A Gaussian–Hermite polynomials function for X-ray diffraction profile fitting

1999 ◽  
Vol 32 (4) ◽  
pp. 730-735 ◽  
Author(s):  
F. Sánchez-Bajo ◽  
F. L. Cumbrera
Keyword(s):  
Hermite Polynomials ◽  
Gaussian Function ◽  
Shape Functions ◽  
X Ray Diffraction ◽  
Stabilized Zirconia ◽  
Profile Function ◽  
Diffraction Profile ◽  
X Ray ◽  
Profile Fitting ◽  
Diffraction Peaks

In recent years, several profile-shape functions have been successfully used in X-ray powder diffraction studies. Here, a new profile function for approximating the X-ray diffraction peaks is proposed. This model, based on a Gaussian function multiplied by a correction factor in the form of a series expansion in Hermite polynomials, can be employed in the cases where there are peak asymmetries. The function has been tested by using samples of α-Al2O3and 9-YSZ (yttria-stabilized zirconia), yielding generally satisfactory results.

A profile-fitting procedure for analysis of broadened X-ray diffraction peaks. I. Methodology. Erratum

1989 ◽  
Vol 22 (2) ◽  
pp. 184-184 ◽  
Author(s):  
S. Enzo ◽  
G. Fagherazzi ◽  
A. Benedetti ◽  
S. Polizzi
Keyword(s):  
X Ray Diffraction ◽  
Correct Equation ◽  
X Ray ◽  
Fitting Procedure ◽  
Profile Fitting ◽  
Diffraction Peaks

Equation (18) of the paper by Enzo, Fagherazzi, Benedetti & Polizzi [J. Appl. Cryst. (1988). 21, 536–542] is in error. The correct equation is: A(2θ) = exp [−a|(2θ − 2θ 0)/cot 2θ 0|].

An Evaluation of Deconvolution Techniques in X-ray Profile Broadening Analysis and the Application of the Maximum Entropy Method to Alumina Data

1994 ◽  
Vol 38 ◽  
pp. 387-395 ◽  
Author(s):  
Walter Kalceff ◽  
Nicholas Armstrong ◽  
James P. Cline
Keyword(s):  
Maximum Entropy ◽  
Maximum Entropy Method ◽  
Profile Analysis ◽  
Domain Size ◽  
Entropy Method ◽  
X Ray Diffraction ◽  
Profile Function ◽  
Diffraction Profile ◽  
X Ray ◽  
Deconvolution Techniques

Abstract This paper reviews several procedures for the removal of instrumental contributions from measured x-ray diffraction profiles, including: direct convolution, unconstrained and constrained deconvolution, an iterative technique, and a maximum entropy method (MEM) which we have adapted to x-ray diffraction profile analysis. Decorevolutions using the maximum entropy approach were found to be the most robust with simulated profiles which included Poisson-distributed noise and uncertainties in the instrument profile function (IPF). The MEM procedure is illustrated by application to the analysis for domain size and microstrain carried out on the four calcined α-alumina candidate materials for Standard Reference Material (SRM) 676 (a quantitative analysis standard for I/Ic determinations), along with the certified material. Williamson-Hall plots of these data were problematic with respect to interpretation of the microstrain, indicating that the line profile standard, SRM 660 (LaB6), exhibits a small amount of strain broadening, particularly at high 2θ angle. The domain sizes for all but one of the test materials were much smaller than the crystallite (particle) size; indicating the presence of low angle grain boundaries.

Experimental Study of Precise Peak Determination in X-Ray Powder Diffraction

1983 ◽  
Vol 27 ◽  
pp. 53-60 ◽  
Author(s):  
T. C. Huang ◽  
W. Parrish ◽  
G. Lim
Keyword(s):  
X Ray Diffraction ◽  
X Ray ◽  
Search Results ◽  
Profile Fitting ◽  
Derivative Method ◽  
Diffraction Peaks ◽  
Peak Search ◽  
Different Characteristics ◽  
Good Agreement ◽  
Statistical Noise

AbstractThe combined derivative method (accompanying paper) was tested with a large number of experimental patterns to illustrate its use in various difficult problems commonly arising in peak search analysis of X-ray diffraction data. Patterns obtained with various step sizes, resolution, counting statistical noise, and profile widths were used. The precision in 2θ determination and overlap resolution are in good agreement with those previously obtained from calculated profiles, raise identification of noise as diffraction peaks was eliminated by using a convolution range proportional to the full width at half maximum. Peak search results (both 2θ and intensity) were also compared to those obtained by profile fitting to illustrate the different characteristics of these two methods.

An Evaluation of Some Profile Models and the Optimization Procedures Used in Profile Fitting

1982 ◽  
Vol 26 ◽  
pp. 73-80 ◽  
Author(s):  
Scott A. Howard ◽  
Robert L. Snyder
Keyword(s):  
Least Squares ◽  
Efficient Method ◽  
Optimization Algorithm ◽  
Optimization Technique ◽  
Silicon Sample ◽  
X Ray Diffraction ◽  
Profile Function ◽  
X Ray ◽  
Profile Fitting ◽  
Time Required

AbstractThis paper examines some of the concerns regarding the development of an algorithm for the refinement of X-ray diffraction profiles. The object of the algorithm is to provide a time efficient method of refinement through the choice of a suitable profile function and optimization technique.Seven profile models were tested using a least-squares error criterion for refinement. Profile parameters were refined using non-linear Gauss-Newton, Marquardt and Simplex algorithms. The profiles were refined on a pattern digitally collected from an NBS 640A silicon sample.The results of this study indicate the repetitive function evaluations are not necessarily the time consuming step in the profile fitting process. As the number of parameters needed to evaluate the profile and the number of points in the profile increases, the time required to perform the mathematics in the Gauss-Newton and Marquardt algorithms increases. Although the Simplex was most memory and time efficient, our Gauss-Newton optimization algorithm provided a more consistent set of refined values which were not as dependent on the initial estimates of the parameters.The most favorable results were obtained by using the split Pearson VII profile with the alpha 2 reflection fixed in position and intensity with respect to the alpha 1 reflsction. This method generated the lowest residual error and was found to avoid some problems resulting from the alpha 1, alpha 2 line overlap.

