The Williamson-Hall plots for ZnO nanoparticles showed parabolic curves. How can I interpret it?
Addition to Dominique Goisset:
The line broadening pertaining to size and strain have both to be Lorentzian for the classical version of a Williamson-Hall plot. This is rarely fulfilled. However, according to Balzar you can separately use Gaussian and Lorentzian contributions of a Voigtian which should add up quadratically and linearly, however, a very high data quality has to be available if you want to do this reliably.
Moreover, strain should be constant over sufficiently large distances (Stokes-Wilson approximation). Isotropy needs NOT to be present. However, then you have perform the WH plots for different hkl types separately (where you often have only two reflections orders; also a big limitation).
In my experience (except for cold-deformed material) you have frequently only size or only strain. Then the WH plot is very valuable for data analysis, because then the limitations with respect to peak shape of the line broadneing do not play a role.
Don't bother. It is not a reliable method for determining strain. Why it is still used I have no clue.
@Lawrence Margulies
This statement is too simple. A Williamson Hall plot is definitely not the best method for size-strain SEPARATION, in particular for cold-deformed matter. But it gives you trends and it is very valuable, if you have one single/dominant source of line broadening. Generally, it is a good step within data analysis. Of course you should be plotting peak-widths adequately corrected for the instrumental resolution.
For parabolic/non linear WH-Plots, check the recent thread "Curved plot in Williamson-Hall"....
In my opinion, no need to worry about the shape of the curve. Just pass linear line using any software ( like origin, excel) it slope and y-intercept is important. From slope you can get Strain, and from y-intercept you can get crystallite size. In case of parabolic shape the slope of linear fit will be very low indicating the lower value of strain.
Definitely you should think about the shape of the curve, at least if you are interested in the physics behind your line broadening. Follow the procedure suggested by Dilwar Ali if your boss ask for numbers and you want to present this numbers to him because you want to obey but you do not care about reality....
The linear version width_on_1/d vs 1/d gives only size and microstrain values under very specific assumptions. In that sense, the critisism of the WH-Plot is justified.... If you get a curved shape, you know that some of the classical assumptions for the WH plot are simply not fulfilled.
Dear Dr. Ferreira:
In the past, 1990, I synthesized ZnO by Pechini’s Method using as soluble, in ethylene glycol, salt precursor at zinc source both nitrate and acetate finds were published at Congress Proceedings of 35th Congress of Brazilian Ceramic Society.
Despite lack of information, taking in account the Zn precursor used, there is a great possibility that ZnO nanoparticles not be those nanoparticles formed during the first or original generation, being in fact the result of a sequence of nanoparticles in some stage of growth.
A priori, first nanoparticles would have 2 nm, see, only by hypothesis in a speculative way, if a coalescence mechanism of growth is operational, the size sequence would be 2, 4, 8, 16 and 32 nm. See, after 5 generation or growth stages, a size of at around 30 nm is reached. Temperature and time can modulate this value.
This powder analyzed by SEM exhibits a zoologic of particles since from 2 nm , and all combination possible as 32x5 = 160 nm, 16x5 = 80 nm, 40 nm and 10 nm …This trend gives a distribution of sizes with center at around 32 nm. See, when temperature increases other mechanisms can be active giving great particles with shape, geometry and texture complex and or heterogeneous, sometimes called aggregates and agglomerates.
As a whole, the nanoparticles that being characterized have the most of size in the range from 25 to 60 nm, typically. The existence of a thermal step type calcinations in an ambient atmosphere and presence of a carbon–chain that can be so short as acetate group gives an exothermic event that generate thermal energy sufficient to activate to particle growth by a mechanism type above or another.
Then, it is very difficult to synthesized ZnO particles really nano-sized at around 1 to 3 nm, as noticed previously. There are real difficulties in a generalized always at crystalline structures of low symmetry also to structures with multiplicity of unitary cell, or both events. Have im mind that by thermodynamics considerations at around 1nm, almost all defect are positioned at “surface” region of nanoparticles, this feature leads to an excess of superficial energy, that exerts significant forces at the “bulk”, being the origin of spherical curvature of nanoparticles.
Starting by technology of the diffractometer, in a general way if the technology is actual, at least a similar trend to the rotating anode should be operational, that allow to access high resolution measurements. Realize all set of calibrations described at operation manual. Proceed to all alignment indicated. Verify if spot and focal spot size of incident beam are optimized. Explore the limit of equipment during measurements, selecting a minor step size of scanning and increasing the integration time.
