sorry guys but you are making a lot of confusion here... physics is the key.

The explanation is simple if you consider the relationship between the object (real space) and the diffraction pattern (reciprocal space). Diffraction is simply a good way to get the Fourier transform of your object (and people rediscovered it with coherent imaging). If you take the Fourier transform of lattice confined in space, you obtain another lattice (the reciprocal lattice): the size and shape of each point in one lattice is the Fourier transform of the boundary of the other space. Remember we talk about size and shape. So for a spherical domain of size D, the reciprocal lattice will therefore have points whose size is inversely proportional to D and an extension which is inversely proportional to the size of each atom.

We usually also have the instrument playing a role, but this effect can be properly taken into account

Now Scherrer equation is not an "empirical formula". It is exact IF the hypotheses under which it is derived are respected. Now this is usually the problem, as it gives the size of a cube-shaped domain (monodispersed) assuming a Gaussian approximation of the corresponding profile. This condition is seldom (if never) met in practice. Specimens are dispersed and the peaks are not Gaussian. Nevertheless people keep using the formula and also citing the 1918 paper of Scherrer (often in a wrong way). Refereeing is usually quite poor when it comes to diffraction (and absent in most cases where Scherrer formula is used quantitatively). We have demonstrated several cases that it is possible to go beyond the limitation of Scherrer formula using a physically-sound whole pattern approach (such as the Whole Powder Pattern Modelling) and to properly consider the shape, size, size distribution and defects.

As for "disorder", we enter into chaos. What is called "disorder" is often just the presence of defects in the material. Point defects usually gives diffuse scattering features. Line defects mainly affect the profile. Stacking defects affect both the profile and the whole pattern introducing also some diffuse features.

As for the surface of the material or the grain boundaries, well... I still would not say anything as it is still a matter of debate: fully ordered and fully disordered models do not work well... but in any case if you know the atomic positions, you can use Debye scattering equation to get the corresponding diffraction pattern: beware, though, as it is the diffraction pattern of just that particular object... and in the specimen you analyze there can be millions with different shape, size, defect, surface etc..

have fun!

Because smaller grain means more boundaries => more defects

Then from a lorentzian you evolve to a Gaussian

Two explanations related to the fundamental physics behind the technique as this effect is notable in perfect crystals even though they are nano-sized. 1) One can see the atoms in the crystal as individual "light sources" or slits in a grating. When you reduce the domain size you get fewer of these "sources" and as with a standard grating for optical experiments the resulting spots become less defined. 2) Even more fundamentally one can see the fact that reduction of the grain size is the same as locating the photons in a confined space. As we know from QM this makes the energy of the photon uncertain i.e. the unsharpness principle, which in turn leads to an inaccuracy in the wavelength and less sharp peaks.

For an approximation of the crystal sizes one can use the Sherrer equation though this is a purely empirical formula.

I would have a different understanding for the background of this question:

the nanoscopic materials will have a different process history (how they are made) compared to the macroscopic material, hence the crystals will be less ordered; plus, there are many more (the smaller the particles, the bigger the proportion) atoms on the surface which can not be ordered the same way as the atoms in bulk, so more disorder.

Bernhard, XrD is not a surface technique though and a general disordering would lead to a lack of peaks, not a widening of them. This widening is observed whenever the crystallite size becomes small, generally below 1000 nm and hence the number of surface atoms is still small compared to the bulk.

sure, XRD is not s surface technique, but nevertheless also collects "information" from the surface, it collects x-ray diffraction from everywhere, from inside the particles and from the surface.

In wikipedia (http://en.wikipedia.org/wiki/Scherrer_equation), one can find this:

"... a variety of factors can contribute to the width of a diffraction peak besides instrumental effects and crystallite size; the most important of these are usually inhomogeneous strain and crystal lattice imperfections. ..."

Above, I talked about "more disorder" being responsible for a peak width broadening, I think the explanation in wikipedia tells the same in other words.

That XRD collects surface information is correct, but this is in most cases negligible as the percentage of surface atoms only becomes large for very small crystals (typically less than 5 nm). The fact that Wikipedia points to other factors influencing the peak width has more to do with the fact that domain sizes from the Sherrer equation should not be taken as definite.

The effect of peak widening because of the points I had above is thinking as this from a physical point of view. Even with perfect crystals of a small sizes one would observe peak widening and this effect is the predominant effect! This is what most textbooks list and is what I would give a 9 out of 10 for answering, a 10/10 would also mention the instrument effects and possibilities of strains in the case of the smallest crystals.

sorry guys but you are making a lot of confusion here... physics is the key.

