In published articles, we typically see lattice constants from X-ray diffraction and subsequent refinement given in precision up to 0.00001 Å (or, in other words, 1 femtometer). Is such a high precision possible due to the measurement process (Bragg scattering) effectively being based on interference?
What is slightly bugging me is that we achieve a resolution of the atomic positions (in a perfectly periodic lattice) which is typically five orders of magnitude smaller than the wavelength of the used photons (around 1 Å for X-rays). So, is the refinement to 0.00001 Å precision really meaningful?
I am curious to hear what experts on X-ray crystallography have to say on this.
Concerning the lattice parameters, it suffices to look at Braggs law, d=lambda/(2sin(theta). Differentiating d with respect to theta, then rearranging a little, and replacing differentials with deltas, you end up with:
delta_d/d = -cotg(theta)*delta_theta.
delta_d/d represents the error in the lattice param, i.e. for a 1 0 0 type lattice plane it will be equivalent to delta_a/a. delta_theta represents your experimental resolution, expressed through (half) the scattering angle, theta. In a home lab setup the resolution is usually limited by beam divergence, whereas with a non-focused collimated synchrotron beam and a high resolution diffractometer it is usually limited by monochromaticity or by the sample mosaic spread (crystal imperfection). For most situations and instruments it will be somewhere in the range 0.1 - 0.001 ( here in degrees).
Anyway, at high scattering angles theta -> 90 deg (back scattering), cotg theta -> 0, and the lattice parameter precision can become very high, irrespective of delta_theta. Ultimately limited by monochromtaticity which, dependent on your X-ray source, monochromator crystal and X-ray energy may approach delta_a/a ~ delta_lambda/lambda ~ 10^-4 - 10^-5.
Therefore, lattice parameters in Angstrom with uncertainties in the 4th digit is reasonable, even with home lab instruments.
Concerning the atomic positions the answer depends on several other factors, like crystal structure complexity, measurement temperature, crystal shape, incident X-ray energy, structure disorder, etc....
Concerning the lattice parameters, it suffices to look at Braggs law, d=lambda/(2sin(theta). Differentiating d with respect to theta, then rearranging a little, and replacing differentials with deltas, you end up with:
delta_d/d = -cotg(theta)*delta_theta.
delta_d/d represents the error in the lattice param, i.e. for a 1 0 0 type lattice plane it will be equivalent to delta_a/a. delta_theta represents your experimental resolution, expressed through (half) the scattering angle, theta. In a home lab setup the resolution is usually limited by beam divergence, whereas with a non-focused collimated synchrotron beam and a high resolution diffractometer it is usually limited by monochromaticity or by the sample mosaic spread (crystal imperfection). For most situations and instruments it will be somewhere in the range 0.1 - 0.001 ( here in degrees).
Anyway, at high scattering angles theta -> 90 deg (back scattering), cotg theta -> 0, and the lattice parameter precision can become very high, irrespective of delta_theta. Ultimately limited by monochromtaticity which, dependent on your X-ray source, monochromator crystal and X-ray energy may approach delta_a/a ~ delta_lambda/lambda ~ 10^-4 - 10^-5.
Therefore, lattice parameters in Angstrom with uncertainties in the 4th digit is reasonable, even with home lab instruments.
Concerning the atomic positions the answer depends on several other factors, like crystal structure complexity, measurement temperature, crystal shape, incident X-ray energy, structure disorder, etc....
I'm agree with Ragnvald, as u close to theta 90 degree, ur measurements getting more precise but in the other hand the intensity getting more and more overlap.
To get the satisfied result you had to know the nature of your sample, how you prepare it and how you major it with XRD (bulk or powder).
I suggest u compare with Neutron diffraction, cause typical X-Ray was diffracted by the electron cloud (refer to atomic scattering factor) so the intensity goes down as theta increase, it wont so good for light element, not yet the other possibilities that Ragnvald mention above...
So u need to know better ur material, and continue with refinement to particularly observe what's going on.
