FWHM is the wrong type of peakwidth to use for this analysis. The physically correct parameter is the integral peak width, beta. beta is defined as the width of a rectangle that has the same height an area as the peak. The relation between beta and FWHM is a simple multiplication factor, beta = K * FWHM, which however depends on the shape of the peak. For a Gaussian K is pi/2 (approx. 1.57). For a Lorentzian K is 0.5*sqrt(pi/ln(2)) (approx. 1). Since the domain size (NOT crystallite size) in the Scherrer equation is inverse proportional to beta, a Lorentzian with the same FWHM will yield a value for the size about 1.5 times higher than a Gaussian of the same FWHM.

Of course, the selected peak shape influences the resulting value of the FWHM, too. 

I think  to calculate the crystallite size using the well known Debye - Scherrer Formulae is definitely using FWHM peak of 2 theta.  So, it is better to compare the result to that of TEM analysis since  those results are really closed. It is meant that crystallite size calculated using FWHM Lorentz or FWHM Gaussian Fits has to be compared to that of TEM analysis. Also the temperature to produce NiO will affect the crystallite size calculation using both fits.

Thank you Rudy for your prompt response. In both fittings, NiO calcined at 650C was used. I will try to do the TEM analysis.

FWHM is the wrong type of peakwidth to use for this analysis. The physically correct parameter is the integral peak width, beta. beta is defined as the width of a rectangle that has the same height an area as the peak. The relation between beta and FWHM is a simple multiplication factor, beta = K * FWHM, which however depends on the shape of the peak. For a Gaussian K is pi/2 (approx. 1.57). For a Lorentzian K is 0.5*sqrt(pi/ln(2)) (approx. 1). Since the domain size (NOT crystallite size) in the Scherrer equation is inverse proportional to beta, a Lorentzian with the same FWHM will yield a value for the size about 1.5 times higher than a Gaussian of the same FWHM.

Of course, the selected peak shape influences the resulting value of the FWHM, too. 

Thank you Daniel. thaw was an interesting explanation..I will check it out..

While calculating the crystallite size from the FWHM, strain in the material is not subtracted, when you use Debye-Scherrer formula. So, you can go for Williamson-Hall plot and separate the strain component and size component. Then, you can compare that calculated crystallite size with TEM. It works. 

U can't use single peak to calculate the crystallite size, unless it's single crystal or epitaxy (beyond common measurement technique)

Did have NIST SRM to get ur instrumental broadening?

βobs-βinst=βsize+βstrain

So u have to subtract the instrumental broadening from ur observed peaks.

With that u can use WH-plot:

βhkl cosθ= k.lambda /D + 4.epsilon.sin(theta)

D is the crystallite size, epsilon is the crystal imperfection and distortion.

U might consider WPPM for advance analysis of line broadening due to size and imperfection (P. Scardi und M. Leoni, „Whole Powder Patter Modelling,“ Acta Crystallographica Section A, Bd. A58, pp. 190-200, 2001).

I agree with Daniel Tobbens, we should use the method of Stokes and Wilson using beta. Notice that the well-known Scherrer formula was derived for a cubic system with a Gaussian distribution of crystallite sizes. I would like to add that after correcting for instrumental broadening, the resulting broadening is a combination of the crystallite size effect and the strain effect. These distributions are usually modeled (without rigorous physical proof) by Lorentzian for the size effect, and Gaussian for the strain effect. These effects should be deconvoluted in the analysis of the peak profile. In the process, one should have a very good signal and a reliable fit, especially in the region of the tail of the peak. Further, one should be careful that the relation provided by Manawan for the broadening related to two contributions is applicable for a Lorentzian line shape; Gaussian line shape involves the square root of the sum of squares of two contributions. Accordingly, both instrumental and observed line shapes should be deconvoluted into Lorentzian and Gaussian components, and the "true" broadening arising from crystallite size and strain effects can be obtained by using the correct algebra for each line shape.   

Sami Mahmood: I use this W-H plot or Stokes-Wilson plot. But I have a doubt. For crystallite size we use Lorentzian function and for strain we use Gaussian separately. Can we use Pseudo-voigt function, which is said to be the overlap of both Gaussian and Lorentzian. What is the error associated when I use this function?