A profile-fitting procedure for analysis of broadened X-ray diffraction peaks. I. Methodology

1988 ◽  
Vol 21 (5) ◽  
pp. 536-542 ◽  
Author(s):  
S. Enzo ◽  
G. Fagherazzi ◽  
A. Benedetti ◽  
S. Polizzi
Keyword(s):  
X Ray Diffraction ◽  
X Ray ◽  
Fitting Procedure ◽  
Profile Fitting ◽  
Diffraction Peaks

Comments on determining X-ray diffraction-based volume fractions of retained austenite in steels

2001 ◽  
Vol 16 (4) ◽  
pp. 198-204 ◽  
Author(s):  
C. K. Lowe-Ma ◽  
W. T. Donlon ◽  
W. E. Dowling
Keyword(s):  
Retained Austenite ◽  
Volume Fraction ◽  
X Ray Diffraction ◽  
Experimental Conditions ◽  
X Ray ◽  
Profile Fitting ◽  
Diffraction Peaks ◽  
Diffraction Patterns ◽  
Heat Treatment Regimes ◽  
Integrated Intensities

Retained austenite is an important characteristic of properly heat-treated steel components, particularly gears and shafts, that will be subjected to long-term use and wear. Normally, either X-ray diffraction or optical microscopy techniques are used to determine the volume percent of retained austenite present in steel components subjected to specific heat-treatment regimes. As described in the literature, a number of phenomenological, experimental, and calculation factors can influence the volume fraction of retained austenite determined from X-ray diffraction measurements. However, recent disagreement between metallurgical properties, microscopy, and service laboratory values for retained austenite led to a re-evaluation of possible reasons for the apparent discrepancies. Broad, distorted X-ray peaks from un-tempered martensite were found to yield unreliable integrated intensities whereas diffraction peaks from tempered samples were more amenable to profile fitting with standard shape functions, yielding reliable integrated intensities. Retained austenite values calculated from reliable integrated intensities were found to be consistent with values obtained by Rietveld refinement of the diffraction patterns. The experimental conditions used by service laboratories combined with a poor choice of diffraction peaks were found to be sources of retained austenite values containing significant bias.

Hydration behavior by X-ray diffraction profile fitting of smectite-bearing minerals in a Plio-Pleistocene mudrock from Eugene Island, Gulf of Mexico

2019 ◽  
Vol 102 ◽  
pp. 86-100 ◽  
Author(s):  
Ruarri J. Day-Stirrat ◽  
L. Taras Bryndzia ◽  
Anja M. Schleicher ◽  
Rieko Adriaens ◽  
Ronny Hofmann ◽  
...  
Keyword(s):  
Gulf Of Mexico ◽  
X Ray Diffraction ◽  
Diffraction Profile ◽  
X Ray ◽  
Profile Fitting

scholarly journals Hydration behavior by X-ray diffraction profile fitting of smectite-bearing minerals

2020 ◽  
Vol 205 ◽  
pp. 04009
Author(s):  
Ruarri J. Day-Stirrat ◽  
L. Taras Bryndzia
Keyword(s):  
Relative Humidity ◽  
Clay Mineralogy ◽  
Wellbore Stability ◽  
Natural Material ◽  
X Ray Diffraction ◽  
Geological Time ◽  
Diffraction Profile ◽  
X Ray ◽  
Profile Fitting ◽  
Term Creep

Clay mineral hydration and dehydration processes are reversible at temperatures <100 °C and strongly affect wellbore stability, fines migration, permeability, and dispersion of pore pressure. The hydration behavior of smectite-rich material as a function of relative humidity (activity of water, aw, controlled by salinity) and temperature was studied using in situ X-ray diffraction on a material retrieved from coring in the Gulf of Mexico. X-ray diffraction profile fitting was used to explore the competition for water between hydratable phases across a range of relative humidity, 2 % to 90 %, and temperature, 25°C to 95°C, conditions. X-ray diffraction profile fitting employed a modified multi-specimen approach in which proportions of minerals were modelled using Ca-exchanged preparations in air-dried and ethylene glycol solvated states. Across the range of hydration states, the mineral proportions and crystallographic parameters remained constant from the multi-specimen approach and only the number of water layers in hydratable phases varied. Quantitative clay mineralogy showed a natural material with a discrete smectite component and a mixed-layered illite-smectite, both capable of hydration/dehydration. Results of this study showed the discrete smectite component and the mixed-layered illite-smectite hydrated at different rates with discrete smectite up-taking more water at lower relative humidity than the mixed-layered illite-smectite. Over geological time this study highlights the non-static nature of smectite hydration with implications of long-term creep and permeability behavior.

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