research papers 252 https://doi.org/10.1107/S1600576719000621 J. Appl. Cryst. (2019). 52, 252–261 Received 5 July 2018 Accepted 11 January 2019 Edited by G. Kostorz, ETH Zurich, Switzerland Keywords: powder X-ray diffraction; materials characterization; angular range of XRD patterns; Rietveld refinement; statistical treatment; quality assurance/quality control applications. Supporting information: this article has supporting information at journals.iucr.org/j The influence of X-ray diffraction pattern angular range on Rietveld refinement results used for quantitative analysis, crystallite size calculation and unit-cell parameter refinement Vladimir Uvarov* The Unit for Nanoscopic Characterization, The Center for Nanoscience and Nanotechnology, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Givat Ram, Jerusalem, 91904, Israel. *Correspondence e-mail: [email protected]This article reports a detailed examination of the effect of the magnitude of the angular range of an X-ray diffraction (XRD) pattern on the Rietveld refinement results used in quantitative phase analysis and quality assurance/quality control applications. XRD patterns from 14 different samples (artificial mixtures, and inorganic and organic materials with nano- and submicrometre crystallite sizes) were recorded in 2interval from 5–10 to 120 . All XRD patterns were processed using Rietveld refinement. The magnitude of the workable angular range was gradually increased, and thereby the number of peaks used in Rietveld refinement was also increased, step by step. Three XRD patterns simulated using CIFs were processed in the same way. Analysis of the results obtained indicated that the magnitude of the angular range chosen for Rietveld refinement does not significantly affect the calculated values of unit-cell parameters, crystallite sizes and percentage of phases. The values of unit-cell parameters obtained for different angular ranges diverge by 10 4 A ˚ (rarely by 10 3 A ˚ ), that is about 10 2 % in relative numbers. The average difference between the calculated and actual phase percentage in artificial mixtures was 1.2%. The maximal difference for the crystallite size did not exceed 0.47, 5.2 and 7.7 nm at crystallite sizes lower than 20, 50 and 120 nm, respectively. It has been established that these differences are statistically insignificant. 1. Introduction Powder X-ray diffraction (PXRD) is an important method in the field of materials characterization and has been success- fully applied to study various natural and synthesized inor- ganic and organic materials. One of the tools in a wide arsenal of PXRD methods is the Rietveld method (Rietveld, 1967, 1969). Initially, the method was developed solely to refine the crystal structure using neutron diffraction patterns obtained from a powder of pure crystalline phases. However, today the method is widely used in analysis of mono- and multiphase samples to solve complicated problems (refinement of atomic coordinates, site occupancies and atomic displacement para- meters) as well as for routine tasks (unit-cell parameter refinement, quantitative analysis, crystallite size determina- tions). The Rietveld method for performing quantitative phase analysis became widely applied after Bish & Howard (1988) modified the previously used computer algorithms. These latter method applications are very important for quantitative analysis and quality assurance/quality control (QA/QC) both in scientific research and in industry (Chauhan & Chauhan, 2014; Feret, 2013; Ufer & Raven, 2017; Zunic et al., 2011). ISSN 1600-5767 # 2019 International Union of Crystallography
10
Embed
The influence of X-ray diffraction pattern angular range on … · choosing the angular range. For example, Jenkins & Snyder (1996) recommend recording the X-ray diffraction (XRD)
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
research papers
252 https://doi.org/10.1107/S1600576719000621 J. Appl. Cryst. (2019). 52, 252–261
The influence of X-ray diffraction pattern angularrange on Rietveld refinement results used forquantitative analysis, crystallite size calculationand unit-cell parameter refinement
Vladimir Uvarov*
The Unit for Nanoscopic Characterization, The Center for Nanoscience and Nanotechnology, The Hebrew University of
samples can be seen in Tables S1–S10 and in Figs. S1–S10 in
the supporting information.
