Question
Asked 1st Dec, 2013
  • Manipal Institute of Technology

Any advice on the calculation of dislocation density using HRXRD rocking curves?

Measuring rocking curves to get the dislocation density by using formula ((FWHM)^2) / (9*b^2)
Where b is Burger's vector.
If you do it, you will end up in quantity with unit deg^2 / cm^2, but in the unit of dislocation density is /cm^2. I am bit confused about this. Does anyone have any advice?

Most recent answer

13th Apr, 2021
Shahid Ali
University of Peshawar
K. K. Nagaraja The dislocation density is a measure of the number of dislocations in a unit volume of a crystalline material i.e. the length of dislocation lines per unit volume of the crystal (m/m3) Edge dislocation Screw dislocation Dislocations distort a crystal lattice, causing elastic stress around the dislocation line, & hence strain energy. For ultimate strength, dislocations are to be eliminated.
In this video, I have discussed the nature of dislocation density and how to calculate the dislocation density from XRD using origin. In the case you want to further ask about it, please do comment on the specific video, I'll respond to it shortly. I have provided the practice as well as calculations files here. Thanks

Popular answers (1)

Dear Dr Nagaraja K K,
Angular mosaic tilt or twist spread of the crystals that measured from FWHM value of omega scan which has to be taken in the units of Radians. In this context, which can be considered as unitless.
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All Answers (17)

Dear Dr Nagaraja K K,
Angular mosaic tilt or twist spread of the crystals that measured from FWHM value of omega scan which has to be taken in the units of Radians. In this context, which can be considered as unitless.
4 Recommendations
2nd Dec, 2013
Obi Kingsley Echendu
Federal University of Technology Owerri
Dear Nagaraja,
Malleswararao's answer is correct. Just convert your FWHM value to Radians (this makes it a number). It is usually measured in degrees originally. Just multiply your FWHM by (pi/180).
2nd Dec, 2013
Edward Andrew Payzant
Oak Ridge National Laboratory
Ravi is correct in noting that you must account for instrumental broadening and not simply convert the rocking curve FWHM to dislocation density.
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2nd Dec, 2013
Ravi Ananth
OnSight Technology USA
Actually one needs to define the "dislocation density" first. So if we are measuring say a "line" in a "volume" perhaps, then it would need to be mm/mm^3=mm^(-2). Yes?
In practice, we do need to compare the FWHM to a "standard" to really come up with its relationship to dislocation density. (Sorry Edward, I re-posted after your comment). The formula Nagaraja has used, isolates the various parameters involved and their general relationship. I'd like to see the math for that equation. I've been working with it since 1979 starting with the group at Rutgers under the tutelage of Sigmund Weissmann et al.
Here in an example of Bragg XRD Microscopy using relative FWHM mapping for a ZnSe wafer (224) Asymmetric reflection, clearly demonstrating dislocation density patterns from the as-grown Nanostructure for the top 1.1um depth (cumulatively).
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3rd Dec, 2013
Sami Mahmood
University of Jordan
The angular spread is dimensionless; it has to be considered in radians. This will give a value which is of the order of ten thousand times smaller than what you would obtain if you don't convert to radians
1 Recommendation
4th Dec, 2013
Iren Leonidovna. Shul’pina
Russian Academy of Sciences
See Acta Met. 1957.V.5.P.548 for high density of disloations and Phyl. Mag. 1998. V.A77. P. 1013 for low density.
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4th Dec, 2013
Ravi Ananth
OnSight Technology USA
Thanks Iren! Please send me PDF files if available. Anybody!
Some cool info while exploring your references on Google:
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16th Dec, 2013
K. K. Nagaraja
Manipal Institute of Technology
Dear Ravi Ananth,
Here are the papers, these papers are cited most
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16th Dec, 2013
K. K. Nagaraja
Manipal Institute of Technology
17th Dec, 2013
Iuliana Pasuk
The National Institute of Materials Physics
Thanks to Irene Shul'pina and Nagaraja K.K. for these papers!
1 Recommendation
14th Mar, 2014
Ravi Ananth
OnSight Technology USA
Dislocation Density, ρ=β2/9b2 for a Gaussian distribution, b=Burger’s Vector, ρ-Dislocation Density, integral breadth, β, is related to the FWHM peak width, H, by β = 0.5 H (π / loge2)1/2,
(If someone can post the original reference for this relationship, it would be very helpful)
Also, β2(Sample)= β2(Experimental) - β2(Instrumental)
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14th Mar, 2014
Ravi Ananth
OnSight Technology USA
Let us take this example of GaAs (004) symmetric reflection. The experimental data is compared with the theoretical expected profile as calculated by Bruker LEPTOS software. The match and "figure of merit" should be exceptionally high in this case. In fact the "deviation" from the theoretical profile may be used to conclude the levels of stacking faults with missing smaller Ga (31) atom rows/planes or larger As (33) atom rows/planes. This would depend on which side the deviation is on, the lower angle or higher angle side of the Bragg peak.
My question then is that if FWHM is related to the average excess dislocation density, then how do we compute the volume of "stacking faults" or "twins" from the Bragg rocking curve profile?
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14th Mar, 2014
Ravi Ananth
OnSight Technology USA
Anyone that may entertain the idea that the "deviation" from theoretical or IDEAL Bragg condition was due to sample alignment/orientation etc. better think again after seeing this example for GaSb (004) on the same diffractometer with identical sample mounting conditions. Besides, samples rescanned on other diffractometers produced similar asymmetry in the RCP (rocking curve profile). These deviations from the theoretical profile are generally below the FHHM and can be directly attributed to stacking fault defects. I do need to find the literature references to this correlation to stacking faults and twins to the profile below the FWHM. If any of you have access to them please post links or forward to me at BraggXRDMicroscopy@gmail.com
There are more diverse example that I'll post as well regarding the "deviation".
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24th Nov, 2020
Leyla Barghamadi
Iran University of Science and Technology
13th Apr, 2021
Shahid Ali
University of Peshawar
K. K. Nagaraja The dislocation density is a measure of the number of dislocations in a unit volume of a crystalline material i.e. the length of dislocation lines per unit volume of the crystal (m/m3) Edge dislocation Screw dislocation Dislocations distort a crystal lattice, causing elastic stress around the dislocation line, & hence strain energy. For ultimate strength, dislocations are to be eliminated.
In this video, I have discussed the nature of dislocation density and how to calculate the dislocation density from XRD using origin. In the case you want to further ask about it, please do comment on the specific video, I'll respond to it shortly. I have provided the practice as well as calculations files here. Thanks