WH plot only give u a mean size, which always smaller than TEM distribution (look for P. Scardi and M. Leoni works). And, what is the particular defect if it's give u only %strain ? Further more, It's very depend on how good is ur refinement, not easy for Line Profile Analysis if the peaks is highly overlapped  and must close to ideal powder sample for Rietveld analysis.

The error quite small for isotropic shape, but not for anisotropic shape (if u not group the corresponding peaks). Below is WH-plot from Barium Hexaferrite. The dash-dot-dot linear plot shows 12% deviation with similar weighted residue to continuous line.

A. Surendra Babu: This is exactly what is usually done in a full profile fitting.  The method of selecting Lorentzian and Gaussian line shapes is carried out for single peak analysis. When applying the two methods  (the single peak fit and the whole profile refinement) you will find differences which could be significant. This is normal due to differences in background fitting for the two methods, and local variations in the diffraction pattern. Notice that when adopting the single peak approach applied to the (hkl) reflection, the calculated size is the average crystallite dimension in the [hkl] direction; the whole pattern fit uses all reflections, and therefore gives an average of the crystallite dimension as a whole. The results of the two methods should, in principle, give consistent results for spherical-shaped crystallites, provided that the quality of peak and background fitting is similar in the two methods. This, however, is not attainable in real samples in which the particles could be rod-like or platelet-like for example.  Reliable results could be obtained when the noise level approaches zero relative to the signal (peak intensity), ad the quality of fit is excellent, especially at the tail of the peak. However, one should keep an eye on: what average dimension am I looking at? 

Note to Daniel Tobbens: I think the relations you have presented between beta and FWHM for the Gaussian and Lorentzian line shapes were inverted.

For Lorentzian: β = FWHM ×π/2 

For Gaussian: β= FWHM×(1/2) √(π/ln2)

For details see: J. Ian Langford and Daniel Louer (1996), Powder Diffraction, Rep. Prog. Phys. 59(2) 131. 

The last relation gives the famous "Scherrer constant" of 0.94 in the well known Scherrer equation (Warren, X-Ray Diffraction, Addison Wesley, Reading, Massachusetts, 1969). 

Note to Lishil Silvester: Waht was the "Scherrer constant" you used in applying the Scherrer equation to the Lorentzian fit? If you used a value of 0.94 (which should be correct for Gaussian fit, NOT Lorentzian) then divide your answer by 1.571 to obtain the correct answer for the adopted line shape. 

I prefer Voight function fit, which represents intermediate function between Gaussian and Lorentzian for identifying the breadth of the beak. The size in this cause becomes more accurate in terms of Sherrer relation for powders.

Yes Prof. Shaaban,  that is what we routinely do. A diffraction peak usually cannot be fitted satisfactorily with one component. However, these two components, as you know, are associated with different contributions to the breadth of the peak, namely, the size effect and the strain effect. The strain gives extra broadening which, if not filtered out in the analysis, would yield crystallite size smaller than it actually is. For accurate determination of the crystallite size we normally consider ONLY the Lorentzian component contribution to the broadening.  This of course has to be corrected for instrumental broadening. 

i agree with Dr. Shaaban and Dr. Mahmood abt the Voigt Function fit. In Origin use the Voigt 2 (peak fitting function) fitting. It gives the FWHM of L as well as G component. Covert them to integral breadths and then use them to separately calculate the strain and size. If you want to be really through do the double-voigt analysis.

Here are some references that may help:

X-ray Diffraction Fundamentals:

• a. http://pd.chem.ucl.ac.uk/pdnn/peaks/peakcon.htm 
• b. http://pd.chem.ucl.ac.uk/pdnn/pdindex.htm#peaks 

http://www.bgmn.de/device.html

Thank you Sami....yes I used 0.94 to calculate using gaussian. Why can't I use 0.94 for Lorentz fit?? Why should I divide finally by 1.571?

Thank You Essam I will try using Voight fit....