I've wondered about this myself - specifically, I've wondered about the reason for reporting the error. I understand the error can provide ideas to the accuracy of you measurements and accuracy of your fit, but when other data are presented with error, it is usually acknowledged as 'fit error' and not 'experimental error'. If someone really wanted to report all error of a crystal's unit cell, it seems as though one would fit (at least) three different samples - prepared in the same way - and then report the standard deviation of those measurements.
There is a nice paper on this issue, F.H. Herbstein (2000) Acta Cryst B56,547-557 "How precise are measurements of unit-cell dimensions from single crystals?"
In particular, the author cites the following story:
"...There...appears to be some confusion about the reproducibility of the results and the accuracy of the data...The exact height of the Emperor of China could be obtained by asking each of the 500 000 000 Chinese to guess his height. It was not necessary for any of his subjects to have seen him, or even his picture, because the application of statistical methods to so many individual observations would give an answer for the Emperor's height to a precision of a few microns or perhaps a few atomic diameters!"
What we obtain in a crystallographic experiment is precise data, precision is provided by statistical methods properly treating numerous observations of many Bragg reflections, that however does not warranty that the obtained values are accurate. Therefore the precision may be meaningful as estimated from multiple observations, but "`precise estimates are not necessarily accurate" (`Statistical Descriptors in Crystallography' (Schwarzenbach et al., Acta Cryst. A45, 1989) )
No, it is an experimental error, as you fit the position of several diffraction effects. That is, you are repeating the measurement of lattice constants several times ...
Yes - but this is on one sample - the error arises from the fact that the sample may have been prepared slightly differently and the unit cell may have been contracted or expanded from one sample to the next. That is, if you take a single crystal and get it's unit cell, the next time you take a single crystal from a different synthesis (or from the same synthesis) - you can obtain a unit cell that is outside of the error of the first measurement.
The cited accuracy is really possible.
The principal answer: the sampling (the number of atoms N) is huge and what you are speaking about is an AVERAGE distance between them (the error is ~1/N^1/2). People use precision determination of lattice parametr of silicon to get accurate Avogadro number. See, e.g., P. Sevfried, P. Becker, A. Kozdon, F. Lüdicke… in Zeitschrift für Physik B Condensed Matter (1992) or
Google (Scholar) for "Accurate determination of Avogadro number" to see more recent work.
I think you are confusing accuracy with precision, as Dmitry already pointed out ...
Sorry Paolo, but I am not confusing accuracy and precision, because I am speaking about sampling of atomic distances, which are directly related to the definition of lattice parameter, whereas judgments about the height of Chinese emperor generate random numbers. By the way,
the story of how tall is the Chinese emperor cited by Dmitry was presented as an argument in the summary on the round-robin test for precision determination of lattice parameters of the same tungsten powder sent to different labs in late fifties.
I believe this summary was given by
W. Parrish, Results of the IUCr precision lattice-parameter project, Acta Cryst. (1960). 13, 838-850.
Edward is right saying that limitations on precision are imposed by how good are our models describing environmental (temperature, stress state, etc) and instrumental (specimen geometry, absorption etc) effects.
I believe Mark is correct that the cited accuracy is possible. In practice though it is rarely achieved because of uncertainties in sample temperature, sample purity/homogeneity, and incorrect estimation of instrumental errors. The 1960 W. Parrish reference is a classic, and well worth reading - thanks for reminding us of it.
Dear R. H. Sceicher, The answer is depend what technique you are using for measuring the lattice constant of materials. Example, using difractometer with Bragg-Brentano geometry the precision of yours measurments are in the level of 0.001 Å. However, if you are using with function of errors Sin^2Bragg angle or Cos^2 Bragg angle or Sin^2 Bragg angle + Cos^2Bragg angle (In the book of X-Ray Diffraction by Cullity you can find more errors functions) the precision can arrive to the order of 0.0001Å. In the electron diffractions microscope the precision can be only 0.01Å. We are using with gold coating to increase the precision. Some time, the writters of the articles are using with wrong precision writting.
