Ravi..  I wrote that companies don't have it.. but the software exists. At least the software I wrote implementing the WPPM does it. Just look for PM2K in my publication list. Those interested in getting a copy just send me an email

Hi!

Maybe is necessary carried out Rietveld refinement but I think is a little complicate, because XRD peak profile shape and width are the result of imperfections in both the experimental setup and the sample.

Instrumental broadening dependent on experimental set up (e.g. sample size, slit widths, goniometer radius), function of 2θ, and determined by measurement of a suitable reference.

On another hand, sample broadening dependent of periodicity in crystals but is not finite. For finite size thus size broadening, most apparent in crystals smaller than 100 nm. Also, lattice imperfections (e.g. dislocations, vacancies, substitutional) raise strain broadening. For this case, is necessary measured using different diffractometers with different optical configurations for observed this phenomena.

Strain is usually quantified as ε0 = Δd/d, with d the idealized d‐spacing and Δd the most extreme deviation from d.

The peak broadening due to strain is assumed vary as: B_strain (2θ)=4ε_0 tanθ.

For this case is mandatory deconvolution for determinate several factors that contribute to the observed peak broadening:

Bobs = Binstr + Bsample = Binstr + Bsize + Bstrain

Hi, @David, Thanks for the answer. But I still did not get the linkage between dislocation density and peak width. I mean, I do realize the imperfections lead to the broadening of the diffraction peak, but how can I quantify this and relate to the dislocation density?

And is the d you mentioned in the answer the lattice parameter?

For division of influence of the size of blocks (L) and micropressure (Δd/d) on physical width of a line (b) crystals use a graphic method.

Build dependences 1/L = (b*cos θ/l) from (sinθ/l). θ - corner of reflection. l - length of a wave of radiation.

On the equation of the received straight line it is possible to find the "cleared" meaning L in a point (sinθ/l) = 0.

Tangent of a corner of an inclination of a straight line will give Δd/d.

Knowing L, it is possible approximately to estimate dislocation density.

For a presence Binstr remove a spectrum of the standard free of defects and with the large blocks (more than 500 нм).

... guys you better not keep reinventing the wheel or some terms (what is micropressure??). Line profile analysis of powders, at least in the case of finite domain size effects and dislocations, is rather well assessed. Unfortunately we still find a lot of people that do not read or that do not consider the limits of some approaches. For instance the method of Wlliamson and Hall (described by D Orozco) is strictly valid when both the size and the microstrain broadening are Lorentzian (otherwise summing the widths is arbitrary) and this is almost never the case...

Dislocations affect not just the width, but also the shape of the peaks (the link between dislocations and peaks is the local strain distribution that depends on the elastic constants and therefore is intrinsically anisotropic).

Calculating a dislocation density without considering the anisotropy is kind of nonsense. Be careful also when using home made "standards". The only standards are those produced by a standardization institute and NIST is the one providing standards for the profile (the SRM 660 series). Those have grains that are sufficiently large and free of defects to show no appreciable broadening, but sufficiently small for the material to be still a powder. And the cell parameter is certified as well (so you can check the alignment of your machine, which is quite important).

You might consider using the Whole Powder Pattern Modelling approach (see my publication list), which is still the state of the art technique for line profile analysis from powder data.

@Huan Wang: if you know of such company/software please let me know... as I doubt you can find any.

In any case it seems the original proposer of the question is not that interested

@Metteo Leoni Hi Prof. Matteo Leoni, thank you for your professional explanation. Sorry for the delay, I did not check the researchgate recently.

When you said to pay attention to the anisotropy, did you mean the texture of the materials? So if I use powder diffraction with random crystallographic preference, then I can do the dislocation calculation, right? Unfortunately, we only have bulk materials which can not be fragmented into powders, can it still work? (The grain size, however, is around 100-300nm, and SAD pattern suggests no specific texture.)

According to your publication list, you are a real XRD expert, I never used XRD to calculate dislocation density, so the information included is kind of overwhelming, but I am sure I will learn a lot from your papers.