See, below, is listed two references that(~60 nm) described further condictions to use W-H successfully. Both, accessible from “Google”.
Williamson-Hall analysis in estimation of lattice strain in nanometer-sized ZnO particles
Journal of Theoretical and Applied Physics 2012, 6:6
X-ray analysis of ZnO nanoparticles by Williamson–Hall and size–strain plot methods
Solid State Sciences Volume 13, Issue 1, January 2011, Pages 251–256
Kind regards
Marcos Nobre
All points above are very valuable. I would add that a WH plot only works if coherence length and strain are isotropic. Depending on how many data points you have the parabolic shape may merely result from the fact that strain or grain size are anisotropic, which means different reflexes are broadened differently. However, reflexes in the same direction, e.g. 100, 200, 300, should still give a straight line.
Hall-Williamson method leads to data on a straight line only if the hypothesis are satisfied: Sherrer-like formula for grain size and elastic behaviour. Moreover, the experimental contribution to linewidths have to be deconvoluted prior analysis!
Addition to Dominique Goisset:
The line broadening pertaining to size and strain have both to be Lorentzian for the classical version of a Williamson-Hall plot. This is rarely fulfilled. However, according to Balzar you can separately use Gaussian and Lorentzian contributions of a Voigtian which should add up quadratically and linearly, however, a very high data quality has to be available if you want to do this reliably.
Moreover, strain should be constant over sufficiently large distances (Stokes-Wilson approximation). Isotropy needs NOT to be present. However, then you have perform the WH plots for different hkl types separately (where you often have only two reflections orders; also a big limitation).
In my experience (except for cold-deformed material) you have frequently only size or only strain. Then the WH plot is very valuable for data analysis, because then the limitations with respect to peak shape of the line broadneing do not play a role.
Dear Dr. Ferreira:
Seems that there is a great number of ideas that can be joined to reach a more efficient or a fast succesfull.
Is fact that peak shape from view point of line broadness have not some kind of influence in a WH plot. But, also is fact that "..., reflection frequently overlap, particularly at high two-theta values....". This phrase was picked up in Book 23 of the serie Oxford Chemistry Primers with Title "Inorganic Materials Chemistry", authored by Mark T. Weller, 1994!!!!!!!
See, a priori, in a cubic primitive structure such kind of carefull is not necessary in the "assignement" of two-theta position of a particular diffraction line. But see, in a face centred cubic material there are some diffraction lines very close or overllapped. Now, the use of theoretical adjust of the set of diffraction lines can be a usefull tools, since will allow to derive further two-theta position which is sine qua non condiction to a high quality WH-analysis. Depending on the equipqment, this analysis can be carried out after acquisition of data. Some two-theta intervals can be selected, and the adjust function to the fitting of diffraction lines enveloped or not can be determined. The set of non-symmetric functions of the class Lorentz and Voigt should be selected as mentined at another analysis. If in the menu, the class Voigt is absent, Pseudo-voigt as a linear combination of Gaussian + Lorentizian should be tryed. As expected, a reliable set of two-theta positions will be derived.
Kind regards
Marcos A. L. Nobre
Thank so much all of you for helping me to understand what happens with my WH plot in which a parabolic curve, for my surprise, showed up. Bellow, I am giving some comments and new question. Please correct me if there are misinterpretations because am I not an expert in this issue. I am just staring to learn about.