Dear all, I think that some distictions are here useful between single-crystal (SC) and powder XRD techniques.
As matter of fact, in powder diffraction the lattice parameters are estimated through the non-linear least-squares fitting of the whole diffraction pattern (such as in the Rietveld method). Therefore, in this case it is mandatory that possible correlations with other instrumental parameters are avoided or properly accounted for, as the latter reduce the precision and may also produce less accurate results. In any case, the better the OVERALL fit, the lower will be the estimated standard deviations of the refined parameters. It is common to obtain very low estimates for the standard deviations of cell edges by powder diffraction methods, especially when the cell angles are symmetry-constrained. To the best of my knowledge, this topic has been addressed in detail since long time (see for example Kaduk, 'Chemical accuracy and precision in structural refinements from powder diffraction data', Advances in X-ray Analysis 40, 1997 (http://www.icdd.com/resources/axa/VOL40/V40_352.pdf) or Hill & Cranswick, J. Appl. Cryst. 1994,27,802), and it has been found that repeated estimates of cell parameters from different samples are roughly normal-distributed, with variances at least one order of magnitude greater with respect to those obtained by Rietveld refinements on individual samples.
In any case, in my opinion Paolo is right when he claims that precision should not be confused with accuracy. Without other estimates of the 'true' value you are looking for, you may resort to other statistical tools, such as the normal probability plot, to detect possible uncorrected systematic errors in your dataset.
In SC experiments, on the other hand, cell parameters are usually deduced from the least-squares fitting against the angular positions of a set of accurately centred reflections spanning a somewhat large volume in the reciprocal space. This implies that great caution should be employed in comparing SC and powder cell parameters, as the former are often less precise than the latter (e.g. estimated standard deviations are of the order of 10^(-3) A using point detectors), but are also likely accurate. In fact, they come from independent measurements of intense reflections whose reciprocal space coordinates are usually known with a high degree of accuracy. I do not affirm here that powder estimates can not be very accurate as well, but that discrepancies between SC and powder results on the cell edges are quite common and are due to intrinsic differences between the two methods in extracting information from the diffraction data.
Hi,
If I understand you, you ask about the accuracy of x-ray spectroscopy. If that true, then I think the accuracy of this technique is technical issue more than academic i.e. if you want to gain a goo data you may need:
1. Heigh efficient detector that can absorb and see the lowest scattered or transferred beams (depending on your method).
2. Choosing the suitable pipe is of importance since the high diameter the more beams package.
3. Even voltage, current and the slit dimensions may play an important role in the measurements.
Lastly, you have to do some trials before deciding to take real measurements since each material has its own settings. Also a good fitting to the resulting curve is a key of the accuracy of your data with the required significants.
Good luck
Ray
Generally, 0.001 A is what is being reported. If one wants to increase the precision,
room temperature should be stabilized atl east to +/- 0.5 deg. This is due to thermal expansion of the sample itself. I feel this is an important point.
The crystal lattice parameters are not homogeneos above the level of (delta d)/d to 10-8 for the best avaible FZ dislocation free silicon crystal. That means that you can measure with x-ray diffraction the lattice parameters of any material with a precision above what is MEANINGFUL.
I guess you can find the answer if you go to the place where those things are the main concern: National Institute for Standards and Technology (NIST).
Have a look e.g. at the NIST SRM (Standard Reference Material) 640d or 660c standards (certified for cell parameters at a given temperature!) and at the procedure employed for the certification. Jim Cline spent a huge amount of time trying to find out all sources of error and to give a reasonable estimate of the lattice parameter error. I guess it's hard to have something more accurate than. The maximum of accuracy can be obtained if you use that internal standard and if you work with a powder that has sufficiently narrow reflections and that has sufficiently large number of peaks to be simultaneously processed. Forget about the figures you are presenting if you deal with defective or nanocrystalline materials.