@Huan Wang Hi, thank you for your answer, I checked Bruker's website, they do offer XRD training courses, I will talk to my boss to see whether I can go there. The software, however, I could not find online, please let me know if applicable.

You can get an estimation of screw and edge dislocation by xrd (though it won't be accurate as compared with TEM) if you know the line direction of dislocation and 'burger vector' b value of particular dislocation. N number of dislocations is given by beta^2(beta is FWHM of corresponding peak)/4.35*b^2.

Check this article, it may help you,

Philosophical Magazine A, 1998, vol 77, No.4, 1013-1025

@Nagboopathy thank you for your answer, I will look into the paper

@Shiteng: well it seems you haven't made progress so far as i see the thread is still alive and with no feedback about success on this. Dislocations are anisotropic defects (the strain field is, due to the anisotropy in the elastic constants). The broadening will therefore be anisotropic i.e. each hkl will have a different breadth).

A first possible indication for the presence of dislocation is a NON monotonic trend of the FWHM of peaks with the diffraction angle.

Now powder diffraction can be employed for any powder in a diffraction sense i.e. any assembly of coherent scattering domains covering the whole space of orientations. So a powder and a polycrystalline bulk are both potentially good. As XRD does not see the grain size, your material is also potentially a "powder" for XRD, provided the coherent domains inside it are less than 10-150 nm max (limit depending on the instrument/setup).

So far the only theory available to describe the shape of a profile broadened by dislocations is the Krivoglaz & Wilkens . It's not easy and it contains a lot of magic (if we can call it like this), but it works and it consider the anisotropic broadening in detail. There is currently only one software that handles it for any material and it is PM2K implementing the WPPM method (and the source code is on my PC).

@Nagaboopathy: I did not see that publication for long time! It is a quite old work on Till Metzger et al. on dislocations in thin films where any other possible broadening effect is very limited. No modelling of the rocking curve shape though and therefore it is impossible to judge how good is the result...

@Matteo, I really appreciate your professional answer, as always. Yes, I am curious about the dislocation density of my materials and I did a series of TEM observation to measure it. However, the Observation area of the TEM is too localized and that's the reason I tried to find an alternative, the XRD.

In terms of the Krivoglaz & Wilkens' theory, can you please tell me some classical literature or instructions? Besides, is that possible that I can get your software PM2K? Your response shed me more light on this subject.

In recent years, the effective way to measure the density of geometrically necessary dislocations is to use the high resolution Electron Backscatter Diffraction (EBSD). See for instance

Experimental lower bounds on geometrically necessary dislocation density

J.W. Kysar, Y. Saito, M.S. Oztop, D. Lee, W.T. Huh

Int. J. Plasticity 26 (2010) 1097-1123

The following formula for dislocation density;

dislocation desity= 1/D*2

where, D is the grain crystal size which can be obtain by the following formula;

D=(κ λ)/(FWHM cosθ )

dear Haardindra, well NO! that's definitely complete nonsense. Size and dislocations have a different effect on the diffraction profiles in terms of shape and breadth.

Shiteng! "How to determine the dislocation density by XRD and subsequent analysis?" Will depend on the type of material you are studying, polycrystalline (as in your case) or monocrystalline? Please post your WAXS and SAXS data. BTW how did you eliminate the presence of "preferred orientation" by SAD? Did you use a 2D detector or rotate the sample about its surface normal?

Matteo has outlined the general limitations & methodology quite eloquently for nanocrystalline powder samples. The WPPM, whole powder pattern modeling, (I'll post the correct reference once I find it) is the most appropriate method. I'd pay heed if I were you! See one example in "Diffraction line broadening from nanocrystals under large hydrostatic pressures" by Michael Burgess, Alberto Leonardi, Matteo Leoni, Paolo Scardi. There are many more references in Matteo's profile besides this that will be helpful. Download and study them all without exception. The ubiquitous Scherrer equation alone would not be very helpful, in my opinion.

