Dear Priyadarshini... if peak intensity decreases, there is no reason why fwhm should decrease as well.. Take some powder, measure it, then take a very small quantity of the same powder and measure it again. You'll see a decrease in the intensity!

As for what Christopher says, it's true that you should always use a combination of techniques whenever possible. However Scherrer formula is quite obsolete for a quantitative analysis of the microstructure via XRD. Using modern analysis techniques (such as the WPPM method), you can get information that are of the same quality and in several cases of better quality and higher statistical value than TEM observations. Unless you believe that observing a handful of particles in section is better then measuring millions and having the directional information! In any case you can obtain dislocation and fault densities by XRD on ranges not accessible by TEM...

the peak intensity depends on the volume of diffracted matter. It is therefore related to the integral of the size distribution and it is not possible from the integral to obtain information on the mean.

My suggestion in any case is that you open a good and recent book on X-ray diffraction. You would immediately see that there is no such a thing like the "Debye Scherrer formula". The formula is due to Scherrer only.

Due to the large number of people from India coming up with the concept of "Debye Scherrer equation", I start suspecting there is some book or reference there that reports it wrongly, or people keep citing something they have heard, without actually caring of what the formula or its meaning is

Thank you for your suggestion Dr. Leoni. Regarding first part i want to clarify myself. Is it possible with decrease of peak intensity, the fwhm will decrease thereby increasing crystallite size?

As a general rule, the peaks in the XRD will broaden as crystallite size decreases. As Matteo Leoni describes, the intensity depends on the X-ray flux and the quantity of crystalline material. The so-called Scherrer equation is commonly used in the literature particularly in the charcterization of nanostructured materials but unless the peaks are fitted to a structure model it is really only an estimate and is often used in conjunction with microscopy measurements for instance. Another general issue with such an analysis is that it provides an estimate of the size only of the crystalline domains. Therefore, in a highly polycrystalline material the use of the equation would not be very helpful in estimating particle size. In my opinion, high resolution TEM provides the best way to investigate crystallite size.

Dear Priyadarshini... if peak intensity decreases, there is no reason why fwhm should decrease as well.. Take some powder, measure it, then take a very small quantity of the same powder and measure it again. You'll see a decrease in the intensity!

As for what Christopher says, it's true that you should always use a combination of techniques whenever possible. However Scherrer formula is quite obsolete for a quantitative analysis of the microstructure via XRD. Using modern analysis techniques (such as the WPPM method), you can get information that are of the same quality and in several cases of better quality and higher statistical value than TEM observations. Unless you believe that observing a handful of particles in section is better then measuring millions and having the directional information! In any case you can obtain dislocation and fault densities by XRD on ranges not accessible by TEM...

Hello Priyadarshini, The measured intensity in XRD is always in arbitrary units. And there are several factors that influence the intensity like the operating kV/mA, crystallinity of the specimen, sensitivity of the detector, etc. So, you should not draw too much out of it. Rather, the most important observable is the FWHM, which is a direct measure of the crystallite size (assuming the instrumental parameters are not changed) and as Matteo said Scherer eqn is obsolete now and better if you try any method that fits the whole pattern from structural and microstructural considerations and results should eb interesting.

Thank you Dr. Leoni and Christopher. I almost clarified myself from your suggestion.

Can you give me some idea on Modified williamson hall equation?

Priyadarshini, you better spend some days reading some literature and some modern (I mean the last 10 years) books on microstructure analysis by XRD.

Modified Williamson Hall (MWH) is easy to explain. Let's suppose you have a bad migraine. You tell a doctor and he gives you a cure for tendinitis. The MWH is just an improved cure for tendinitis!

The MWH is an arbitrary combination between Scherrer formula (valid if peaks are Gaussian) with an anisotropic model or microstrain (valid or Lorenzian peaks). So if your migraine is related to tendinitis you have the cure otherwise you better look for a better doctor

Williamson-Hall equation is just addition of size broadening part and strain broadening part. The linear fit of the plot will give us crystallite size and strain. But when all the points will not come under linear fit then we will go for MWH, as u have told improved cure for tendinitis i.e inflammation of tedons. I don' t know if i am properly catching your suggestions? If yes, then this MWH consists of some additional parts like average dislocation density, Burger' s vector of dislocation and contrast factor of dislocation along with size broadening part. Then how to calculate these factors for any system for ex. gold?