The data presented in the tables show that the differences in
the values of all the parameters calculated using the Rietveld
refinement are extremely small. This concerns the values of
unit-cell parameters, as well as the crystallite sizes and the
phase concentrations. Moreover, these differences are
comparable to the errors that the TOPAS software calculates
for the corresponding parameters. In graphical form, this can
be seen in Figs. 1–4 and Figs. S11–S17. We stress that we did
not observe any one-valued tendency (for example, mono-
tonic decrease or increase) in the calculated parameters.
research papers
J. Appl. Cryst. (2019). 52, 252–261 Vladimir Uvarov � Influence of XRD pattern range on Rietveld refinement 255
Figure 2Graphical representation of Rietveld refinement results of the iridium–iridium oxide (1:2) mixture (the refined Bragg peak positions are shown byvertical bars) (a) and the values of unit-cell parameters, percentage and crystallite sizes of iridium calculated by Rietveld refinement for different angularintervals (b).
Figure 1Graphical representation of Rietveld refinement results of LaB6 (the refined Bragg peak positions are shown by vertical bars) and the values of unit-cellparameters calculated by Rietveld refinement for different angular ranges (inset).
In addition, we see that the calculated errors of parameters
decrease with increasing angular range. This is easily
explainable: when the angular range increases, the number of
peaks (and, correspondingly, the number of measurement
points) that are used for calculating the parameters increases,
and therefore the statistics of the error calculations are
improved. Similarly, for all the samples, we observe a relative
increase of Rwp value by 8–15% with the extension of angular
range. Hill (1992) reported similar observations for a Rietveld
refinement round robin test. It is commonly accepted that
high-angle X-ray powder diffraction data have poorer
counting statistics, owing to the combined effects of a decrease
in the scattering coefficient with increasing sin� /�, Lorentz–
polarization factor and thermal vibrations (Hill, 1992; Lang-
ford & Louer, 1996). The value of the Rwp factor is often
associated with the quality of the Rietveld refinement.
Therefore, the values of the Rwp factor of the order of 30–35%
given in the tables may cause readers some unease. However,
as we showed previously (Uvarov & Popov, 2008, 2013), this
value can easily be decreased several times by increasing the
counting time within the same angular range. Moreover, the
Rwp value does not affect the results of crystallite size calcu-
lation and phase content quantification. At the same time,
there is an opinion (Toby, 2006) that the character of the
difference curve (the difference between the experimental and
calculated profiles) is the best indicator of quality in Rietveld
refinement. As is clearly seen in Figs. 1–4 the difference curves
indicate a good quality of Rietveld refinement.
research papers
256 Vladimir Uvarov � Influence of XRD pattern range on Rietveld refinement J. Appl. Cryst. (2019). 52, 252–261
Figure 3Graphical representation of Rietveld refinement results of the calcite–gypsum (1:2) mixture (the refined Bragg peak positions are shown by vertical bars)(top) and the values of percentage (a), unit-cell parameters (b), (c), and crystallite size of calcite calculated by Rietveld refinement for different angularintervals (bottom).
Let us briefly discuss the factors affecting the accuracy of
the determination of the unit-cell parameters, the crystallite
size and the phase percentages, and also the behavior of these
parameters, observed in this work:
Unit-cell parameters. The accuracy of calculating the unit-
cell parameters depends on the correct determination of peak
positions and on the instrument alignment. First, we note a
very small difference between the value obtained in the
present study for the LaB6 unit-cell parameter [4.156333
(46) A] and its certified value of 4.156950 (6) A (SRM 660,
1989). This difference of 0.0148% is quite small, especially
considering that the certified lattice parameter was deter-
mined in a wider angular region, namely from the reflections
that were in the range 15–160� 2�. Thus we assume that our
diffractometer gives a systematic error of about 0.0007–
0.0008 A, which can be taken into account when performing
ultra-precise measurements. Our data are also in good
agreement with the data reported by Chantler et al. (2007) [a =
4.15680 (5) A]. In recent work devoted to the accuracy of
determining the unit-cell parameters by the Rietveld method,
Tsubota & Kitagawa (2017) obtained a = 4.15811 (22) A and
a = 4.15655 (1) A at angular ranges of 18–92 and 18–152� 2�.