In general, the group at Rutgers (RU) under the tutelage of Dr. Sigmund Weissman has successfully related XRD data to dislocation density since the 1950's. The key is the ability to deconvolute the "strain effect" versus "particle size effect". The method is known as XRD rocking curve analysis. The RU group has successfully accomplish this for large crystals (>4um) by measuring the FWHM of individual crystal's Bragg reflection. In general the FWHM is related to the excess dislocation density in the reflecting grain. It is also feasible to interpret the Bragg peak asymmetry in terms of "twins" and "stacking faults" for individual reflections. A thorough reading of the classic book by Andre Guinier's (XRD from Crystals, Imperfect Crystals & Amorphous Bodies http://www.flickr.com/photos/85210325@N04/10515372183/) will shed more light on this issue. Here are a bunch of references from that RU group that may be helpful for materials of larger particle size and monocrystalline materials:

All of Dr. Sigmund Weissmann’s relevant references:

1. R. Yazici, W. Mayo, T. Takemoto & S. Weissmann (1983). Defect structure analysis of polycrystalline materials by computer-controlled double-crystal diffractometer with position-sensitive detector. J. Appl. Cryst. 16, 89-95.

2. W. Mayo, R. Yazici, T. Takemoto & S. Weissmann (1981). Defect structure analysis of polycrystalline materials by computer -controlled double-crystal diffractometer and position-sensitive detector. Acta Cryst. A37, C253

Photographic Film/Topography and RC Analysis:

3. Rocking Curve Analysis - Dr. Sigmund Weissmann (Rutgers) Acta Cryst. (1954). 7, 729-732 [ doi:10.1107/S0365110X54002216 ]An X-ray diffraction method for the study of substructure of crystals J. Intrater and S. Weissmann http://scripts.iucr.org/cgi-bin/paper?S0365110X54002216

4. Acta Cryst. (1954). 7, 733-737 [ doi:10.1107/S0365110X54002228 ]An X-ray study of the substructure of fine-grained aluminium S. Weissmann and D. L. Evans; http://scripts.iucr.org/cgi-bin/paper?S0365110X54002228

5. J. Intrater & S. Weissmann (1954). An X-ray diffraction method for the study of substructure of crystals. Acta Cryst. 7, 729-732.

6. S. Weissmann, Z. H. Kalman, J. Chaudhuri & G. J. Weng (1981). Determination of strain concentration and strain interaction by intensity measurements of X-ray topographs. Acta Cryst. A37, C247.

7. J. Chaudhuri, Z. H. Kalman, G. J. Weng & S. Weissmann (1982). Determination of the strain concentration factors around holes and inclusions in crystals by X-ray topography. J. Appl. Cryst. 15, 423-429.

8. H. L. Glass & S. Weissmann (1969). Synergy of line profile analysis and selected area topography by the X-ray divergent beam method. J. Appl. Cryst. 2, 200-209.

9. T. Imura, S. Weissmann & J. J. Slade Jnr (1962). A study of age-hardening of Al-3.85% Cu by the divergent X-ray beam method. Acta Cryst. 15, 786-793.

10. Z. H. Kalman, J. Chaudhuri, G. J. Weng & S. Weissmann (1980). Determination of strain concentration by microfluorescent densitometry of X-ray topography: A bridge between microfracture and continuum mechanics. J. Appl. Cryst. 13, 290-296.

11. Z. H. Kalman & S. Weissmann (1983). On the X-ray reflectivity of elastically bent perfect crystals. J. Appl. Cryst. 16, 295-303.

12. Z. H. Kalman & S. Weissmann (1979). Determination of strain distribution in elastically bent materials by X-ray intensity measurement. J. Appl. Cryst. 12, 209-220.

13. H. Y. Liu, G. J. Weng & S. Weissmann (1982). Determination of notch-tip plasticity by X-ray diffraction and comparison to continuum mechanics analysis. J. Appl. Cryst. 15, 594-601. .

14. B. A. Newman & S. Weissmann (1968). Strain inhomogeneities in lightly compressed tungsten crystals. J. Appl. Cryst. 1, 139-145.