So it is clear to me that WH method is quite realistic for separating strain broadening and size broadening in the presence of isotropic microstrain, however, is not suitable in case of crystalline material that present inherent anisotropic nature (Dr. Leineweber has discussed well about it in his note on “Parabolic microstrain-like line broadening induced by random twin faulting, Philosophical Magazine, 92:14, 1844-1864.”). Recently, several authors have used an non-traditional, i.e. a modified-WH method, to separate size and microstrain broadening in XRD lines, mainly in ZnO nanoparticles, taking into account the anisotropic nature of the crystal (thank you Dr. Marco Nobre for supporting me the good references presented above). These modified-WH methods (please correct me if I cannot name it in this way) consider a “uniform deformation stress” or “deformation energy density” assuming that the “anisotropic microstrain” is the “uniform stress deformation”. Therefore, if it is assumed that deformation stress is uniform (has same value in all crystallographic directions), it is possible state that the density of energy is anisotropic due to the anisotropic behavior of the elasticity’s modulus. Since stress (σ) and strain (ε) are given by σ = Yhkl.ε and the energy by u=(ε2 Yhkl)/2, thus an βhkl cosθ vs 4 sinθ (2/Yhkl)1/2 plot is linear and yields u from the slope and crystallite size from y-intercept. It works wonderfully well when data points have a low or very low scatter; however, the accuracy of the method for estimating size and strain contribution of XRD lines is reduced with increasing of scatter. However, it occurs that WH plot for all my samples present parabolic curves and if we consider all the data points in WH plot for analysis (more internal and more external plans) as suggested by Dr Dilawar Ali, it seems that we have an example of high scatter points so that whether linear fits to these nonlinear data are considered to calculate average crystallite size, the results may carry very large inherent uncertainties (Can I say “the strain anisotropy effect will be ignored”?). But if we split the WH plot in two regions of analysis and choose one of them to plot a linear fit, we have a problem since the obtained slope and intercept values do not show the “real value” (i.e. if we take the slope of the plot for more internal plans and determine from this fit the microstrain, this obtained value may not be consider the effective “microstrain value” since ±ε is possible, right?). Another question I would like to highlight (though do not know how relevant this is) that, after account the instrumental broadening and last cycle of refinement, calculated crystallite sizes are smaller for more internal plans (plans with smaller interplanar distances) and larger for external plans (first diffraction planes families). Hence, the parabolic behavior of the WH plot indicates that the existent residual stress is not uniform in all crystallographic direction (there is a distribution of both tensile and compressive forces in the crystal lattice of the nanoparticles). Can I state that? Anyway, taking in mind all these consideration, if it is true, I may follow two ways from now on:
1) Consider the Dr Leineweber’s suggestion regarding to use Gaussian and Lorentizian contribution of a Voigtian. As a matter of fact, crystallite size and strain profiles are described by the Lorentzian and the Gaussian function, respectively. In this case, a better evaluation of the size and strain parameters can be obtained by considering the Halder-Wagner method (C .Halder and C.N. Wagner. Acta Cryst. 20 312, (1966)).
2) Consider the Dr Nobre’s suggestion regarding to “zoologic of particles size/shape”. TEM analysis from my ZnO particles presents the same behavior of the ZnO particles of Dr Nobre, which were analyzed by SEM (unfortunately I did not find his work: “Nobre, M. A. L. ; D. Gouvea ; E. Longo ; J. A. Varela . Citratos como Precursores de Varistores. In: 35º Congresso Brasileiro de Cerâmica, 1991, Belo Horizonte - MG. 35º Congresso Brasileiro de Cerâmica, 1991. v. 1. p. 24-30”). If we take into account the existence of different shapes and geometries of the particles, perhaps I can also try the Ungar method (T. Ungár, G. Tichy, J. Gubicza and R.J. Hellmig: Correlation between subgrains and coherently-scattering-domains, J. Powder Diffraction, 20, 366 (2005).)
"As a matter of fact, crystallite size and strain profiles are described by the Lorentzian and the Gaussian function, respectively."
Yes, this is frequently adopted, but also an assumption. Indeed, Lorentzian microstrain broadening as required for classical WH-method is incompatible with the Stokes Wilson approximation and with the equations for elastic strain energy you mentions. But I also know good examples contradicting Gaussian strain. This is in particular dangerous if you apply single line methods, which rely on Lorentzian size (also an approximation!) and Gaussian microstrain.
Bcos(teta)=0.9 (landa)/d + (η)sin(teta)
Thus, when (B cos(teta) ) is plotted against (sin(teta)), a straight line is obtained with the slope as: η=2*strain and the intercept as 0.9 (landa)/d . From these, you can calculate the crystallite size, d and the lattice strain η.
X-ray peak broadening methods are most appropriate for crystallite sizes in the range of 10-100 nm
Following papers may be useful for you,
Transition metal (Co, Mn) co-doped ZnO nanoparticles: Effect on structural and optical properties
http://www.sciencedirect.com/science/article/pii/S0925838816341482
Analysis of structural, optical and magnetic properties of Fe/Co co-doped ZnO nanocrystals
http://www.sciencedirect.com/science/article/pii/S0272884217305886
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