# When using the Debye-Scherrer equation for calculating particle size (D=Kλ/(β cos θ) does one have to halve the FWHM(β)? Must all angles be in Radians?

The Debye-Scherrer method is used to obtain X-ray diffraction measurements in powders.This method is applicable to crystallites ranging from 1.0 to 0.01 μm in diameter, but the grains must have good crystallinity.

## Popular Answers

Matteo Leoni· Università degli Studi di TrentoMy suggestion is always: if you want to drive a car on a public road, first learn to drive.

So please, before using ANY tool, first open a book or browse through the Internet to find out what is all about and to find out the relevant literature IN the field (too many people look for literature on Scherrer equation in papers that have nothing to do with X-ray diffraction). You can find the derivation of Scherrer equation in any X-ray (powder) diffraction book and if you look at that derivation, you find the answer to your question as well (any student know that angles, especially in trig formulas are in radians). Plus β and FWHM are two different things (the integral breadth versus the full width at half maximum) that may or may not be the same ofr a given peak.

This said, Scherrer equation give you a number related to the distribution of length of columns composing your specimen. It CAN be related, under quite strict hypotheses, to the mean domain size. In general, knowing the domain shape (that you can hardly guess from Scherrer formula), the value you obtain is a ratio between high order moments (4th over 3rd) of the size distribution. For some discussion:

https://www.researchgate.net/publication/200045699_Line_broadening_analysis_using_integral_breadth_methods_a_critical_review

(Scherrer formula is the size part of the Williamson-Hall method).

So forget about "particle size". And forget about being able to characterize anything above 100 nm using Scherrer equation (the equation does not include instrumental and strain broadening effects always present in a pattern). Moreover, talking about "crystallinity" is also not appropriate as you characterize the crystalline part of your material only.

If you are still convinced that Scherrer formula is of any quantitative use in 2013 (it is good for qualitative comparison), then:

* D and λ have the same unit of measurement (e.g. nm as per SI)

* FWHM is the full width at half maximum of the peak (not half of it) in rad. As you have it in deg from the machine, just multiply by pi/180

* theta is half of Bragg angle in rad

If you want to work quantitatively then first open a good book on diffraction and then have a look at more modern analysis techniques (e.g. Whole Powder Pattern Modelling, https://www.researchgate.net/publication/11527348_Whole_powder_pattern_modelling) that can give quantitative information compatible with the microstructure of the material.

## Article: Line-Broadening Analysis Using Integral Breadth Methods: A Critical Review

ABSTRACT:Integral breadth methods for line profile analysis are reviewed, including modifications of the Williamson-Hall method recently proposed for the specific case of dislocation strain broadening. Two cases of study, supported by the results of a TEM investigation, are considered in detail: nanocrystalline ceria crystallized from amorphous precursors and highly deformed nickel powder produced by extensive ball milling. A further application concerns a series of Fe-Mo powder specimens that were ball milled for increasing time. Traditional and modified Williamson-Hall methods confirm their merits for a rapid overview of the line broadening effects and possible understanding of the main causes. However, quantitative results are generally not reliable. Limits in the applicability of integral breadth methods and reliability of the results are discussed in detail.## Article: Whole Powder Pattern Modelling

ABSTRACT:A new approach for the modelling of diffraction patterns without using analytical profile functions is described and tested on ball milled f.c.c. Ni powder samples. The proposed whole powder pattern modelling (WPPM) procedure allows a one-step refinement of microstructure parameters by a direct modelling of the experimental pattern. Lattice parameter and defect content, expressed as dislocation density, outer cut-off radius, contrast factor, twin and deformation fault probabilities), can be refined together with the parameters (mean and variance) of a grain-size distribution. Different models for lattice distortions and domain size and shape can be tested to simulate or model diffraction data for systems as different as plastically deformed metals or finely dispersed crystalline powders. TEM pictures support the conclusions obtained by WPPM and confirm the validity of the proposed procedure.Matteo Leoni· Università degli Studi di TrentoSitarama, I think most people do not care about the instrument correction simply because they found that someone else was using Scherrer formula that way and they decided to do the same. I cannot count the number of wrong citations to Scherrer (1918) and I am pretty sure most people out there never saw or read it (they just keep citing something they found in other publications). Believe or not, in that article there is no mention on how to derive the formula!

## All Answers (37)

Matteo Leoni· Università degli Studi di TrentoMy suggestion is always: if you want to drive a car on a public road, first learn to drive.