15. S. Weissmann & D. L. Evans (1954). An X-ray study of the substructure of fine-grained aluminium. Acta Cryst. 7, 733-737.

16. S. Weissmann, V. A. Greenhut, J. Chaudhuri & Z. H. Kalman (1983). Quantitative analysis of intensitities in X-ray topographs by enhanced microfluorescence. J. Appl. Cryst. 16, 606-610.

17. L. D. Calvert, T. C. Huang, M. H. Mueller, P. L. Wallace & S. Weissmann (1993). New ICDD metals and alloys indexes, a powerful materials research tool. Acta Cryst. A49, c443-c444.

18. R. J. Schutz, S. Weissmann & J. Yaniero (1981). The fabrication of a dual-element X-ray divergent-beam target. J. Appl. Cryst. 14, 352-353.

The example below is the dislocation density map for a single crystal wafer of ZnSe using the modern 2D version of the XRD rocking curve analysis known as Bragg XRD Microscopy. The YouTube video play list has many examples ranging for substrates, epitaxial films and polycrystalline materials:

http://www.youtube.com/watch?v=dFCQS8oUyT0&list=PL7032E2DAF1F3941F

"XRD companies will supply the software for the users to analyze the XRD spectrum", sure they'll sell you anything if you pay for it. But, that does not make it appropriate. In the past 3 decades many of them have been peddling shear pabulum to gullible experimentalists desparate for solutions!

@Matteo Leoni. I'm sorry because actually i was not referring the relationship between the grain crystal size and dislocation. I was just stating the formula to calculate to dislocation from XRD with the following reference.

Haarindra! That is the exessively used Scherrer formula which has many limitations. Many of us in the XRD community are guilty of using the elegant Scherrer relationship incorrectly or more so incompletely.

Here is another important reference for smaller powders (certainly not Nano powders yet): Yacici, R, & Kalyon, D - "Microstrain and Defect Analysis of CL-20 Crystals by Novel X-Ray ..." http://www.hfmi.stevens.edu/publications/203.pdf

In fond memory of my friend and fellow grad student at RU, the late Dr. Rahmi Yacici of SIT.

Is there any software available to determine the dislocation density from XRD

Karthikeyan! Many think "no such software exists yet"! I'm not aware of any thus far :-)

In my opinion, it would be easier to determine "relative dislocation density" based on "known standards" rather than determine absolute values. Especially with "powders" where several other factors affecting the FWHM are convoluted in the XRD observations.

Ravi..  I wrote that companies don't have it.. but the software exists. At least the software I wrote implementing the WPPM does it. Just look for PM2K in my publication list. Those interested in getting a copy just send me an email

Thanx Matteo! I see you are always paying attention unlike many of the queriers :-)

I was just going through and analyzing the reference posted by Haarindra. I found several inconsistencies:

• They still are using JCPDS in 2013. As archaic as I was. The data hasn't changed much, I guess?
• "θ is the Bragg’s angle of the peak and β is FWHM" - incorrect. This would change only the absolute value, I suppose :-) It would still work for relative analyses. But, is a common blatant error in publications!
• As Scott mentioned earlier the "K" factor is system specific and needs to be better understood. These folks have used 0.94. Not sure what that means in this case. But they may be off a bit?

Conference Paper RIETVELD AMORPHOUS QUANTIFICATION WITHOUT PAIN: THE K-FACTOR APPROACH

Hello,

There is a simple analysis to determine the dislocations density from XRD measurements. You need only to measure the full width at half maximum (FWHM) of your layer peak at different reflections (004, 115, 117, 444,..), than you plot the graph FWHM^2=f(tan(theta)^)), then, you determine the slope and the intersection, From these latter, you can determine the dislocations density. For more details, you can read this paper:

http://www.sciencedirect.com/science/article/pii/0022024894907277

Best regards,

D. Benyahia

The dislocation densities estimated by Wiliamson-Hall equation in X-RD profiles using help of PANalytical’s X-RD software...I have calculated using that.

You can refer my paper.Article Characterization of Microstructure, Mechanical Properties an...