The other suggestion if you are interested in those issues is the APD IV conference (Accuracy in Powder Diffraction IV, http://www.nist.gov/mml/apdiv_conference_2013.cfm). The progress in improving the accuracy of powder diffraction in the last 10 years will be shown. In my talk I will present something about accuracy in cell parameter determination in nanocrystalline powders: if you do not do things properly (the Rietveld method is not the Holy Grail), you can easily go out of more than 0.01A!
To the termine the lattice constants. It is important first to register the XRD diagram up to high 2theta angle with short steps and long time. After you can get the lattice parameter by uson the LeBail method
I don't belief it that the precision can't make the measurement to 0.00001A with a x-ray wavelength of only 1.0A.
That much precision will make the measurement to be unfaithful, who ever may do the measurement with whatever expertise in the field.
Such precision is also useless to express.
Again, it seems to me that some one of you is confusing precision with accuracy ....
...this last comment will definitely make it to the Accuracy in Powder Diffraction IV conference (I will hide the name of the scientist)! Frankly, this is the most absurd thing about X-ray diffraction (and physics in general) that I read in ages!
Prof. Leoni is of course correct. Precision of 1 part in 50000 has been considered "routine" for many decades. For Si that is about ±0.0001Å. That is ROUTINE.
In Adv. X-Ray Analysis, vol. 10 (1967) p.354, King and Preece showed that 1 part in 150000 (i.e., ±0.00003 for Si) could be achieved by double-scanning on a modified Siemens diffractometer with a Cu x-ray tube and sample cryostat. Needless to say, advances in instruments and analysis since then make precisions of ±0.00001Å a reality. Yes, you need to be extremely careful and control the temperature, etc. The only absurdity comes in carelessly applying high precision methods to poor quality samples.
In reality that much precision is just an absurd thing to be recognized by the scientific community.
Do you think people in NIST spent a lot of time vainly measuring accurately any of
line positions/line shapes of diffraction standards and certified cell parameters?
From other side, the better cell parameters we know - the more precise atomic positions we can determine... Sometimes, it is difficult to recognise Zwilling twins
with CCD-eqipped diffractometers (suggesting higher metrics of apparent cell), while accurate dimensions of real cell are easily effordable with PXRD... The list of cases when we PRACTICALLY need the most accurate knowledge of cell could be expanded enormously.
The precision in determination of lattice parameter is directly influenced of the precision of wavelength of X-ray used.
Sorry, but I must disagree with yours answer Constantine, what is with the geometry effect.
The precision of wavelength of beam is the one of the other factors that influences the precision in determination of lattice parameters. Other factor can be: temperature, precision in determination of Bragg angle, silts size, value of Miller index. Therefore, be happy Mr. Paul, the geometry influences also the precision.
For those interested: http://dx.doi.org/10.1154/1.3409482 is the paper on the certification of the NIST SRM 640d standard. The 640d is THE current standard for line position used by the powder diffraction community.
It is a nice reading for those interested in some error analysis related to the generation of a standard. It is just by analyzing the sources of errors that we can understand how to improve our instruments and increase accuracy and precision.
You can see that systematic errors are those contributing to most part of the measurement error. Lot of factors influence both precision and accuracy, including wavelength and instrument. There is some reason if we keep building large scale facilities (for instance where Andrew Payzant works..) and more precise and accurate instruments.
Remember that the stone age did not end because we run out of stones... but because people started to use their brain and experiment, even things that, to the majority, look absurd.
The fact that someone does not need a high accuracy or precision, or is not able to attain it, is definitely something that has nothing to do with science and is, in my opinion, just a sign of scientific myopia.