So please, before using ANY tool, first open a book or browse through the Internet to find out what is all about and to find out the relevant literature IN the field (too many people look for literature on Scherrer equation in papers that have nothing to do with X-ray diffraction). You can find the derivation of Scherrer equation in any X-ray (powder) diffraction book and if you look at that derivation, you find the answer to your question as well (any student know that angles, especially in trig formulas are in radians). Plus β and FWHM are two different things (the integral breadth versus the full width at half maximum) that may or may not be the same ofr a given peak.

This said, Scherrer equation give you a number related to the distribution of length of columns composing your specimen. It CAN be related, under quite strict hypotheses, to the mean domain size. In general, knowing the domain shape (that you can hardly guess from Scherrer formula), the value you obtain is a ratio between high order moments (4th over 3rd) of the size distribution. For some discussion:

https://www.researchgate.net/publication/200045699_Line_broadening_analysis_using_integral_breadth_methods_a_critical_review

(Scherrer formula is the size part of the Williamson-Hall method).

So forget about "particle size". And forget about being able to characterize anything above 100 nm using Scherrer equation (the equation does not include instrumental and strain broadening effects always present in a pattern). Moreover, talking about "crystallinity" is also not appropriate as you characterize the crystalline part of your material only.

If you are still convinced that Scherrer formula is of any quantitative use in 2013 (it is good for qualitative comparison), then:

* D and λ have the same unit of measurement (e.g. nm as per SI)

* FWHM is the full width at half maximum of the peak (not half of it) in rad. As you have it in deg from the machine, just multiply by pi/180

* theta is half of Bragg angle in rad

If you want to work quantitatively then first open a good book on diffraction and then have a look at more modern analysis techniques (e.g. Whole Powder Pattern Modelling, https://www.researchgate.net/publication/11527348_Whole_powder_pattern_modelling) that can give quantitative information compatible with the microstructure of the material.

## Article: Line-Broadening Analysis Using Integral Breadth Methods: A Critical Review

ABSTRACT:Integral breadth methods for line profile analysis are reviewed, including modifications of the Williamson-Hall method recently proposed for the specific case of dislocation strain broadening. Two cases of study, supported by the results of a TEM investigation, are considered in detail: nanocrystalline ceria crystallized from amorphous precursors and highly deformed nickel powder produced by extensive ball milling. A further application concerns a series of Fe-Mo powder specimens that were ball milled for increasing time. Traditional and modified Williamson-Hall methods confirm their merits for a rapid overview of the line broadening effects and possible understanding of the main causes. However, quantitative results are generally not reliable. Limits in the applicability of integral breadth methods and reliability of the results are discussed in detail.## Article: Whole Powder Pattern Modelling

ABSTRACT:A new approach for the modelling of diffraction patterns without using analytical profile functions is described and tested on ball milled f.c.c. Ni powder samples. The proposed whole powder pattern modelling (WPPM) procedure allows a one-step refinement of microstructure parameters by a direct modelling of the experimental pattern. Lattice parameter and defect content, expressed as dislocation density, outer cut-off radius, contrast factor, twin and deformation fault probabilities), can be refined together with the parameters (mean and variance) of a grain-size distribution. Different models for lattice distortions and domain size and shape can be tested to simulate or model diffraction data for systems as different as plastically deformed metals or finely dispersed crystalline powders. TEM pictures support the conclusions obtained by WPPM and confirm the validity of the proposed procedure.Alejandro Gómez-Pérez· University Foundation San Pablo CEUAnil Choubey· Christian College of Engineering & TechnologyAli Wako· University of the Free StateAli Wako· University of the Free StateMatteo Leoni· Università degli Studi di TrentoI would like to see a paper where people want to halve the FWHM... that's scary

Sitarama Raju Kada· Deakin UniversityI did see so many people reporting crystallite size in nano for an ultrafine grained powders. This just happens by not subtracting the instrument broadening from FWHM in scherrer equation. People do not bother about the correction for instrument while using Scherrer equation just because the word nano sounds crazy.

C. Meneghini· Università Degli Studi Roma TreMatteo Leoni· Università degli Studi di TrentoSitarama, I think most people do not care about the instrument correction simply because they found that someone else was using Scherrer formula that way and they decided to do the same. I cannot count the number of wrong citations to Scherrer (1918) and I am pretty sure most people out there never saw or read it (they just keep citing something they found in other publications). Believe or not, in that article there is no mention on how to derive the formula!

Paul Rozenak· Hydrogen Energy Batteries, LTDS.J. Picken· Delft University Of TechnologyIn some cases you may know a-priori what sort of defects are possible & then the interpretation becomes considerably easier.

