sorry for the apparently rude answer, but have you considered reading a book on diffraction? At least you have some verified and detailed answer, avoid useless headaches or waste of time and get also some formulas to play with... It would also learn the right terminology and start separating single crystal from powder diffraction, cause things are getting a bit odd here... the "repeatabiity", is actually called multiplicity and the "structural factor" is the structure factor. The multiplicity enters into account just for powder diffraction! For a single crystal you don't have it... the structure factor can already be different for different peaks and it is the one that give the intensity according to atom types and position/symmetry. You would get the same value if you would actually take the Fourier transform of the 3D lattice numerically.

From the mathematical point of view things go well beyond Bragg and Laue equations that give you just the geometrical part of the problem, whereas the structure factor, with the multiplicity for powders give the intensity in the static case (and in the easy case of 3D periodic objects and kinematical approximation). Then you would need thermal effects etc... In my opinion the older books are still the best ones for a start (and you can find them for a very low price). For the most recent, you can get the whole picture of Crystallography and Diffraction from the International Tables for Crystallography from vol A to vol H. Powder Diffraction is in the latest volume H.

Concerning the older books, I would suggest two: the one of Zachariasen is the most complex/complete from the mathematical point of view, whereas that of Warren is complete but much simpler in the treatment. I still use it a lot. For the basics they are good. For the most advanced stuff they are a bit obsolete, but you better start to learn how to walk before you run! You should be able to calculate the theoretical pattern e.g. for NaCl. As I already said in my previous comment, all the formulas you find are an easy way to predict the Fourier transform of your 3D periodic lattice without calculating it explicitly.

Diffraction seems easy...

As we know that the intensities in the XRD pattern depend on the various factors. According to that we see different intensity distribution in the pattern. Find the attachment below.

X-ray powder diffraction intensities of crystalline materials can be different from the randomly oriented pattern because of the nature of materials and crystallographic preferred orientation.

The effect of preferred orientation can be minimised by mill-micronising the samples, and mounted the samples into the XRD sample holder by side-drifting techniques. Then, use the sample spinner during the data collection.

During the Rietveld refinement of all the XRD data sets, the March model or generalised Spherical harmonics approach can be used to correct the intensities due to the effect of preferred orientation (O'Connor et al 1991 & 1992, Adv. X-Ray Analysis; Sitepu (2002). J Appl Cryst; Sitepu, O'Connor and Li (2005). J Appl Cryst;; Sitepu (2009). Powder Diffraction Journal).

I hope the above description help you in conducting your research.

With best wishes,

Husin Sitepu

Prof Husin , thanks for your answer.

But what I want to know is that for each diffraction all the centres contribute constructively. Still they have different intensities. For example for pure alpha iron all the peaks are generated by constructive addition from all the centres. It may be in powder form or single crystal form. So without direct result from formula , I want to know the physical reason. Wanting for reply. -- Regards

Dear Vinci

Thank you for your valuable answer and providing the valuable reference.

But the way I want to think is a different one. As we know that the optical grating is basically a Fourier transform device. The crystal , in the same sense be thought of as a three dimensional Fourier device in the X Ray region. So I want to keep on a unified way with a single mathematical operation , operating on the incident beam and transforms into the observed distribution. In that sense can you think the XRD as an mathematical operation that gives the intensity distribution ?

Regards

Nityananda Das yes, diffraction is Fourier transform. What you do with the diffraction experiment is to measure some parts of this Fourier transform to obtain the information on the object. So you measure some intensity (the phase information is already lost) in reciprocal space. The diffractometer, sample (single crystal vs powder) and detector determine which slice in reciprocal space you are considering.

In the general case you would need the whole reciprocal space to reconstruct the object (see what people nowadays do with the TEM for structure solution from single crystals) and you would then start discovering symmetry, differences in atom types etc.. That's why we try to have a compact view (Bragg and Laue formulas) and introduce concepts like periodicity, symmetry and atomicity to simplify the task of obtaining some info on the object from the measured data. For your periodic object, the difference in intensity between the peaks is due to the symmetry and again it can be explained with the Fourier transform. You introduce the Fhkl cause in the rest of the space (in the ideal case) the intensity is zero. It's a smart way of doing the Fourier transform and a way of linking the atomic positions with the data.

X-ray diffraction techniques works on the basis of diffraction of X-rays by the planes of atoms in the solid material (Bragg's law). The diffraction peaks would arise only when these diffracted x-rays gets addedup (constructive interference), while destructive interference leads to absence of peaks, and this only gives you the position of reflection, which depends on the different planes of atoms arranged in the solid. The intensity of diffraction peaks purely depends on scattering factor of the atoms present in the plane, number of such planes resulting to contructive interference (Intensity directly proportional to the square of the structure factor, Ihkl = square of Fhkl, where as Fhkl = summation of (product of scattering factor and exponential of (2pi(hx+ky+lz) for each atoms) ).

From the above equation also, Intensity of the diffraction peak is depends on the scattering factor (atomic number) and position of the atom in the crystal.

Dear Marappa, I know the intensity variation from the angle of scattering factor and structure factor .

I am thinking in a different way of understanding. From the elegant derivation of Laue , we know that a constructive diffraction occurs in a direction when all the diffracted X Ray from all the scattering centres added constructively. I mean if there are total of N no of scattering centres then for each peak all these centres contribute in constructive manner. Still there is intensity difference. From this angle of thinking , how can I explain the intensity variation ?

Dear Nityananda

The scattering power(form factor) of the centers is a function of angle in X-rays unlike in neutrons where scattering length does not change with angle.

