Figure 2 shows an example of the broadened peak of the specimen annealed at K, and the peak of the reference specimen. For all peaks a linear background was subtracted before the Fourier coefficients were calculated. The thus obtained Fourier coefficients of the structurally broadened profiles were scaled such that the first Fourier coefficient correlation distance zero was equal to one.
The methods discussed in Sec. A good fit is obtained up to a correlation distance of about 15 nm. Figure 4 shows the first and second order strain coefficients constructed using Eqs 4 , 9 , and 10 solid lines. Figure 5 shows the size coefficients, obtained by dividing the first and second order Fourier coefficients by the constructed strain coefficients solid lines. The solid line is a fit of Eq. Markers are the results of the VB-V analysis. Dotted lines are fits of Eq. Size Fourier coefficients of the specimens annealed at K, K, and K.
Markers connected by dotted lines are the result of the VB-V analysis. The size and strain coefficients were also determined with the VB-V analysis. In Fig. The size coefficients determined with both methods agree well. Figure 4 shows the first and second order strain coefficients obtained with the VB-V analysis circles. These strain coefficients were obtained by dividing the measured Fourier coefficients after elimination of instrumental broadening by the VB-V size coefficients.
Again, good agreement exists between both methods of data analysis. Figure 4 also shows fits of Eq. Figure 6 shows the plot proposed below Eq. The data follow reasonably the expected straight line. Right-hand side of Eq.
For all specimens, the results of the different methods of data analysis correspond well. From the good agreement of the different methods of data analysis, we conclude that the assumption underlying the VB-V analysis, i.
For clarity, the error bars of the results using the method on the basis of Eq. The errors in these values are typically equally large as those observed for the other methods of data analysis. Extrapolating the first few data of the size coefficients with a straight line to the correlation distance axis yields the so-called apparent grain size [ 7 ].
For the specimen annealed at K, a value of about 15 nm is obtained see Fig. Figure 5 shows that during annealing at K some grain growth has occurred. The size coefficients for the specimen annealed at K are close to one; i. In the specimen annealed at K, the dislocation density has decreased considerably. In summary, the following can be concluded about the evolution of microstructure of the nickel layers.
Up to an annealing temperature of K, relatively small changes in the microstructure of the specimens occur. In the specimen annealed at K, the grains have grown out slightly Fig. Consequently, the annealing temperatures not exceeding K are high enough to allow for some dislocation rearrangement within the grains, but are too low to establish large scale recrystallization.
In the specimen annealed at K on the other hand, recrystallization has occurred. The specimen consists of large grains with low dislocation density.
Because of the much lower dislocation density, the dislocation interaction is small and there is less possibility to minimize the strain energy by means of dislocation rearrangement.
The determination of the contrast factor is in general a difficult task and is beyond the scope of the present paper. In Refs. It follows that the dislocation densities and outer cut-off radii, before recrystallization, are about 10 16 m —2 and 18 nm, respectively.
It is stressed here that these values should be considered as rough estimates. Further, it should be noted that dislocations might be generated during cooling down. Due to the different thermal expansion coefficients of silicon and nickel a thermal tensile strain develops in the nickel layer during cooling down.
For the specimen annealed at K, this strain is approximately 0. X-ray diffraction strain measurements revealed that the strain in the nickel layer annealed at K, at room temperature is close to this value, which suggests that the thermal strains are hardly plastically relaxed, and consequently, not many dislocations are formed during cooling down.
Thus, in this case, possible dislocations formed during cooling down can be neglected for the layers annealed at temperatures up to K, that show much larger dislocation densities. However, for the specimen annealed at K, it cannot be excluded that the dislocations are in part generated during cooling down. A last example concerns the limitations of the above analyses. For the nickel layers treated above it has been assumed that a more or less homogeneous grain size and dislocation distribution was present in the specimens.
However, the specimens annealed at K and K have partly recrystallized. The microstructure of these specimens therefore consists of a mixture of small grains with large dislocation density i. For such inhomogeneous specimens, the methods treated above are not useful. In this case, an analysis of the diffraction line broadening is still possible. The microstructures of the specimens annealed at K and K are considered to be a mixture of the microstructures of the specimens annealed at K and K.
Then, the diffraction peaks of the specimens annealed at K and K are simply the sums of the diffraction peaks of the diffraction peaks of the specimens annealed at K and K, scaled with their respective diffracting volume fractions.
The same holds for the Fourier coefficients of the diffraction peaks. The specimen can quite well be characterized as a mixture of the microstructures before and after recrystallization.