Scott Speakman· PANalytical B.V.For an excellent paper describing the Scherrer formula and the Scherrer constant, I recommed J.I. Langford and A.J.C. Wilson, “Scherrer after Sixty Years: A Survey and Some New Results in the Determination of Crystallite Size,” J. Appl. Cryst. 11 (1978) pp 102-113.

Ali Wako· University of the Free StateMatteo Leoni· Università degli Studi di TrentoThe meaning of the result does not change if you use 1 or 0.89. The result you obtain is in both cases not what you think it is! Moreover most people do not even consider the fact that that peak is often made up of two wavelength components (separated, thus giving broadening), that have an intrinsic Lorentzian breadth. You should also consider those before dealing with K....

Ali, what do you mean? usually peaks split when you have a reduction in symmetry of the lattice (e.g. the 001 010 and 100 are superimposed i.e. make one peak in cubic and can be 2 or 3 if you have tetragonal or orthorhombic distortion)

Anil Choubey· Christian College of Engineering & TechnologySitarama Raju Kada· Deakin UniversityAnil Choubey· Christian College of Engineering & TechnologyEdward Andrew Payzant· Oak Ridge National LaboratoryAs Matteo and other have noted, the description "JCPDS" (Joint Committee on Powder Diffraction Standards of the ASTM) has long been superseded by "ICDD" (the International Center for Diffraction Data).

Ravi Ananth· OnSight Technology USAhttp://www.flickr.com/photos/85210325@N04/8434970459/in/photostream

Aluminum foil and scotch tape are reasonably good calibration standards:

1. Kevlar+Al foil: http://www.flickr.com/photos/85210325@N04/7975835813/in/set-72157632728981912

2. Energetic composite + Al Foil:

http://www.flickr.com/photos/85210325@N04/7988760159/in/set-72157632728981912/

Thanks Scott Speakman! I use your PP presentation generously:

a. Basics of XRD: http://web.pdx.edu/~pmoeck/phy381/Topic5a-XRD.pdf

b. Often used slide: http://www.flickr.com/photos/85210325@N04/8592218974/in/photostream

Matteo Leoni! Thanks for your excellent comments. I've learnt a lot from this discussion. Thanks to all contributors!

"Plus β and FWHM are two different things (the integral breadth versus the full width at half maximum) that may or may not be the same ofr a given peak." Helpful link:

http://pd.chem.ucl.ac.uk/pdnn/peaks/gauss.htm

Ali Wako! "Peak Splitting" in mono-crystals occur due to sub-grain structure:

http://www.flickr.com/photos/85210325@N04/8294404556/sizes/h/in/photostream/

Bragg XRD Microscopy - YouTube play list for many materials with various nano-structural morphologies:

http://www.youtube.com/watch?v=ykSjd2ZZn64&list=PL7032E2DAF1F3941F

Please feel free to join us and contribute with your knowledge & expertise. You are all pre-approved and welcome to join "X-ray Diffraction Imaging for Materials Microstructural QC" group -

http://www.linkedin.com/groupAnswers?viewQuestionAndAnswers=&discussionID=218925637&gid=2683600&commentID=128396954&trk=view_disc&ut=3Xe63G3z1I5RI1

Presently we are discussing: " XRD Methods in "Materials Analyses": How many of you have used or are currently using and/or would like to use in the future? XRD methods are generally portable, real time, non-contact, NDE & in situ."

Ravi

BraggXRDMicroscopy@GMail.com

P.S: I'm still reviewing some of all your comments/links/references and continuing to learn.

Ravi Ananth· OnSight Technology USARavi

BraggXRDMicroscopy@GMail.com

http://www.linkedin.com/profile/view?id=63121884&trk=hb_tab_pro_top

Ravi Ananth· OnSight Technology USAI've been still living in the past and using JCPDS. I shall henceforth switch to ICDD.

Ravi Ananth· OnSight Technology USAQuartz Bragg XRD Micrograph: http://www.flickr.com/photos/85210325@N04/8449083122/in/set-72157632724063041/

In this case the incident beam contained both K Alpha 1 & 2. Only a Ni foil K Beta filter was used. Each spatial pixel yields a RCP (Rocking Curve Profile) with the well separated K Alpha 1 & 2 peaks. We used this "INVARIANT" separation to recalibrate each RCP along the Omega scale. Very precise!

Ravi Ananth· OnSight Technology USAIn most X-ray rocking curve analyses the incident beam is absent the K Alpha 2 component and a very low inherent FWHM (instrumental). However, in the case of Quartz above, we actually turned "lemon to lemonade" by using the separation between the two topographs (K Alpha 1 & 2) to re-calibrate the Omega scale.