Thank you Loveleen

Thank you Ravi for the links

For Lorentzian line shape, the integral broadening is beta = (pi/2)*FWHM = 1.571*FWHM, and for a Gaussian line shape beta = 1.06*FWHM. The particle size is, according to Stokes-Wilson formulation, = lambda/(beta*cos(theta)). The 0.94 factor appearing in Scherrer equation is simply (1/1.06), which is the constant needed to convert the integral breadth to FWHM for a Gaussian distribution. For the Lorentzian distribution, however, the relation between the integral breadth and FWHM is different as I illustrated in my previous answer, where beta = 1.571*FWHM in this case. Accordingly, the constant (k) in Scherrer equation should be 1/1.571 = 0.64 for Lorentzian line shape rather than 0.94..

Thank you Sami....The answer is very clear...I really appreciate your active and helping mind.

Dear Dr. Mahmood, can you please give a reference for the value of k = 0.64 for Lorenzian line sahpe

Dear Loveleen

See for example J.I. Langford and D. Louer, Eq. 2.6, A.2, A.13 [1], which report on the relations between the integral breadth and FWHM for the various distributions. In that reference, the constant corresponding to that in Scherrer equation is termed (phi).

[1] J.I. Langford, D. Louer, Powder diffraction, Reports on Progress in Physics 59 (2) (1996) 131.

Hi Sami Mahmood,

Here is your sentences "Accordingly, both instrumental and observed line shapes should be deconvoluted into Lorentzian and Gaussian components, and the "true" broadening arising from crystallite size and strain effects can be obtained by using the correct algebra for each line shape.   " My question is that when I got the deconvoluted Lorentzian and Gaussian for both samples and instrument, and then use FWHM_true_gauss=SQRT(FWHM_sample_gauss^2-FWHM_instr_gauss^2) and FWHM_true_lorentz=FWHM_sample_lorentz-FWHM_instr_lorentz. For next step, what algebra, formula or what plot should I fit to get the size and stain values ?

Thanks

Dear J. Wang

The procedure discussed and used to separate the instrumental broadening from the observed peak profile may be useful as a quick and easy way to obtain a crude estimate of the crystallite size with significant uncertainty,  The deconvolution procedure, I am afraid, is more complicated. When analyzing a given peak profile, the observed peak function h results from the convolution of the sample broadening function f and the instrumental broadening function g.  The function g contains all contributions resulting from slit width, poor beam focusing, unresolved α1 and α2 lines and the widths of these lines, etc. This g line shape can be obtained from the pattern of a standard sample with very large crystallite size so that size effects are eliminated.  The sample contribution f is determined by deconvolution using Fourier analysis in the Stokes correction procedure. The Fourier components of the f-curve are given in terms of the Fourier components of the observed h and g curves, where

F(n) = H(n)/G(n)

These F(n) components are used to construct the f-curve, which is free of instrumental effects. The resulting curve is then fitted to obtain the integral breadth β = (peak area/maximum intensity). Using this integral breadth (not the FWHM), the crystallite size is determined from Stokes and Wilson formula

D = λ/βcosθ

This formula has the advantage that you do not have to worry about the line shape and use the correct constant such as K = 0.939 for a Gaussian distribution used in obtaining the Scherrer formula. (Note: I believe that the constants quoted in Tobbens answer for the Lorentzian and Gaussian line shapes are switched).

Notice that if the f-curve has a Lorentzian line shape, then its breadth is associated with the grain size effect. If, however, the curve is fitted by a combination of Lorentzian and Gaussian lines, the breadth of the Lorentzian line is used in the Stokes and Wilson formula to determine the average crystallite size.

Try to use your estimate of the Lorentzian breadth in Stokes-Wilson formula above, but be careful to use the integral breadth rather than the FWHM, where β = (pi/2)*FWHM. For a more accurate result, however, a proper deconvolution procedure should be adopted.

Further, it should be noted that the f-curve could be fitted a Voigt function when the ratio FWHM/β is between the limiting values of 0.637 (=2/pi) for a Lorentzian, and 0.939 for a Gaussian. Also, other functions such as Pearson VII function can be used to obtain β.

Do remember to separate (deconvolute) "size effect" and "strain effect" on the FWHM/β for precision :-)

Yes of course. This is normally done by fitting the peak function (which is first freed from instrumental broadening by the deconvolution procedure discussed above) by a Lorentzian representing crystallite size effect, and a Gaussian representing strain effect.

" Lorentzian representing crystallite size effect, and a Gaussian representing strain effect" -