You can determine the detection limit based upon the signal to noise of your measurement. If you measure infinitely long you can detect an infinitesimly low phase fraction, since noise is the square root of the measured intensity.

The limit of a detectable fraction is only a rule and depends on the phases which are mixed. For strong scatterers like W, Mo etc the limit can go clearly lower. But without seeing the diffractogram I would say that 0.056% looks to me at least one order to small. But show us the refinement, and we say more. As long as there is no significant signal for this phase I wouldn't trust any result simply because of a given number. 

You can determine the detection limit based upon the signal to noise of your measurement. If you measure infinitely long you can detect an infinitesimly low phase fraction, since noise is the square root of the measured intensity.

You can not determine a specific detection limit or a rule for that, because it all depends on the exact type of crystalline phases, their crystallite size, their chemical composition and the chemical composition of your batch sample. But you may have good chances to go as low as 0.1%(wt) for some materials... or maybe even lower, depends on the apparatus and measuring conditions you are using.

you have found the volume fraction of a certain phase is 0.056%. have you refined it with more than one phase?? your question seems to be incomplete.

Did you do some calibration with samples of known composition and a standard?

Maybe I'm wrong, but as far as I know this is the only reliable way to determine accurate phase fractions.

Thank you all for great answers. I am analyzing a mixture of ferrite and (Nb,Ti)(C,N). I have attached a photo that shows my fit results along side the peak indices. I have also attached the pattern I am analyzing in two different formats.

@Gert: Thanks for the answer. Is there a way to quantify the detection limit?

@ Lawrence Margulies: Thanks for the great answer. That is exactly what I need to do. How does S/N relate to detection limit?

@ Frenec Kristály: How do I calculate the detection limit?

@Alfred Amon: Thanks for your answer. I have done LaB6 with this XRD machine. Can I use that for calibration?

I agree with all the comments here. From my experience with Rietveld your number looks too low. Also, looking at your diffractogramm, the point scatter is very high so I would even trust the number around 1%. I just wonder what kind of instrument you are using to obtain such diffractogramm?

In my lab, we've done some calibration curves to evaluate the accuracy of Rietveld: it was witin 0.5-1.0% for a few known mixtures of two inorganic salts.

We've participated in Round Robin on Rietveld organised by ICDD - unfortunately, I do not know how to attach this file here, but I can send it to you by e-mail if you like.

@ Yuriy Ososkov: I used Rigaku Ultima IV X-ray Diffractometer with a copper anode using both Kalpha1 and Kalpha2 wavelengths. I'd appreciate any help. My email is [email protected].

1. what i meant was not a calibration of the instrument but a quantitative calibration curve like in basic analytical chemistry. i.d. measuring mixtures of your phases with known composition and  mixed with a standard (e.g. LaB6 or any other suitable material). then you can relate the peak intensities of your phases to the amount and create a calibration curve. afterwards you measure your sample of unknown composition with admixed standard. obviously you should keep the amount of standard constant (by weight fraction). just as you probably did in the basic analytical undergraduate courses.

having said that:

2. as i remember, the statistical limit of DETECTION is defined as 3x the estimated standard deviation (esd) of your blank (≈ esd of your background). the limit of QUANTIFICATION  however is defined as 9x the esd.

I didn't analyse your data, but from graphical inspection it seems that you hardly reach the limit of detection (with statistical certainty), and what you fit in your rietveld refinement is mostly NOISE.

I guess one can argue that the (Nb,Ti)(C,N) phase is present. but personally I would not go so far as to state a number based on this data. You should keep in mind that a rietveld analysis will always give you some value. that doesn't necessarily that it is correct.

Maybe some metallographic investigation is more suited to determine the approximate phase amount.

Alfred: this is correct. Some publications even suggest S/N=5. Actually, I meant excatly what you've meantL not calibration of the instrument, but calibration curves using known mixtures. Still cannot get why the spectrum is so noisy...

@Alfred Amon and Yuriy Ososkov:  Thank you both. I guess I should get a cleaner XRD next time with less noise for such low volume fractions.

Just a short additional comment: Lawrence, in essence, you are correct, but I think only in theory. To my knowl;edge there are two limits in general: LOD (which you calculate as you say) and MDL (method detection limit) which would be significantly affected by instrumental and other limitations. I've never heard about any of the XRD method able to detect below 0.1% (in a very ideal case).

@ Yuriy: do you think it is very noisy? My conclusion would be that it is just a very small amount of the second phase. also maybe the sample preparation was not optimal. I see quite some peak broadening

I agree with the recommendations regarding standard mixtures. Full pattern refinement has been introduced in order to coming over the former procedure of standard mixtures, and it works extremely nice. But one should always take into account that this mathematical approach has a lot of uncertainties...whereas the experimental technique can never be completely applied to complex mixtures as analysed with a full pattern refinement. On the other hand, it can make sense to switch back to the older technique if the conditions are hard on the edge as in your case. As written before; a mathematical algorithm usually will always give you a number. Reliable software has some thresholds which simply ignore especially tiny results. Evaluating your XRD pattern I would also say, that it is more the noise which you interpret. Did you know that this phase is existing? Did you checked it e.g. in SEM (cf. A. Amons answer above)?   

@Gert Yes I knew this phase existed before doing XRD. I have seen it in TEM and SEM. They are very fine (6nm to 400nm) dispersed particles in the matrix of ferrite. What I need to know is their amount. TEM (carbon extraction replica) and SEM do not allow for an accurate and reliable measurement.