Obviously, the presence of two superimposed topographs may appear to be confusing but when viewed in the Omega space the individual topographs are easily deconvoluted. The inherent FWHM is at least an order of magnitude higher in this case compared with the mono-chromated K Alpha 1 only case. But since the Quartz was of such high quality the peaks were amply well separated for deconvolution. Here the RCP profile shape was more akin to Pearson VII m=5 (just tested it): http://pd.chem.ucl.ac.uk/pdnn/peaks/pvii.htm

Here are some examples of other RCP shapes that we encountered:

http://www.flickr.com/photos/85210325@N04/8561945111/sizes/k/in/photostream/

Ravi

BraggXRDMicroscopy@GMail.com

Anil Choubey· Christian College of Engineering & TechnologyN. Kamarulzaman· Universiti Teknologi MARABo = Bi+Bst+Bs

where Bo = total peak broadeing (=observed peak broadening), Bi=broadening due to instrument, Bst=broadening due to lattice strain, Bs=broadening due to crystallite size. So many people just use the FWHM directly and plugged the value into the Scherrer equation! This gives inaccurate results which undestimate the actual value.

Ravi Ananth· OnSight Technology USAThat is correct N. Kamarulzaman! The "particle size effect" and the "strain effect" must be deconvoluted besides the instrumental factors for correct results. It is tough to do this with the Debye-Scherrer technique using film. Multiple (hkl)'s need to be analyzed to potentially separate the effects of the two factors on the FWHM. Lots of computation! The particle size effect should be independent of the (hkl) chosen unless some preference in shape exists. The strain effect would certainly be (hkl) dependent.

Ali Wako· University of the Free StateAli Bakly· University of BabylonI think, everybody doing very well in xrd tunnel

Arbab Rehan· Högskolan VästMay be I am wrong but what it seems like that the question was about sherrer equation that it contains Beta.cos(theta). However the diffraction pattern contains 2theta on x-axis and the Broadening (FWHM) for a peak i.e delta 2teta would not be the same as (FWHM) of teta. As sherrer's equation includes only teta not 2teta. Can anyone explain what should be used for Beta (FWHM) or (FWHM/2) ??

Ravi Ananth· OnSight Technology USAAli! You ought to at least edit your initial question and change the "Debye-Scherrer equation" to Scherrer equation after all the energy invested. Use the "Edit" option in the "scroll down" to the right of the question itself or use the icon right after the main query.

Arbab! If "May be I am wrong" doesn't stop you, it's ok :-)

For, if it had stopped TOM Edison, NO LIGHT BULBS!The question of correlation between Omega (ω) decoupled scan versus the Omega-2Theta (ω-2θ) coupled scan? I'd like to know too! I'd think, when measuring 2Theta, you'd use HWHM and when measuring Omega (Theta, θ) on the ordinate axis, FWHM. However, you'd convert to "integral breadth" Beta (β, always in Radians) to include in the Scherrer equation.Note:

BetaNOT EQUAL TOFWHMβ≠FWHM:-)Arbab Rehan· Högskolan Västthanks for sharing your thoughts...

Arun Karthick· Anna University, Chennaithe FWHM what you measure from XRD is of unit degree (theta), to make it unit less u need to convert it to radians to make the euquation unit results in angstrom or nm.

S. K. Sharma· Bhabha Atomic Research CentreRavi Ananth is correct. It should be HWHM from the theeta-2theeta scans.

file:///home/internet-user/Desktop/Deby-scherrer%20formula.pdf

See the derivation for the formula.

Saroj Kumar Das· Indian Institute of Technology GandhinagarI was just curious to understand the unit balance in Scherrer equation: where D=(0.9*λ)/(B*Cosθ), Here λ is in nm, B is in radian. How come the crystal size D will be in nm as well. According to formula it's unit must be nm/radian. I am very new to XRD. Kindly share your opinion with an appropriate reference.

Matteo Leoni· Università degli Studi di Trentowell if you are not happy about the radians, you can just say that Scherrer constant (0.9 in your case) is in rad as well and this sorts things out.

In any case, if somebody is interested inusing Scherrer formula for QUALITATIVE analysis of their patterns, they must use the FWHM of the peak measured in 2theta. No other factors, no HWHM. Scherrer formula can be written both with FWHM and beta (integral breadth). The functional from of the two equations is the same

Ravi Ananth· OnSight Technology USACan you help by adding an answer?