In W-H plot, if u plot beta cos theta vs. sin theta, then u get strain from the slope and the size from the intercept. What Matteo Leoni suggested by MWH eqn is incorporation of the strain component in the Scherer eqn. If all the points don't lie on the same linear fit (i.e. a scatter) then it indicates anisotropy and is often the case when dealing with metallic systems. The reason for this is the anisotropic elastic constants in metals and alloys. I hope you are not dealing with a deformed metallic system and can hope that the scatter may not be very prominent, although not very sure about it. If you are dealing with gold thin film then the issues of residual stress is expected to be prominent and this can alter the slope and you might even get a negative slope (indicating compressive strain). Now the MWH eqn that you talk about is based on the methodology Tamas Ungar and his coworkers and is based on the assumption that strain anisotropy is due to dislocations and the WH eqn has to be corrected in such cases. See some publications of Prof. Ungar which might give u some idea on the modification of the conventional WH eqn with average dislocation contrast factors (Chkl). Chkl and Ch00 are related by simple eqns for a cubic system and Ch00 has to be determined from the elastic constants of your system.

To conclude, like Matteo did, I would also advise you to read some good recent text books.

Dear P. Sahu Thank you. I agree with your opinion.

''Thin Film Analysis by X-Ray Scattering'' by Mario Birkholz can be a good book for you. This of course doesn't mean there are no other books.

well it seems my answer was not clear enough...

- Scherrer equation was developed for GAUSSIAN peaks and CUBE shaped domains

- Williamson Hall plot SUMS the broadening due to Scherrer equation and strain in the GAUSS (size) - LORENTZ (strain) approximation. Mathematics says that you can sum the two only if the peaks are GAUSSIANS

- Scherrer equation does NOT give you the average domain size (it's easy to demonstrate.. and you get a value related to a ratio between high moments of the domain size distribution) https://www.researchgate.net/publication/11820234_Diffraction_line_profiles_from_polydisperse_crystalline_systems

- the term due to dislocations is calculated assuming LORENTZIAN peaks with the anisotropy due to the contrast factor

- it can be demonstrated that the peak due to dislocations is not properly a Lorentzian and it's not even a Gaussian (check any of the publications on dislocation broadening e.g. )

- the WH or MWH analysis is usually done by considering thepeaks as VOIGTIAN (i.e. intermediate between Gaussian and Lorentzian. Little changes if they are pseudo Voigt or Voigt)

Now can you give me any physical reason why this mess whould give some realistic number? Do you think the MWH really adds that much physics to the whole picture?

Article Diffraction Line Profiles from Polydisperse Crystalline Systems

One approach is to use "known" standard samples to compare with.

Determining peak shape is important. Our experience with multitudes of materials shows several shapes including pseudo-Voigt, Gaussian, Pearson VII. We've even observed changes topographically on mono-crystalline samples. Most likely due to the variations in the defect types topographically.

Examples of Bragg Profiles with various shapes (pseudo-Voigt, Gaussian, Pearson VII):

1. Mono-crystalline substrates: http://www.flickr.com/photos/85210325@N04/8446683472/sizes/h/in/photostream/

2. Epitaxial film: http://www.flickr.com/photos/85210325@N04/8561945111/sizes/h/in/photostream/

3. Powder diffraction: http://www.flickr.com/photos/85210325@N04/8434970459/in/photostream

I've always felt more comfortable using a known standard sample to compare with. This potentially eliminates the challenge of variations based on each instrument used (Beam Size/Shape, Monochromation, Collimation, Slits, Detector, Optics, Dimensions etc.).

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Original elements of Priyadarshini Dash’s question include:

1. Bragg Peak Intensity (function of thickness, diffracting volume, DD-diffracting domain size, preferred orientation)

2. Bragg FWHM (DD size, instrumental effects, strain effect, etc.)

3. Poly-crystalline thin film (substrate? Interface strain? Miss-match?)

4. Correlate to derive formula leading to crystallite size (possible to create a chart using “known standards”)

Logically, one would have to understand the various components involved.

a. What is your material and how was it produced? Film & substrate?

b. Are the constituents crystalline, amorphous, what?

c. Do you have TEM or other data/images?

d. Do you have a model for what you expect the film to look like?

There are several variables that would affect the XRD peak intensity & FWHM in this scenario including:

1. Film thickness

2. “Crystallite Size” (need to define)

3. Degree of order (spatial correlation factor)

4. Amorphous component content & dispersion

5. Preferred orientation.

Below is an example of a real time 2D transmission XRD pattern form Aluminum foil ("thick film" about 200µm). You will realize that using the 1-D Conventional Diffractogram with a 0D scintillation (point) counter (as you are most likely using) will “lead you nowhere in a hurry” with the presence of preferred orientation.

Aluminum Foil (1D Conventional Diffractogram compared with 2D real time XRD):

http://www.flickr.com/photos/85210325@N04/7944785794/in/set-72157632728981912

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"this MWH consists of some additional parts like average dislocation density, Burger' s vector of dislocation and contrast factor of dislocation along with size broadening part" This is where I get lost in the whole "convolution" business.

As Priyadarshini queries, would the Burger's vector be a function of the (hkl) we are analyzing? How's it determined?

The key is, the correct deconvolution of the individual contributing components.

I could use any helpful suggestions and references in these regards.

I think this link can help

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