Note that for this procedure, neither correction for instrumental broadening, nor determination of the strain broadening was necessary, in contrast to the methods above. Therefore, possible errors made by the removal of the instrumental broadening are avoided. Solid lines are the sum of 0. In the last example it was possible to perform a diffraction line shape analysis despite the inhomogeneity of the specimens under consideration.
In general however, this may not be possible and investigations to dislocation distributions using x-ray line profile analysis, for inhomogeneous microstructures may become very difficult. Several methods can be used to determine dislocation distribution parameters from diffraction line broadening measurements in specimens for which both strain broadening caused by dislocations and size broadening occurs. If the strain broadening can be described with the model due to Wilkens, dislocation distribution parameters can be determined from the ratio of the Fourier coefficient of diffraction line profiles from two orders of reflection, as well as using the Van Berkum-Vermeulen analysis.
The use of the Warren-Averbach method is in this case dissuaded. For thin nickel layers on silicon, the analysis on the basis of the ratio of the Fourier coefficients of two orders of reflection and the Van Berkum-Vermeulen analysis yield equal results within experimental precision.
Annealing thin nickel layers with large dislocation density and small grain size, at temperatures up to K does not lead to large scale grain growth and changes in dislocation density. However, the outer cut-off radius decreases somewhat, which suggests that strain energy is minimized at these temperatures by means of movement of the dislocations within the grains. Annealing at K and K leads to an inhomogeneous microstructure that consists of large grains with low dislocation density and small grains with high dislocation density.
Complete recrystallization occurs during annealing at K; after annealing the specimen consists of large grains with low dislocation density. The authors are indebted to E. Delhez is acknowledged for critically reading the manuscript. This research was carried out under project number MS.
About the authors: J. Kamminga is a research fellow at the Netherlands Institute for Metals Research. National Center for Biotechnology Information , U. Published online Feb 1. Article notes Copyright and License information Disclaimer.
Each domain is represented by columns of cells along the 03 direction [ 61 ] see Fig. The crystal has orthorhombic axes with the direction 03 normal to the diffracting planes 00 l. The experimentally observable diffraction power may be ex-pressed as a Fourier series. The A n coefficients are the product of two terms.
The first term depends only on the column length size coefficient ; the second depends only on distortion in domains distortion coefficient :. It is more convenient to express the distortion coefficient in terms of the strain component.
To obtain the strain component, it is necessary to approximate the exponential term. For not too large L. Now, Eq. Warren and Averbach [ 22 ] derived this relationship in a similar way. It separates size and strain contributions to the broadening, and allows for their simultaneous evaluation.
Representation of the crystal in terms of columns of cells along the a 3 direction [ 59 ],. If the size coefficients are obtained by applications of Eq. Surface-weighted domain size is determined: a by the intercept of the initial slope on the L-axis; b as a mean value of the distribution function.
To separate size and strain broadening by using integral breadths, it is necessary to define the functional form for each effect. In the beginning, size and strain contributions were described by Cauchy or Gauss functions.
Using Eqs. Equation 24 uses the Haider and Wagner [ 62 ] parabolic approximation for the integral breadth of the Voigt function expressed by Eq. Experience shows, however, that neither Cauchy nor Gauss functions can model satisfactorily size or strain broadening in a general case.
Langford [ 26 ] introduced the so-called multiple-line Voigt-function analysis. Both size-broadened and strain-broadened profiles are assumed to be Voigt functions. This approach disagrees with the Warren-Averbach analysis, that is, the two methods give different results see Sec. There are cases where only the first order of reflection is available or higher-order reflections are severely suppressed extremely deformed materials, multiphase composites, catalysts, and oriented thin films.
Many methods exist to separate size and strain broadening from only one diffraction peak. However, it was stated in Sec. Consequently, single-line methods should be used only when no other option exists.
The single-line methods can be di-vided in two main parts: Fourier-space and real-space methods. Fourier-space methods are based on the Warren-Averbach separation of size and strain broadening following Eq. Then, Eq. All Fourier-space methods have the serious problem that the Fourier coefficients. All real-space methods [ 74 , 75 , 76 ] are based on the assumption that the Cauchy function deter-mines size and that the Gauss function gives strain.
The most widely used method of de Keijser et al. In this field, very few studies exist. Williams et al. Using a GSAS Rietveld refinement program [ 78 ], both size and strain broadening were modeled with the Gauss functions for the neutron-diffraction data [ 79 , 80 ], and with the Cauchy functions for the x-ray diffraction data modified method of Thompson, Cox, and Hastings [ 81 ].