Respected Masina,

Please give some basic idea of neutron scattering so that I can catch up the idea of neutron scattering and compare with XRD .

Regards

Nityananda Das

Dear Dr Das,

With regard to neutron powder diffraction, which gives you the bulk information of your samples instead of surface infornation from XRD data, it might be useful to read the below article entitled

“Texture and structural refinement using neutron diffraction data from molybdite (MoO3) and calcite (CaCO3) powders and a Ni-rich Ni50.7Ti49.30 alloy”

that can be downloaded at the link - https://www.researchgate.net/profile/Husin_Sitepu/publication/259418708/inline/jsViewer/5697dbd808ae34f3cf1f22fb

Hopefully, it is usefu!

With best wishes,

Husin Sitepu

Respected Dr. Husin Sitepu, your answer is really helpful . Thanks for the useful answer.

Regards

Nityananda Das

Dear Nityananda Das,

Peaks intensities in XRD patterns can be considered using the structural factor F. It is true.

However, the simplest way to understand “why?” is the repeatability factor m.

It considers a contribution of all reflecting planes of the {hkl} set, into the concrete hkl peak.

The set includes all planes (hkl) with identical indexes taking into account their permutations and signs (±). So, for the (111) plane of FCC lattice (for ex.) the m factor is equal to m=8, for the (220) plane m=12 and for the (200) plane m=6 and so on.

in a perfect crystal lattice, Interplanar distances between the (hkl) planes of the same {hkl} set are identical. Thus, the intensity of the hkl reflex mutually depends on total number of the {hkl} planes in the reflecting crystal taking into account their repeatability factor m.

As well, it is important to take into account package of atoms within planes of the concrete {hkl} set.

Dear Shishonok ,

Thank you for your valuable answer.

Can you suggest me any good reference for studying the concept of repeatability ?

Regards

Nityananda Das

sorry for the apparently rude answer, but have you considered reading a book on diffraction? At least you have some verified and detailed answer, avoid useless headaches or waste of time and get also some formulas to play with... It would also learn the right terminology and start separating single crystal from powder diffraction, cause things are getting a bit odd here... the "repeatabiity", is actually called multiplicity and the "structural factor" is the structure factor. The multiplicity enters into account just for powder diffraction! For a single crystal you don't have it... the structure factor can already be different for different peaks and it is the one that give the intensity according to atom types and position/symmetry. You would get the same value if you would actually take the Fourier transform of the 3D lattice numerically.

From the mathematical point of view things go well beyond Bragg and Laue equations that give you just the geometrical part of the problem, whereas the structure factor, with the multiplicity for powders give the intensity in the static case (and in the easy case of 3D periodic objects and kinematical approximation). Then you would need thermal effects etc... In my opinion the older books are still the best ones for a start (and you can find them for a very low price). For the most recent, you can get the whole picture of Crystallography and Diffraction from the International Tables for Crystallography from vol A to vol H. Powder Diffraction is in the latest volume H.

Concerning the older books, I would suggest two: the one of Zachariasen is the most complex/complete from the mathematical point of view, whereas that of Warren is complete but much simpler in the treatment. I still use it a lot. For the basics they are good. For the most advanced stuff they are a bit obsolete, but you better start to learn how to walk before you run! You should be able to calculate the theoretical pattern e.g. for NaCl. As I already said in my previous comment, all the formulas you find are an easy way to predict the Fourier transform of your 3D periodic lattice without calculating it explicitly.

Diffraction seems easy...

Thanks. I absolutely agree with your comments but it was certainly rud enough. No sense..I know all you wrote.You were in a hurry. I was going to continue tomorrow.

The reason is my English...I have missed a little.

I mentioned that F factor is general approach ...But the multiplicity factor m can give students initial and simple knowledge

about .It is just start to understand. By the way...if you rotate a single crystal during XRD analysis, the pattern will show all peaks.You will see the same story. I have done it with diamond crystal. Rotation is common technique for many diffractometers.

I continue in Russian, ....господин Das является студентом и он должен начтнать с очевидного...

Всех благ...

Powders and crystals will show different XRD patterns..it is clear. I have referred to tabulated data..

When intensities are..1:0.2 :0.7...and so on..

https://link.springer.com/article/10.1007%2Fs11106-012-9386-5

Dear Nityananda Das, please go through this paper. It will give initial info about.

A lot of books are there even in www. Try to use keywords to find necessary information.

With regards

Please see the references below:

1-https://www.researchgate.net/post/How_to_explain_the_differences_of_XRD_peak_intensities_the_same_material.

2- https://www.researchgate.net/post/X-ray_Rocking_Curve_Analysis_of_Super-lattice_SL_Epitaxial_Structures_What_are_some_of_your_practical_experiences_with_this_method.

3- Basics of X-Ray Powder Diffraction.

4- Diffraction Line Profiles from Polydisperse Crystalline Systems.

Best regards

The intensity is calculated by

^2*Integrate[4*Pi*r*(rho(r)-rho_0)*sin(Q*r)/Q,{r,0,Infinity}]+

f is the scattering factor and represents the weighted average of all the elements in the sample.

rho(r) is the density at the shell with radius of r.

The scattering factor is an important reason of different peaks' heights.

... perhaps I am missing something Yizhou Lu.... You show Debye scattering equation and it calculates directly the powder pattern without calculating the scattering factor (i.e. it does not do the calculation considering the lattice points in reciprocal space, so the notion of Bragg peak is somehow artificial there)...