Dear Ahmad Bahmani,

I think explaning exactly how to perform a quantitative phase analysis is beyond the scope of an simple answer here and I recommend you to look this up in an modern x-ray diffraction textbook (on the internet you can also find several good introductions to this topic). In the following just some general remarks concerning your questions:

Basically, there exist the traditional methods (such as RIR) and the quantitative phase analysis using Rietveld refinement. In my opinion, there is no best method in general, but the method should be chosen for each individual problem (e.g. I would prefer Rietveld if strong peak overlapping occurs). The accuracy of the method strongly depends on the sample. In an ideal sample (no texture; no overlapping peaks) I would estimate an error of the calculated phase fractions of slightly less than 1%.

In an texture-free sample the intensity of a Bragg peak of one phase divided by its material scattering factor Rhkl is proportional to the phase fraction. Thus, to determine the phase fraction you need either one peak of each phase in the sample (assuming that all phases are crystalline) or one peak of the phase to be determined and one peak of a phase of known fraction (internal standard). If the sample is texured, analysis becomes much more difficult and you might average over several peaks for the traditional methods or add parameters for the description of texture in Rietveld refinement.

FWHM does not directly contribute to the phase fraction. However, it should be clear that when we talk about intensity concerning quantitative phase analysis, we talk about integrated intensity. Thus, a peak with fixed maximum intensity has a larger (integrated) intensity, the broader the peak is. Besides, large FWHM may decrease the accuracy of quantitative phase analysis due to peak overlapping or badly defined peaks because of overlap with the scatter of the background.

Best regards,

Timo Müller

Dear Ahmad,

Timo has given some hints above to solve your problem.

I would suggest a simple empirical approach which is based on the assumption that  the peak integrals, or the sum of peak integrals (Timo pointed already to the fact that you should use peak integrals instead  of peak height in order to get rid of FMHM dependences) are proportional to the concentration of the appropriate phase.

So please take the sum of peak integrals of each phase and perform a ratio of both sum values.  The relative concentration of phases which belongs to that ratio can be taken from a calibration curve. This calibration curve can be performed by yourself by mixing powder samples having known relative concentrations of your phases. So you will have the ratio of sum peak integrals as a function of the relative concentration of your phases. Thats it.

Preferred orientation issues  may affect your sample (Timo mentioned already above). So please mill your sample to a powder one or at least use a spinning sample stage.

Good luck.

Dear Ahmed Bahmani,

I have written a small Turbo-Basic program named 'analyze', which uses some calculated peak intensities (from crystal structure)  to extract the phase fractions.Your measured intensities of selected peaks can be related to the calculated peaks. I used this program years ago for superconductor research. If you are interested, I can send you the program and an example.

 Hermann Otto

Applause! Great answers to a rudimentary question :-)

Here are some other practically challenging issues to consider during quantitative analyses using powder XRD:

Using "known standard samples" for calibration is a reasonable empirical solution. Nanostructural model assumption and numerical methods are excellent options due to available present-day computing power.

https://www.researchgate.net/post/What_is_the_best_method_used_to_align_the_diffractometer_plane_axis_with_respect_to_the_incident_beam_direction?_tpcectx=profile_questions

https://www.researchgate.net/post/How_do_I_calibrate_the_diffractometer_2Theta_angles_with_this_powder_diffractogram_of_corundum_standard_from_a_Bruker_D8_XRD_instrument

Dear Ahmad Bahmani,

Look the link, may be useful.

http://www.lambdatechs.com/documents/224.pdf

Regards, Shafagat

Dear Ahmad,

XRD generally give you a Qualitative analysis. You can calculate and find Quantitative analysis but with some error.

FWHM just related to Crystallite size

These errors are generally due to incorrect assumptions and models.

The first common example being the use of FWHM in place of "integrated breadth" (beta) by many XRD exploiters :-)

As I have discussed before, I just upload the source code analyze.bas and an example data-file as part of a short Researchgate note wit the title

Quantitative X-Ray Phase Analysis with the program 'analyze' ......

Yon can simply use the compiler QB64. I wish you many success.

Hermann Otto from Bavaria, Germany

Hi, dear Bahmani,

According to the below reference, you can calculate R-value for each peak, and then by division of intensity (I) to the R-value, the peak gets normalized. At the end:

1 Vol% = sigma (I1/R1)/(I1/R1)+(I2/R2)+(I3/R3)+ ...

J. Robertson, Elements of X-ray diffraction by BD Cullity, (1979).