Interestingly, both the neutron and x-ray data gave identical values for the isotropic strain 0. Singh et al. They separated size and strain parameters by means of Eq. Size broadening was found to be negligible, but isotropic microstrains range from 0. Using the Gauss-Gauss approximafion, they found strains of 0. We are aware of only two more unpublished studies [ 84 , 85 ] involving size-strain analysis in high- T c superconductors.
The probable reason is that any analysis is very difficult because of weak line broadening and overlapping reflections. This precludes application of reliable analysis, such as the Stokes deconvolution method with the Warren-Averbach analysis of the broadening. Instead, simple integral-breadth methods are used, which gives generally different results for each approach.
Moreover, for x-ray diffraction broadening, application of the Gauss-Gauss approximation does not have any theoretical merit, although reasonable values, especially of domain sizes, may be obtained [ 86 ]. We showed [ 87 , 88 , 89 ] that reliable diffraction-line-broadening analysis of superconductors can be accomplished and valuable information about anisotropic strains and incoherently diffracting domain sizes obtained.
Roshko, using a freeze-drying acetate process [ 90 ]. Acetates of the various cations were assayed by mass by calcining to the corresponding oxide or carbonate. The appropriate masses of the acetates for the desired compositions were dissolved in deionized water. The acetate solutions were then sprayed through a fine nozzle into liquid nitrogen to preserve the homogeneous cation distributions. After drying, the powders, except the La 1.
Because BaCO 3 is difficult to decompose, the La 1. The bulk specimens were surface polished, if necessary, and mounted in specimen holders. Silver and tungsten powders were dry ground with a mortar and pestle. X-ray-diffraction data were collected using a standard two-circle powder goniometer in Bragg-Brentano parafocusing geometry [ 92 , 93 ] see Fig. A flat sample is irradiated at some angle incident to its surface, and diffraction occurs only from crystallographic planes parallel to the specimen surface.
Soller slits in the diffracted beam, 0. Optical arrangement of an x-ray diffractometer. Adapted from Klug and Alexander [ 24 ]. The diffractometer was controlled by a computer, and all measurements were stored on hard disc.
Data were transferred to a personal computer for processing. We used computer programs for most calculations. This program allows a choice of the fitting function and gives refined positions of the peak maximums, intensities, and function-dependent parameters.
It also has the ability to convolute the predefined instrumental profile with the specimen function to match the observed pattern. Choice of the specimen function includes Gauss and Cauchy functions. In the fitting procedure, for every peak in the pattern, the program first generates the instrumental profile at the required diffraction angle. The instrumental profile is determined from prior measurements on a well-annealed standard specimen see Sec.
Then it assumes parameters of the specimen profile. For an exact Voigt function, parameters are peak position, peak intensity, and Cauchy and Gauss integral breadths of the Voigt function. By convoluting the instrumental profile with the specimen profile, and adding a background, the calculated pattern is obtained [ Eq. Parameters of the specimen profile are varied until the weighted least-squares error of calculated and observed patterns Eq.
This process avoids the unstable Stokes deconvolution method. It is possible that the refinement algorithm is being trapped in a false minimum [ 96 ], but it can be corrected by constraining some parameters. Refined parameters of the pure-specimen profile are input for the size-strain analysis of the broadening. A program for this analysis was written in Fortran.
A program in Fortran was written to apply corrections to observed peak maximums by using NIST standard reference material LaB 6 as an external standard. When instrumental and specimen contributions to the observed line profile must be modeled separately, adopting a specimen function is a critical step.
Yau and Howard [,] used Cauchy, and Enzo et al. Benedetti, Fagherazzi, Enzo, and Battagliarin [ ] showed that modeling the specimen function with the pseudo-Voigt function gives results comparable to those of the Stokes deconvolution method when combined with the Warren-Averbach analysis of Fourier coefficients.
De Keijser, Mittemeijer, and Rozendaal [ ] analytically derived domain sizes and root-mean-square strains for small averaging distance L in the case of the Voigt and related functions. The aim here is to study more thoroughly the consequences of assumed Voigt specimen function on the size-strain analysis of the Fourier coefficients of the broadened peaks.
Moreover, the discrepancy between the integral-breadth methods and Warren-Averbach analysis results from different approximations for the strain broadening and the background experimental errors.