IMHO, one can meet not much papers where JUST lattice parameters determinations are reported from separate accurately measured peaks using specially calibrated instrument. So, original posted question could be rephrased as
" Is the refinement to 0.00001 Å precision really meaningful TO ME?"
accuracy on XRD related to the Instruments resolutions , more sofisticate instrument the resolution is better. From brag formula, 2 d sin(teta) = n (lamda), indicate that the precion of d measurement depend on the how resolution radiation wave length can be achieved and how the resolution in measurement of Teta (angle of difraction), in this matter represnt by angle step size. IMHO, the precision can be define precisely.
More precise, more meaning, .......
accuracy, precision and resolution are three different things. The angular resolution has nothing to do with accuracy. You can have a diffractometer with a 0.0001° resolution and double encoder but if you do not calibrate it properly you you always have non-accurate measurements.
I tried to find some article explaining things in an easy way and I guess: http://www.tutelman.com/golf/measure/precision.php can be a good reference!
By the way, Artem: the original question is "How precise can one really determine crystal lattice constants with X-ray diffraction?". The one you refer to is part of the discussion about the question.
The initial question (and some of the answers that followed) seems to be related to the Abbe limit idea (Ralph correct me if I am wrong). I mean given a wavelength lambda, with a traditional MICROSCOPE we can hardly resolve objects of the order of lambda/2. Well, even with diffraction we cannot directly measure interplanar distances smaller than lambda/2. Abbe limit cannot be applied here as we are working with diffraction (i.e. we use the interference of the waves) and not with microscopy (where we don't want this interference!).
So the answer to the initial question is: yes, the high precision is due to the measurement process and yes, a precision of 0.00001 Å can be meaningful if you do things in controlled conditions, but can hardly be obtained in the average XRD laboratory
Dear Colleagues,
I and Edward Payzant have already mentioned (20 days ago) that our discussion copies the one that was held in the community in late 1950’s - early 1960’s. I was reading those papers in early 1980”s and then have interpreted (to myself) the difference between precision and accuracy as follows.
Actually, the question about the difference between accuracy and precision relates not only to the determination of lattice parameters, but to any mathematical model of the physical phenomenon. It has a clear answer in the theory of statistical inference:
the precision coincides with accuracy until the model is valid.
We can judge about the validity of a model using a Chi^2 criterion.
Then, we can roughly say that = (Chi^2)^(1/2)*PRECISSION.
Coming back to determination of lattice parameters, the mathematical model can be invalid in description of physical (temperature, external stresses, etc) and instrumental (sample shape and position, absorption of x-rays, etc) effects.
My experience: the relative accuracy of 10^-5 is routinely achievable with the standard equipment.
I am sure there is no valid reason why to go for such non realistic approach by any way may it be experimental or model. I am sure no instrument have that precise resolution.
Several responses have given at this question, therefore I consider that this subject is finished.
I agree both with Dmitry and Ragnvald and thankful for such a nice comment
Every thing that matters is the data refinement but it should not go to that resolution that one is expecting, what ever refinement it may be.
Precision is a measure of how reproducible is a measurement (hence independent of the actual size of the object to be measured). Whereas accuracy refers to the true size of the object to be measured (hence how accurately this can be determined by the measuring tool). This distinction was properly addressed by Dimitry Chernyshov.
Regarding accuracy, the answer given by Ragnvald Mathiesen is quite exhaustive, the crux of the matter is resting in how close the measured diffractions are to the back-scattered direction (theta = 90 degrees) whereby the limit rests on the accuracy with which the X-ray wavelength can be determined.
Hence cell parameters (likewise all measurements), can be very precisely determined but could be affected by a low degree of accuracy, less frequently vice-versa.
The aim should be that of getting the same level of accuracy that the precision of the measuring tool can afford.
Romano is quite right. But, as I said before, if you take dislocation free FZ silicon single crystals, the most perfect and homogeneous material, it will not be homogenous above (delta d)/d to 10^-8. That is, the latice parameter will not be constant along the crystal. This is the precision limit where the measurement has a meaning and is independent of the method you used. The most precise method to measure lattice parameter is x-ray interferometry that does not depend of the x-ray wavelenght.