Similar questions and discussions

Using MAUD for dislocation density calculation, Conversion between D-spacing and 2Theta ?
Question
5 answers
  • Diween HawezyDiween Hawezy
Hi all,
I would like to start by thanking Prof Luca Lutterotti on the continuous development of the MAUD software and apologizing if any of my questions seem basic. This is my first foray into Rietveld refinement. My fitting I think is good given that I'm using a detector with 160banks.
I currently have the data extracted from MAUD and would like to do some dislocation density estimates/calculations based on the FWHM of each peak.
I have used the simple formula:
((FWHM)^2) / (9*b^2)
linked here:
However as the first figure attached shows, my x-axis is d-spacing not 2theta. Which then leads to a dislocation density values shown in the table (second figure):
I'd like to first check if I am using the best approach and secondly to convert this value to be in microns or meters - I am aware the formula linked was based on degree^2/cm^2 which in my case I would assume to be d-spacing^2/cm^2 ?
Final question - I have coloured in an orange circle highlighting what I believe to be an intermetallic phase in the first figure. Online at https://materialsproject.org/materials/mp-20738/ I can view the diffraction peaks of different intermetallics in 2theta which would make identification of what the intermetallic phase much much easier if I was able to compare it to a 2theta plot of my data.
I'd like to add that our beamline scientist was not familiar with texture/dislocation density calculations so this topic has been difficult for me to breach.
Thank you,
Diween

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Etch pit techniques have been used to investigate the dislocations in cronstedtite. The Burgers vector of the dislocations is determined by relating the dislocation arrangement revealed by the etch pits to the known stacking faults which have been found by x-ray diffraction.
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