The normalized Fourier transform of a Voigt function is easily computable [ 46 ]:. Equation 32 is a good approximation even for large specimen broadening. It is important to keep Fourier interval limits identical for all multiple-order reflections; otherwise serious errors in the subsequent analysis will occur [ ]. Experience shows that Cauchy and Gauss functions can not satisfactory model specimen broadening.
Balzar and Ledbetter [ 64 ] postulate that the specimen function includes contributions of size and strain effects, both approximated with the Voigt functions. Because the convolution of two Voigt functions is also a Voigt function, Cauchy and Gauss integral breadths of the specimen profile are easily separable:. Langford [ 26 ] separated the contributions from size and strain broadening in a similar way.
See Eqs. Note, however, that Eqs. Because Fourier coefficients are a product of a size and a distortion coefficient, from Eqs. Wang, Lee, and Lee [ ] modeled the distortion coefficient, and Selivanov and Smislov [ ] modeled the size coefficient in the same way. To obtain size and distortion coefficients, at least two reflections from the same crystallographic-plane family must be available.
Surface-weighted domain size is calculated from the size coefficients following Eq. From Eq. Therefore, surface-weighted domain size depends only on the Cauchy part of the size-integral breadth. The second derivative of the size coefficients is proportional to the surface-weighted column-length distribution function, Eq.
The volume-weighted column-length distribution function follows similarly [ ]:. By differentiating Eq. Because the column-length distribution function should always be positive [ 59 ], the Cauchy part must dominate. Inspection of Eq. This is a widely encountered problem in the Fourier analysis of line broadening.
It results in overestimation of effective domain sizes and underestimation of the RMSS [ 36 ]. However, Eq. Figure 6 shows that negative values of the column-length distribution functions set to zero , do not affect the shape, but shift the entire distribution toward larger L values.
If the column-length distribution functions are known, it is possible to evaluate mean values of respective distributions:. Integrals of this type can be evaluated analytically [ ]:. The volume-weighted domain size follows:.
Theoretically, k can change from zero to infinity. However, the minimum value of k is determined by Eq. Hence, the ratio of domain sizes can change in a limited range see also Fig. This is a case of pure Cauchy size broadening, described by Haider and Wagner [ 62 ] and de Keijser, Mittemeijer and Rozendaal [ ].
It is possible to imagine a more complicated column-length distribution function [ 27 ] than Eq. The ratio of volume-weighted and surface-weighted domain sizes as a function of the characteristic ratio of Cauchy and Gauss integral breadths k.
In Sec. Comparing with Eq. Therefore, mean-square strains MSS decrease linearly with averaging distance L. This behavior is usually observed in the Warren-Averbach analysis. Rothman and Cohen [ ] showed that such behavior would be expected of strains around dislocations. Adler and Houska [ ], Houska and Smith [ ], and Rao and Houska [ ] demonstrated for a number of materials that MSS can be represented by a sum of two terms, given by Cauchy and Gauss strain-broadened profiles.
This is a limiting case of pure-Gauss strain broadening, described by de Keijser, Mittemeijer and Rozendaal [ ]. To calculate domain sizes and strain, it is necessary to define size and distortion integral-breadths angular order-dependence.
However, from Eq. If we compare Eq. If we substitute these expressions into Eqs. This is expected because the distortion coefficient is approximated with the exponential 47 ].
Delhez, de Keijser, and Mittemeijer [ 23 ] argued that, instead of Eq. These two approximations differ with fourth-order terms in the power-series expansion. In terms of this approach, Eq. This means that even if the strain-broadened pro-file is given entirely by the Gauss function, the MSS depend on distance L see Fig. In this approximation no simple relation for the distortion integral-breadths angular order-dependence exists.
For not so large L , however, Eq. Generally, it was shown that in the size-broadened profile the Cauchy part must dominate. No similar requirement for the strain-broadened profile exists. However, experience favors the assumption that it has to be more of Gauss-type.
The Warren-Averbach approach is exact if strain broadening is purely Gaussian, so Eqs. In any case, both approaches, given by Eqs. If at least two orders of reflection from the same plane hkl are available, we can use Eqs. Subsequent application of Eqs.
This approach is more straightforward and much simpler than the original Warren-Averbach analysis. Great care should be given to the possible systematic errors. If they show negative values for small L see Fig.
This is because i only the positive values of the column-length distribution functions are numerically integrated or ii the intercept on the L -axis of the linear portion of the A S vs L curve is taken see Fig.