See "Windisch D and Becker P, Silicon Lattice parameters as an absolute scale of length for high precision measurements of fundamental constants, Phys. Status Solidi A 118, 379–388 1990."
I'm crashing the "party" here! And, I intend to make "waves"! How about measuring in the FEMTOMETER scale with XRD?
This is precisely what is accomplished in Bragg XRD Microscopy. Commonly known as X-ray Rocking Curve Analysis. Conventionally this was conducted using a 0D "point counter". We have developed the method into a practical tool for measuring deviations from IDEAL BRAGG CONDITION at a spatial resolution in the 10-100um range for up to 25mm diameter sampling area simultaneously.
Here for example is the essence of the analysis after crunching through TERRA BYTES of data in real time video mode. We've been able to achieve precision down to fractions of Arc Sec in Omega resolution (
Ravi,
I could not understand spatial resolution of 10 - 100um and it can be achieved with good resolution in Small Angle X-ray Scattering (SAXS) if the above is 10- 10nm.
SAXS is the potential technique other then microscopy with good averaging all the time.
Dillip! I should have clarified. I meant spatial resolution in the real space. The 10-100nm for SAXS is through the reciprocal space (2θ). Using the XRD Rocking Curve Analysis or XRD Microscopy it is possible to resolve FEMTOMETERS in strain-field variations.
When I say 10um, I mean the average pixel size/spacing in the 2D array of "individual detectors" (pixels). For SAXS analogy here is a table converting to δ2θ:
http://www.flickr.com/photos/85210325@N04/7977700644/in/set-72157632728981912
At 900mm SDD (Sample to Detector Distance) you could achieve about 0.0005 degree per 8um spatial pixel.
Here's how the rocking curve technique is used to resolve nm to 10's of nm:
From the good folks at Caltech - X-ray Rocking Curve Analysis of Super-lattice Epitaxial Structures: http://authors.library.caltech.edu/10206/1/SPEjap84.pdf
Ravi,
I am not still not satisfied with your explanation and believe scattering experiment is superior then microscopy so for as structural (spatial) resolution.
Dillip! I'm reffereing to XRD (scattered beam) Microscopy and not X-ray (Radiographic) Microscopy. The method has been used for more than half a century. Also known as rocking curve analysis. I'm actully in concurrance with you in the statement "scattering experiment is superior then microscopy" only if you mean radiograph when mentioning "microscopy".
Dillip! I'd love to delve in to this further and clarify my understanding of the subject better.
With SAXS it is certainly possible to resolve nearest neighbor distances matching the figures you've quoted. Meaning the measurements with SAXS is the average parameter for the diffracting volume (sampling size). With the micro-focus system the sampling size may be minimized further. If you were to translate this beam across the sample then it may be feasible to use it as well in the microscopic mode. The challenge is how to display the various reciprocal space (2θ or 2Sinθ/λ) parameters measured by SAXS for each spatial VOXEL of sample (real space).
To answer the question "How precise can one really determine crystal lattice constants with X-ray diffraction", my response - with in FEMTOMETERS with XRD Microscopy or X-ray Rocking Curve Analysis. We've just used it in 2D Real Time Microscopic Mode. I learnt it from the group at Rutgers way back in the 1980's. (Sigmund Weissmann et al).
Here is an example of its use in the case of a 0.5µm MBE Super Lattice Epi of InAs/InAsSb on (001) GaSb with 25nm layer period:
http://www.flickr.com/photos/85210325@N04/8619377536/in/photostream
Through proper calibration we can detect epi thickness in the "µm", SL periodicity in the "10's of nm" and crystallographic lattice strain variations in the FEMTOMETERS.
Currently "spatial resolution" is limited by the detector pixel size at about 1-5µm. Theoretical limit is the wavelength of X-rays. May be we should start using the term "nano-structure resolution" for your definition of resolution of SAXS?
Ravi,
I understood everything that you explained and the nano order structural resolution in incorporated in the SAXS data.
