Dear Anup,

please have a look at the  Scherrer (not Debye-Scherrer) equation ( see 1st wikipedia link) and the Williamsen-Hall expression (2nd wikipedia link)  at the section 'size and strain broadening'. When setting the strain eta to zero the Williamsen-Hall equation reduces to the Scherrer equation.

Thus Williamsen-Hall equation is more suited for crystallite size determination because XRD peak broadening due the strain is in addition taken into account.

https://en.wikipedia.org/wiki/Scherrer_equation

https://en.wikipedia.org/wiki/Powder_diffraction

Dear Anup,

please have a look at the  Scherrer (not Debye-Scherrer) equation ( see 1st wikipedia link) and the Williamsen-Hall expression (2nd wikipedia link)  at the section 'size and strain broadening'. When setting the strain eta to zero the Williamsen-Hall equation reduces to the Scherrer equation.

Thus Williamsen-Hall equation is more suited for crystallite size determination because XRD peak broadening due the strain is in addition taken into account.

https://en.wikipedia.org/wiki/Scherrer_equation

https://en.wikipedia.org/wiki/Powder_diffraction

Thanks Sir

The Williamson Hall formula is a more advanced form of theScherrer equation that also provides strain in crystals.

The Williamson Hall method is more accurate and better.

@ Mohsen Mehrabi: First you should read the already made contributions. Nevertheless, I am open for news but I never heard about Debye's equation, only Scherrer equation. Where I can find anything about Debye's equation which is related to crystal strain? Is this the alternative to Scherrer's equation for the average of crystal size?

The William Hall plot is a more accurate and advance for crystalline size determination. Here, the peak broadening in XRD occurs due to strain which is additionally taken into consideration. Therefore the Williamson Hall method much better than scherrer formula.

@ Gert: .... and again:   ..... keep smiling ....

@Gert Nolze: Thank you for your good reminder. The following file is suitable for completing the discussion.

Thanks a lot for this paper, Mohsen Mehrabi. It again points out that what Gerhard Martens already pointed out partially in his first comment. There is a Debye-Scherrer TECHNIQUE, a Scherrer equation for the estimation of crystal (no particle!) size, a Debye equation which has nothing to do with size and strain in crystals, and a Williamson-Hall plot (equation, method) which enables also for VERY special assumption (like Scherrer) an extraction of two numbers which give some indication for size and strain. Better than nothing but only in very special cases exact. Therefore, I am not convinced that the Scherrer equation is much worse than the Williamson-Hall plot. Both are limited models which try to describe two complex features of a polycrystalline material by two simple numbers. 

@Gert Nolze

It is true. Do you think there is a better way (from the two methods above) to determine the particle (or crystal) size (nano and micro)?

@ Mohsen

Thanks for that holzward2011 paper. Nice look into on the history.

@ Mohsen: In fact it is not the question of the crystal size Who is really interested in the crystal size and what does a number mean if you know that crystals are never isometric, e.g. a sphere? We try to define correlations using models which are MODELS. It is up to us to decide whether these models match to a problem or not. Maybe, the crystal size has primarily nothing to do with the effect we are observing. It seems to be a reflex nowadays to apply XRD in order to define the lattice parameters and then ... possibly since we do not get any other values... we simply apply existing formulae without any meaning to the present problem and start to calculate size and strain. It is available, and it looks professional. But is it therefore useful if there is no primary correlation? Presently, I don't know a better technique for crystal size determination, but I doubt how useful it is since you always have a size distribution and not a single value...even for a single grain a number only displays an equivalent approximated from 2D to 3D. And there is still the particle distribution which can be determined by Laser techniques and might show a better correlation to an observed phenomenon. 

Thanks Gert for pointing to the fact that the crystallite  size  (either derived from Scherrer equation or from Williamsen-Hall) is some kind average number, which does not reflect crystallite size distribution or even crystallite shape and crystallite shape distribution. Everyone dealing with that number either in deriving it from XRD pattern or interpreting  the result should be aware of this fact; and therefore that number should not be stressed too much. Thanks again for that hint.

@Gerhard Martens

You're welcome. It was also very interesting for me.

Important or not, depend on your sample, data quality, calculation or model that you use and how you verify it (TEM). Scherrer equation is not enough for high quality data if there's size-strain effect present. A rule of thumb: "garbage in garbage out", or it can be "good data in garbage out"

Despite what Gertz pointed out; it just a mathematical model that try to represent the real phenomenon. It is our task to go through all those literatures before make such assumtion. Scherrer (1918) noted that the line profile width is inversely proportional to crystallite size, Laue (1926) realized that using the integral breadth (IB) gives approximate independent effect of the distribution in size and shape, Stokes & Wilson (1944) introduce strain broadening e0 represents the "mean"deviation from the undistorted state, Bertaut (1949) proposed an interpretation to consider size broadening without making assumptions on crystallite size and shape distributions. Jenkins & Snyder (1996) discussed about Bragg extinction; crystallites larger than 2mm typically have a sufficient number of planes to display its Darwin width. Balzar (1999) modeling of microstructure effects; crystallite size and microstrain comprise Lorentzian and Gaussian component convolutions varying in 2theta, Scardi & Leoni (2001, 2004), Scardi et al. (2010) the WPPM approach for microstructure analysis. And there's a lot more.

The interpretation of size values is also difficult, as different methods may report different quantities such as volume weighted mean, area weighted mean and number weighted mean.

You could try to refine the data from IUCr projects on Size-strain round robin ceria: http://www.iucr.org/resources/commissions/powder-diffraction/projects

Excellent discussion exposing the potential equivocation while interpreting conventional XRD data. Good historical perspective.

Choosing the right "reference standard" and the right Nano structural model are key to successfully employing any of these tried and tested XRD methods.

Tamas Ungar's work on line profile analysis will be helpful as well. There are a series of lectures (I-IV) by Prof. Ungar at Indian Institute of Technology - Kanpur

https://www.youtube.com/watch?v=kGQb8EiSQAw

From both the equations we observed that if strain (eta) which occurs due to peak broadening is zero then Willimsen hall equation converted into Scherrer formula. Therefore, the Williamsen-Hall equation is nothing but it is the mother of Scherrer formula. So we can say that Willimsen Hall equation is more precise to calculate the crystallite size.