An analogous discrepancy exists between integral-breadth and variance methods [ ]. In such cases, correction methods for truncation can be applied [ 35 , , ], but the best procedure is to repeat the pattern fitting with the correct background. In the Fourier analysis it is usually observed that the mean-square strains diverge as the averaging distance L approaches zero. This also follows from Eqs. However, because the MSS dependence on distance L is not defined in Warren-Averbach analysis, it was suggested [ 27 , , ] that local strain can be obtained by taking the second derivative of the distortion coefficient, or by a Taylor-series expansion of local strain.
Therefore, we obtain from Eq. It is evident that this relation is wrong. It holds only for a special case of pure-Gauss strain-broadened profile, when the MSS are equal for any L. If the main origin of strains is dislocations [ ], strains are defined after some distance from the dislocation cutoff radius to be finite.
Averaging strains over a region smaller than the Burgers vector is probably not justified. For instance, Eq. Errors in size and strain analysis of broadened peaks are relatively difficult to evaluate. Following Langford [ 26 ], sources of the systematic errors include choice of standard specimen, background, and type of analytical function used to describe the line profiles.
The first two errors should be minimized in the experimental procedure. Errors caused by inadequate choice of specimen function would systematically affect all derived results, but they can not be evaluated. Random errors caused by counting statistics have been computed by Wilson [ , , ] and applied to the Stokes deconvolution method by Delhez, de Keijser, and Mittemeijer [ 23 ], as well as by Langford [ 26 ] and de Keijser et al.
Nevertheless, the approximate error magnitude can be calculated from estimated standard deviations e. Each line profile has four parameters varied independently: position, intensity, and Cauchy and Gauss integral breadths of the Voigt profile.
In least-squares refinement, e. The main source of errors is integral breadths. Errors in peak position, peak intensity, and background are much smaller and can be neglected in this simple approach.
Alter-natively, to see how errors depend on the Fourier coefficients, errors can be estimated from the Warren-Averbach relationship [ Eq. Errors in Fourier coefficients increase with L , while factors in Eqs.
In general, errors of domain sizes and strains are of the same order of magnitude as errors of integral breadths [ 86 ]. Before specimen broadening is analyzed, instrumental broadening must be determined. It is then assumed that its broadening may be attributed only to the instrument.
The usual procedure is to anneal the specimen. However, in some instances that is not possible, because either the material undergoes an irreversible phase transition on annealing, or the number of defects can not be successfully decreased by annealing. Another possibility is to measure the whole diffraction pattern of the material showing the minimal line broadening, and then to synthesize the instrumental profile at the needed diffraction angle.
This approach re-quires the modeling of the angle dependence of the instrumental standard parameters. Cagliotti, Paoletti, and Ricci [ ] proposed the following function to describe the variation of the full width at the half maximum of profile with the diffraction angle:. Although this function was derived for neutron diffraction, it was confirmed to work well also in x-ray diffraction case [ , ].
A more appropriate function for the x-ray angle-dispersive powder diffractometer, based on theoretically predicted errors of some instrumental parameters [ ] may be the following [ ]:. This function may better model the increased axial divergency at low angles and correct for the specimen transparency [ ]. However, contrary to the requirement on the specimen function, most important for the instrumental function is to correctly describe the angular variation of parameters, regardless of its theoretical foundation.
When specimen broadening is modeled with a Voigt function, the simplest way to correct for the instrumental broadening is by fitting the line profiles with the Voigt function, too.
Cauchy and Gauss integral breadths of the specimen-broadened profile are then easily computable by Eqs. Another approach is to model the instrumental-broadening angle dependence by fitting the profile shapes of a standard specimen with some asymmetrical function; split-Pearson-VII [ 95 ] or pseudo-Voigt convoluted with the exponential function [ ]. The instrumental function can then be synthesized at any desired angle of diffraction and convoluted with the assumed specimen function to match the observed profile by means of Eq.
Refined full widths at half maximum FWHM and shape factors m for both low-angle and high-angle sides of the profiles are fitted with second-order polynomials. A split-Pearson VII profile. The two half profiles have same peak position and intensity. The basic requirement on the standard specimen is, however, to show as small a line broadening as possible. To minimize physical contributions to the Une broadening of the standard specimen, a few moments were emphasized as follows.
Because diffraction-line width depends strongly on degree of annealing, it is preferable to use some reference powder-diffraction standard. Furthermore, asymmetry in the peak profiles is introduced by axial divergence of the beam, flat specimen surface, and specimen transparency [ 24 ].
Choosing a standard specimen with low absorption coefficient would cause transparency effects to dominate. If the studied specimen has a large absorption coefficient compared to the standard , this might produce a fictitious size contribution and errors in microstrains. According to Fawcett et al. Second-order polynomials were fitted through points.