Dillip! I appreciate the opportunity to synergize with you. Looking forward to more. I'm new to Research Gate and I see a lot of good technical content in discussion here.
Please feel free to join "X-ray Diffraction Imaging for Materials Microstructural QC" group and share your expertise:
http://www.linkedin.com/groupItem?view=&gid=2683600&item=ANET%3AS%3A218925637&trk=NUS_RITM-title
All of you are pre-approved and invited! Come join us and help us understand XRD better. Thanks!
Dear Ralph,
Just one more comment to the second half of your question. I am not expert in XRD, but I have worked with atomic resolution holography, and worked a lot on the question of resolution and accuracy. Forgetting the disturbing instrumental and physical effects :), the accuracy of measurements using interference (like diffraction) does not defined by the wavelength itself, but by the phase differences coming from different parts of the sample. i.e. if you have two atom very far from each other, then a very small movement of one atom causes large relative phase shift, thus your interference image changes. So, on paper infinite accuracy can be achieved if you have an almost non-scattering infinite perfect single crystal. The reality is written above.
0.00001 å ｉｓ ｃｌｏｓｅ ｔｏ ｔｈｅｒｍａｌ ｅｘｐａｎｓｉｏｎ ｉｎ ａ ｆｅｗ ｄｅｇｒｅｅ， ｉ ａｌｓｏ ｂｅｌｉｅｖｅ ｔｈｉｓ ｍａｙ ｎｏｔ ｂｅ ｍｅａｎｉｎｇｆｕｌ ｆｏｒ ｓｏｍｅ ｍａｔｅｒｉａｌｓ．
This paper answers your question:
https://www.researchgate.net/publication/242284045_UNCERTAINTY_ESTIMATION_OF_LATTICE_PARAMETERS_MEASURED_BY_X-RAY_DIFFRACTION
Article Uncertainty estimation of lattice parameters measured by X-R...
see this paper:
https://www.researchgate.net/publication/242284045_UNCERTAINTY_ESTIMATION_OF_LATTICE_PARAMETERS_MEASURED_BY_X-RAY_DIFFRACTION
Article Uncertainty estimation of lattice parameters measured by X-R...
M.F. Campos! Please forward the PDF copy of the article when convenient. Thanks!
The theoretical precision problem was elegantly addressed by Klaus-Dieter Liss. Another matter is how to quote the values obtained from measurements of lattice constants (through x-ray diffraction) in terms of accuracy. That is how far are we from the real dimensions of those lattice parameters. This I feel is by and large dependent of the technique and, as Ragnvald Mathiesen rightly points out on the scattering angle at which the data are colleted. By persoanl experience some 45 years ago (! - in my thesis work) a back-scattering Weissenberg camera, once care is taken about film shrinkage, can yield measurements with accuracy and precision well within the 4th decimal digit in Angrstroms!
Rinaldi,
Of course you are right but optimization the problem is the major concerned.If any of us will work on it seriously then it will be an asset for x-ray cryptographers.
Thanks for your valuable comment on the issue.
It seems to me, that absolute accuracy is the problem of yur X-ray diffractometer.
But it is not correct ansver! Temerature stability, temperature expansion of your sample.
ICDD card usually connect a,b,c size with +20 degree. Sometimes with +24o. - What is the better?
Trust to your relative measurements, use internal standard and try to connect your results with ICDD card of this standart, calculate corrections- amendment. Accuracy of those procedures is your error.
By the way: sometimes your standart (for example Si) has dopants: it may be B or P! The size of crystal unite cell differs. Hidrogenium and etching affect the results.
There is no perfection in the world. You can not get perfect crystal and perfect conditions of measurements.
