Surprising; but via' Scherrer' and 'Williamsen-Hall 'you are on the right way.

You might share your diffractograms....and share what kind of sample do you deal with (thus a little bit  more information than 'nano structured')

Thanks in advance

The paper at the link may help you.

Article Modified Scherrer Equation to Estimate More Accurately Nano-...

G. Martens is righ from my point of view. At least there is no better approximation presently. However, I would in maximum call it grains size estimation. Calculation suggests that the value derived is exact which it isn't. In case your peaks really become less sharp (but this need a prove showing the real data) it will certainly underline your conclusion, anyway how logic it appears to you at the moment. Actually it would only express how limited our understanding and models are. But first lets see both diffractograms.

Thank you for your answers

As is known in the nanostructured material, grain size have trend to increases with increasing sintering temperature due to the grain growth that occur to decrease the system energy.

Well the crystallite size must be constant if a stability is realized or increase if the grain growth occurs.

So after doing my calculation and find a decrease in size I suggest that the variation in pic’s width is not due to the variation in crystallite size but in the microstrain that have trend to reducing due the relaxation.   

 So I am looking for a formula that take this in consideration.

Best regards 

So Williamsen-Hall is that approach which could do that job. because strain/stress broadening is addressed too (also when strain/stress is lowere/reduced). But you may have not enough peaks.

Sharing the XRD pattern could help .....

Thank you sir Gerhard Martens 

my sample is a nanostructured Aluminium 

in attachment the xrd pattern at 400 and 500°C

@ Gert; you may have a look at these files?

 I have not access to that format, sorry.

They are open

I reattached as image, i don't know if it can help!

Thanks

Thanks for sharing the diffractograms.

At a first glance (printing your pattern and comparing the peak widths simply by a ruler) are quite identical; difference less than 10%

How do you calculate the peak widths and what differences ( in %) do you get?

..and why such non-linear y-scale? (For having all-over the same noise?)

as Gerhard mention, noise background is appeared, your measurement time (s) is quite good but the peak resolution is not good enough (v-shape) appeared in the peak tip. first you have to get a quite good xrd pattern. instrumental broadening must be taking into account using LaB6 (even if it commercial powder from any vendor selling chemicals). crystallite size, microstrain and dislocation will cause broadening in peaks. please focus on the calculation of dislocation density. please check out these papers, where you can find a modified equations to calculate these parameters. 

The crystallite-size dependence of structural parameters in pure ultrafine-grained copper

X-ray analysis of ZnO nanoparticles by Williamson-Hall and size-strain plot

methods

X-Ray and Neutron Diffraction Measurements of Dislocation Density and Subgrain Size in a Friction-Stir-Welded Aluminum Alloy

X-ray peak broadening studies of nanocrystalline hydroxyapatite by Williamson–Hall analysis

DISLOCATION MODEL OF STRAIN ANISOTROPY (tamas Ungar) 

PM2K: A flexible program implementing Whole Powder Pattern Modelling (Matteo Leoni)

Good Luck

From Scherrer equation,

ᴦ = K λ / β cosѲ

where, β = FWHM = From XRD graph select any pick ( mostly highest) and calculate FWHM value ( use Origin soffware)

2Ѳ = angle of the selected pick

K = 0.9

λ = 1.54 A0

I am grateful for your help @ Mohammed M. Obeid @ Gerhard Martens

For Scherrer equation, crystallite size was calculated based on the measurement of a(hkl) peak using the following equation:

L =Kλ / Bsize cos θ

where L is crystallite size, K is a dimensionless shape factor (0.9), Bsize is line broadening at half of the maximum intensity (FWHM) in radian, λ is the X-ray wavelength for example for Cu Kα radiation (1.5406 Å) and θ is Bragg angle in degree.

Meanwhile, Williamson–Hall plot was used to estimate the crystallite size and lattice strain of the samples using the following formalism:

Btot = Bstrain + Bsize = 4Cε tanθ + Kλ/ L cosθ

where Cɛ is the lattice strain, Βsize is the particle size broadening, Βstrain is the strain broadening, L is the crystallite size, K is a dimensionless shape factor (0.9), λ is the X-ray wavelength for Cu Kα radiation (1.5406 Å) and θ is Bragg angle in degree.

Then Eq. 2 is multiplied by cosθ to yield:

Btot cosθ = 4Cε sinθ + Kλ/L

Hence, by plotting the graph of Βtot cosθ against 4 sinθ, the lattice strain, Cɛ of the sample can be obtained from the slope (gradient) while the crystallite size can be estimated from the intercept (Kλ/L).

Dear Dr Bendoumia,

The below reference might be useful to read:

D. Balzar, N. Audebrand, M. Daymond, A. Fitch, A. Hewat, J.I. Langford, A. Le Bail, D. Louër, O. Masson, C.N. McCowan, N.C. Popa, P.W. Stephens, B. Toby, Size-Strain Line-Broadening Analysis of the Ceria Round-Robin Sample, Journal of Applied Crystallography 37 (2004) 911-924

Additionally, it might be a good idea to take a look at the examples on how to calculate crystallite size and microstrain provided at the High Score Plus software.

Moreover, please use the MAUD program - http://maud.radiographema.eu/

Have a great day,

Husin