Dear Khodaie. I think that it is very imprecise to measure by XRD such a small amount of Nb carbonitrides . What I suggest to you is to report the theoretical value, as a approximation of the amount of precipitates that could be present in the sample.

My best regards. Ana

@ Ms. Rivas. Yes I think that is a good idea.

Just a comment. A number of times in this discussion the value has been referred to as volume fraction. Quantitative phase analysis using the Rietveld method calculates a weight fraction rather than a volume fraction.

@ Alfred: Thanks for the comment. I mostly referred to the noise in the inset. Overall spectrum looks njot bad, but when it comes to a very low amounts of some phases, to fit peaks as shown in the inset may be quite dangerous. I also agree about peaks broadening - it seems to be logical if the expected particles size is between 6nm and 400nm.

@ Ian: the usage of weight fraction is not better, only more convenient for practical reasons since you can weight a phase but hardly count the mols or atoms or measure their volume.  Finally for diffraction the number of scattering "particles" is important and not their weight or volume. In so far I (personally) prefer the use of at% as long as possible. Nevertheless, for the scaling of the diffracted intensity of phase mixtures their unit cell volumes have to take into account as expression of a normalized "scattering power" of each phase, whereas the mass of the phase does not appear at all since it does not contribute to the diffraction result. In so far the usage of vol% makes actually more sense to me than the practically preferred w%.

Finally it doesn't matter since all can be easily transferred into each other.

BTW: there are many phases which doe not have a completely solved structure since the positions of hydrogen atoms, OH groups or the position of crystal water are not clear, unknown, not refinable by XRD. In that case every scaling factor is wrong since the predicted "scattering power" is wrong since the volume of the unit cell of this phase is correct indeed, but structure elements are missing so that the scattering power per volume element i underestimated. The result is an overestimation of this phase and an underestimation of all other phases. This can reach up to ten vol% or w%. A nice example is ettringite. 

Thanks for that clarification Gert. Interesting point about ettringite. I use a structure for this quite frequently. The structure supplied with TOPAS des not fit well at all on natural materials, so I searched the various cif files and played around with refinements. I believe I now have a useful str file for this. However, having read your note I will go back and see if I can prove the accuracy of the values. This is important stuff!

Hi Ian, I am not sure how TOPAS is "correcting" or taking the missing structure elements like crystal water into account. Usually what is not part of the structure file, from the softwares point of view it simply doesn't exist. In order to get more reliable quantitative results it might be a solution to put it somewhere in the unit cell where it does not effect the intensity that much, but it is hard to give a general recommendation were this place might be :-).  A better way is perhaps to use the original structure files (CIF, str cel) but do a re-scaling after refinement taking the real scattering power into account. From my point of view this should work best. The theoretical structure data cannot be very accurate anyway, since during their refinement the amount of OH or crystal water has been neglected as well. The structure data have been optimized only with respect of the best fit to the experimental diffractogram. In this case it is "good" since finally it means that for mixtures only the scaling factors need to be "adapted" in a post-poned step.

Sorry Nasseh for hijacking your thread! If you prefer we can take this offline.

Hi Gert, the TOPAS str files do have O and H positions in the lattice for the ettringite structure. So it is using these atoms in the refinement, they do have atomic co-ordinates. It is just that I could never get any fit using this structure, so I took the cif from Goetz-Neunhoeffer F & Neubauer J, 2006. Powder Diffraction, 21, 4-11. Attached. I refined this using a natural specimen containing a high proportion of ettringite, introduced a preferred orientation on 010 plane. This does seem to work on my materials. I would welcome you comments on this.

http://rruff.geo.arizona.edu/AMS/CIF_text_files/19007_cif.txt

Not at all Ian. This is very educational. Please continue.

Ian: thank you for posting the file. Could you tell how you did the attachment? I could not still figure this out. Yuriy.

Yuriy there is a small paper clip picture under the box you write your comment. Click that. Maybe your browser don't show it.

Thanks now I see it 

This has been pretty well discussed, but I want to vote "yes" on this:

A. L. Rivas · 10.96 · 9.04 · Simon Bolívar University

Dear Khodaie. I think that it is very imprecise to measure by XRD such a small amount of Nb carbonitrides . What I suggest to you is to report the theoretical value, as a approximation of the amount of precipitates that could be present in the sample.

My best regards. Ana

I suggest adding that the data is consistent with the theoretical quantity.  That is to say that the data supports that value, but does not prove it. 

Also, you have one peak that is well fit, but the smaller peak is not fit because it is hardly there.  If you have any preferred orientation, the one peak you see is not reliable to quantify the amount of the phase.  The peak that is hardly there may be representative of the average or the larger peak.

I agree with Martin. For instance, ASTM for retained austenite content in steel requires Mo tube and not Cu one. Yuriy.

@Martin Yes I agree that examining Fe by Cu should give very high backgrounds due to fluorescence. But looking at Nasseh's original diffractograms the background level appears to be very low at only 20 or so (counts/cps?) while the signal to noise is very good. Normally when I run Fe with Cu I get counts in excess of 400 in the background and odd shaped backgrounds. I tend to use window discrimination in the detector to remove it. Although we are in the process of getting a Co tube. I am puzzled by the very low backgrounds if this was Cu on Fe with our any energy discrimination. 

Hi Ian, I guess he  used a monochromator or an energy discriminating detector. This signal is definitely not the original Cu signal.