"It's all relative", said Einstein
Dear Ralph,
I am actually not an X-ray physicist, but there is a theoretical answer to the question: the resolution (in the real space) depends on the wavelength you use in scattering experiments, but the accuracy does not. Moreover, the accuracy is not depending on the resolution of the instrument (Here I disagree with Ragnvald H. Mathiesen). The instrumental resolution gives an upper limit of the accuracy of the lattice spacing, but if you know well the resolution of the instrument (not the width, but the shape also) then the accuracy comes from the fitting (depends on the statistics). I do not think that Rietweld would lie. Of course if you do not take into account other effects (change of themperature, impurities, vacancies, inelastic signal in neutron diffraction etc) then you can get "accurate" but wrong data.
As I said before:
Cesar Cusatis · 31.80 · 114.61 · Universidade Federal do Paraná
The crystal lattice parameters are not homogeneos above the level of (delta d)/d to 10-8 for the best avaible FZ dislocation free silicon crystal. That means that you can measure with x-ray diffraction the lattice parameters of any material with a precision above what is MEANINGFUL.
With deails:
The distante between atoms in a crystal depends of several parâmentes: structural defects like point (intersticial, substitucional, vancanies, present even on perfect single crystals), linear (dislocations), planar, espacial, inomogenitie, etc. that causes strain and consequently short and long range stress, that is, variation os laticce spaciing. Because that two crystals of the same substante grown in the same manner will not have EXACTLY the same average lattice space.
So, reporting to the original question:
"How precise can one really determine crystal lattice constants with X-ray diffraction?"
X-ray interferometry is probably the most precise method to measure lattice constante (at least on the x-ray scale). But nature does not produce PERFECT single crystal.
That is, there is no meaning trying to mesasure the size of and elefante witth high precision (1 mm?).
The precision is limited by the quality of the crystal and not by the method used to measure the lattice constant.
It is a problem that has not an practical application. The precision in estimation of lattice parameter is limited of accuracy of knowledge wave X-ray used in analysis.
Leonardo Lo Presti above is right: The initial question was
How precise can one really determine crystal lattice constants with X-ray diffraction?
Did not specified powder diffraction or single crystal diffraction. As perfect single crystals are the more perfect material on Earth cell parameter determined in this material is can be the best accurate result. See, for instance,
The lattice parameter of the 28Si spheres in the determination of the Avogadro constant
E Massa1, G Mana1, L Ferroglio1, E G Kessler2, D Schiel3 and S Zakel3
Published 22 March 2011 • 2011 BIPM & IOP Publishing Ltd
Yes, there is a confusion in this discussion between accuracy and precision.
To my post 3 years ago:
In my precise experiments I use Bond diffractometer - Poland and the same designe diffractometer in Russia.
Kristall und Technik 13.5.1978 561-567 K. LUKASZEWICZ, D. KUCHARCZYK, M. MALINOWSKI, A. PIETRASZKO - Wroslaw, Poland
http://onlinelibrary.wiley.com/doi/10.1002/crat.19780130516/pdf
New Model of the Bond Diffractometer for Precise Determination of Lattice Parameters and Thermal Expansion of Single Crystals
Two registerd plus and minus X-ray peaks - we got 4 Theta degrees....
- More precize! Temperature stability!
We were restricted by half width of inicial peak, only.
- To Cesar Cusatis comment 3 day ago:
Each crystal has its own peculiarities associated with its crystal structure. We do not investigate the absolute crystal, we will examine the crystal from which the device with specific properties is made. We associate cell dimensions with these properties.
Thank you for your message. I quite agree with you, specially about temperature. The main point is: if the object to be measulred is not define with the aimed precision there is no point in train to measure it with that precision. That limit actually is delta d/d ~ 10^-8 for the best silicon crystal in the world.
I can not agree with the answer of mr.Preston Guynn!
His reccomendation deals with his article, not with thte question of Ralph H. Scheicher! - 27 pages of mr. Guynn's reasoning about the internal properties of matter and advertising his articles.
Dear Ralph, read the book of Bowen & Tanner "High resolution X-Ray diffractometry and topography".
Preston Guynn's "answer" has been reported to RG as irrelevant and blatant self-advertisement. - Thanks for the recommendation Michael!
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