The explanation is simple if you consider the relationship between the object (real space) and the diffraction pattern (reciprocal space). Diffraction is simply a good way to get the Fourier transform of your object (and people rediscovered it with coherent imaging). If you take the Fourier transform of lattice confined in space, you obtain another lattice (the reciprocal lattice): the size and shape of each point in one lattice is the Fourier transform of the boundary of the other space. Remember we talk about size and shape. So for a spherical domain of size D, the reciprocal lattice will therefore have points whose size is inversely proportional to D and an extension which is inversely proportional to the size of each atom.

We usually also have the instrument playing a role, but this effect can be properly taken into account

Now Scherrer equation is not an "empirical formula". It is exact IF the hypotheses under which it is derived are respected. Now this is usually the problem, as it gives the size of a cube-shaped domain (monodispersed) assuming a Gaussian approximation of the corresponding profile. This condition is seldom (if never) met in practice. Specimens are dispersed and the peaks are not Gaussian. Nevertheless people keep using the formula and also citing the 1918 paper of Scherrer (often in a wrong way). Refereeing is usually quite poor when it comes to diffraction (and absent in most cases where Scherrer formula is used quantitatively). We have demonstrated several cases that it is possible to go beyond the limitation of Scherrer formula using a physically-sound whole pattern approach (such as the Whole Powder Pattern Modelling) and to properly consider the shape, size, size distribution and defects.

As for "disorder", we enter into chaos. What is called "disorder" is often just the presence of defects in the material. Point defects usually gives diffuse scattering features. Line defects mainly affect the profile. Stacking defects affect both the profile and the whole pattern introducing also some diffuse features.

As for the surface of the material or the grain boundaries, well... I still would not say anything as it is still a matter of debate: fully ordered and fully disordered models do not work well... but in any case if you know the atomic positions, you can use Debye scattering equation to get the corresponding diffraction pattern: beware, though, as it is the diffraction pattern of just that particular object... and in the specimen you analyze there can be millions with different shape, size, defect, surface etc..

have fun!

you can see answer this question in this book:

A. R. West, Solid State Chemistry and its Applications, John Wiley & Sons, (2003) 173-175.

The Scherrer equation in X-ray diffraction and crystallography, is a formula that relates the size of sub-micrometre particles, or crystallites, in a solid to the broadening of a peak in a diffraction pattern:

So the Particle Size = 0.9λ/βcos(θ)

where:

λ is the X-ray wavelength;

β is the line broadening at half the maximum intensity (FWHM), after subtracting the instrumental line broadening, in radians.

θ is the Bragg angle.

Hence β (FWHM) which is inversely proportional to particle size and if former decreases latter increases.

Hope this helps :)

@Abhinav, please don't talk about "sub-micrometre particles, or crystallites" as this is conceptually wrong. And please do not use the term "particle" when talking about Scherrer formula. Diffraction gives information on the coherently scattering regions of the specimen and the maximum coherence length is what limits it. Scherrer formula translates into some values (see my comment above) a profile width: it is therefore a consequence of the fact that "size" and width are roughly inversely proportional and not the reason! The physics of the problem is the reason

The reason is that smaller grain means more boundaries and more defects

Then from a lorentzian you move to a Gaussian

Lots of great input to study thoroughly! I'll review each and comment. Thanks in advance to all contributors for freely sharing!

@Matteo; Sir very nice explanation.

Thank you Matteo to fully answer the question with your detailed answer above. I would like to reiterate what Matteo has written in the second paragraph. Just because we may see much written about the 'Scherrer equation' in the open literature doesn't mean its all correct. Refereeing is often poor in this area - beware!

Tim, thanks for the nice words. Refereeing is 50% of the story... the other 50% is people not reading nor participating to conferences or asking for collaboration with diffractionists on this matter.

The result is like the answer of Hatem (no offense intended) . The non expert might consider it as a key answer as there is some jargon there. Unfortunately it is completely nonsense. Peaks are neither Gaussian nor Lorenzian in a real case, especially in 2theta! The confusion is total: XRD does not measure grans (but domains) and it could be that domains remain the same when reducing the grain size. Moreover if you synthesize a nanopowder, you can get to cases where the defects tend to zero.s

Start with a distribution of domains around e.g. 40 nm and then simply start adding smaller and smaller domains.. the average will decrease, nevertheless the peaks will start to assume what people call a "superLorentzian" shape... line profile analysis CANNOT be reduced to a line width or to Gauss or Lorentz peaks

@Reza: you can find the answer in papers of the beginning of last century from the early diffractionists (and in the 40s this was clear also from a Fourier point of view)... i guess the book of Wilson (X-ray optics) is great from this point of view. I think I do not violate any rights pointing you to https://archive.org/details where those old books can be found. Full respect for prof. West who is a great chemist, but if I need some information on diffraction I prefer to ask a diffractionist!

 Dear Mohsen ,

reasons:

temperature, decrease grains in nanoscale  than bulk, Non-uniform strain , instrumental errors.

Regards,

Abbas

I think this link can help

http://www.instanano.com/characterization/theoretical/xrd-size-calculation/