with different particle size and cooling rate. Both density and compressive strength were increased with decreasing particle size or cooling rate, while porosity was not related to.

Effect of a crystallite size distribution on X-ray diffraction line profiles and whole-powder-pattern fitting

2000 ◽  
Vol 33 (3) ◽  
pp. 964-974 ◽  
Author(s):  
J. I. Langford ◽  
D. Louër ◽  
P. Scardi
Keyword(s):  
Crystallite Size ◽  
Line Profile ◽  
Profile Analysis ◽  
Diffraction Line ◽  
Powder Pattern ◽  
Line Profile Analysis ◽  
Line Profiles ◽  
Diffraction Line Profile ◽  
Transmission Electron ◽  
Using Data

A distribution of crystallite size reduces the width of a powder diffraction line profile, relative to that for a single crystallite, and lengthens its tails. It is shown that estimates of size from the integral breadth or Fourier methods differ from the arithmetic mean of the distribution by an amount which depends on its dispersion. It is also shown that the form of `size' line profiles for a unimodal distribution is generally not Lorentzian. A powder pattern can be simulated for a given distribution of sizes, if it is assumed that on average the crystallites have a regular shape, and this can then be compared with experimental data to give refined parameters defining the distribution. Unlike `traditional' methods of line-profile analysis, this entirely physical approach can be applied to powder patterns with severe overlap of reflections, as is demonstrated by using data for nanocrystalline ceria. The procedure is compared with alternative powder-pattern fitting methods, by using pseudo-Voigt and Pearson VII functions to model individual line profiles, and with transmission electron microscopy (TEM) data.

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