To study the applicability of method described in Sec. Tungsten has very narrow line profiles, allowing us to obtain the upper limit of domain sizes that can be studied. Silver is easily deformed, which provides a possibility to apply the method to broad line profiles.
To test the case of relatively complicated patterns and weak line broadening, the method was also applied to La 1. In this section only the mechanical aspects of the line broadening are discussed. Discussion about the origins of broadening of superconductors can be found in Sec. Figure 11 shows observed and refined peaks of tungsten untreated and silver ground powders. In Table 2 are listed results of fitted pure-specimen Voigt profiles for silver and tungsten specimens; Table 3 and Fig.
Untreated tungsten powder shows relatively weak broadening. Instrumental profile FWHMs at angle positions of and tungsten lines are 0. Results in Table 3 reveal that small broadening is likely caused by domain sizes, because microstrains have negligible value. This pushes the limit for measurable domain sizes probably up to — A. However, one must be aware that weak specimen broadening implies higher uncertainty of all derived parameters.
Moreover, the choice of the instrumental standard becomes more crucial. Observed points pluses , refined pattern full line , and difference pattern below : W untreated upper ; Ag ground lower. Fourier coefficients for the first- pluses and second-order crosses reflection, and size coefficients circles : [ ] Ag untreated upper ; [ ] Ag ground lower. Parameters of the pure-specimen Voigt function, as obtained from profile-fitting procedure for tungsten and silver powders.
Both silver and tungsten line profiles become more Cauchy-like after grinding, which probably increases dislocation density in the crystallites. This is consistent with the presumption that small crystallites and incoherently diffracting domains separated by dislocations within grains affect the tails of the diffraction-line profiles [ 60 , ].
Figure 13 illustrates the dependence of MSS on the reciprocal of the averaging distance L. Errors in integral breadths allow estimation of errors in strain and size parameters Sec. However, in Eqs. Errors of domain sizes and strains are of the same order of magnitude as errors of integral breadths. The possible source of systematic error is potential inadequacy of Voigt function to accurately describe specimen broadening.
This effect can not be evaluated analytically; but it would affect all derived parameters, especially the column-length distribution functions. The logical relationship between values of domain sizes see Table 3 for different degrees of broadening indicates that possible systematic errors can not be large. Equation 41 allows computation of volume-weighted and surface-weighted average domain sizes if respective column-length distribution functions can be obtained.
Figure 14 gives surface-weighted and volume-weighted average column-length distribution functions following Eqs. The broader the distribution, the larger the differences, because small crystallites contribute more to the surface-weighted average. That is much more evident comparing the surface-weighted column-length distribution with the volume-weighted.
If they have similar shape and maximum position, as in Fig. Conversely, if the surface-weighted distribution function has a sharp maximum toward smaller sizes, differences are larger see Fig. Surface-weighted and volume-weighted column-length distribution functions, normalized on unit area: [ ] Ag untreated upper ; [ ] Ag ground lower.
Surface-weighted and volume-weighted column-length distribution functions for [] La 2 CuO 4 , normalized on unit area. If experimental profiles are deconvoluted by the Stokes method, even for considerable specimen broadening, size coefficients A s usually oscillate at larger L values [ 59 , ], preventing computation of the column-length distribution function.
Few techniques were used to deal with this problem: successive convolution unfolding method [ 32 , ], smoothing, and iterative methods [ , , , ]. However, they follow from size coefficients A s that depend on the accuracy of the approximation for the distortion coefficient, given by Eq. Equation 47 is exact if the strain distribution is Gaussian, but in general holds only for small harmonic numbers n , if strain broadening is not negligible. To test the applicability of the discussed method to more complicated patterns, two compounds with lower crystallographic symmetry were studied.
La 2 CuO 4 is orthorhombic at room temperature. Both compounds show slight line broadening and relatively highly overlap-ping peaks see Fig. J Mater Sci 20, — Download citation. Received : 17 April Accepted : 23 May Issue Date : April Anyone you share the following link with will be able to read this content:. Sorry, a shareable link is not currently available for this article.
Provided by the Springer Nature SharedIt content-sharing initiative. Skip to main content. Search SpringerLink Search. Abstract X-ray peak broadening has been used to study the milling behaviour of a number of commercial alumina powders. References 1. Google Scholar 2. Google Scholar 3. Google Scholar 4. Google Scholar 5. Idem, Keram.
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