However, this is not of fundamental importance for the
purposes of this work. For all the tested samples, the differ-
ence in the values of the unit-cell parameters was a few ten
thousandths of an angstrom (rarely a few thousandths) or
about a few hundredths of one percent in relative numbers.
For example, for LaB6, the difference between the unit-cell
parameters calculated for intervals 10–65 and 10–115� 2� was
0.0001 A or 0.0027%. In the case of low phase concentration
(for example, the impurity of muscovite in kaolin) or for
nanosized phases (for example, magnetite), the difference
research papers
J. Appl. Cryst. (2019). 52, 252–261 Vladimir Uvarov � Influence of XRD pattern range on Rietveld refinement 257
Figure 4Graphical representation of Rietveld refinement results for the copper–cuprite (1:3) mixture (the refined Bragg peak positions are shown by verticalbars) (a) and the values of unit-cell parameters, percentage and crystallite size of copper calculated by Rietveld refinement for different angularintervals (b).
Table 4Results of Rietveld refinement of the copper–cuprite (1:3) mixture.
between the calculated values of the unit-cell parameters was
sometimes up to several hundredths of an angstrom. Note that
we did not find any obvious tendency in change of the unit-cell
parameters with increasing angular interval. Sometimes the
parameters slightly increased, and sometimes they slightly
decreased (see Figs. 1–4 and S11–S17).
Crystallite size. The accuracy of crystallite size calculation
depends on the accuracy of profile fitting. In this case, the
overlapping of peaks and the fact that the used radiation is not
monochromatic (Cu K�1 and Cu K�2 are present) may have a
significant effect at high angles. In most cases, the calculated
sizes of crystallites varied on average by 5% with a change in
the angular range. The calculated error smoothly decreased as
the angular interval increased. The calculated crystallite sizes
generally decreased with increasing angular interval.
However, for the anatase and rutile in the P90 sample and for
uricite from the artificial mixture, the calculated crystallite
sizes were larger for the extended processed angular range.
We recall that PXRD allows correct estimation of a crystallite
size only up to about 100–120 nm for conventional diffract-
ometers (Uvarov & Popov, 2013). Therefore, the calculated
crystallite size values exceeding 100 nm were included in the
tables only to demonstrate the trend; they should not be
understood as true sizes, related to actual physical dimensions.
Percentage of phases. The accuracy of the quantitative
analysis depends on the accuracy of calculating the ratio of the
intensities of the peaks from different phases. Therefore, in
this case, possible preferred orientation of crystallites should
be taken into account. The results of calculations of the phase
percentages for six artificial mixtures are very good. For
illustrative purposes, the results of calculation of the phase
percentage for some artificial mixtures are shown in Fig. 5. The
maximal difference of percentage values calculated for
different angular intervals did not exceed 3%. These results
show that the dispersions obtained are similar to those of the
round robin on determination of quantitative phase abun-
dance from diffraction data that was carried out by the
International Union of Crystallography Commission on
Powder Diffraction (Madsen et al., 2001; Scarlett et al., 2002).
At low concentrations (for example, impurity of muscovite
and traces of quartz in kaolin) the error that the TOPAS
software gives for minor phases reached 25% or more.
Furthermore, we need to find out whether the results
obtained by Rietveld refinement for different angular inter-
vals are independent. From a practical point of view, they
probably are independent. Let us suppose that one participant
recorded the XRD pattern in the interval 10–70� 2� and after
the Rietveld refinement obtained some results. Another
participant recorded the XRD pattern for the same material in
the interval 10–120� 2�, performed Rietveld refinement and
obtained another result. We can assume that they worked
independently, so their results were independent as well.
McCusker et al. (1999) believe that, from a purely statistical
point of view, each measurement is an independent observa-
tion, and the intensities measured at different points of the
same peak are simply two independent measurements of the
intensity of this peak. However, the situation could be
considered in another way. When performing the Rietveld
procedure (to refine the unit-cell parameters, determine the
crystal size, calculate the percentage of components) we use
diffraction peaks lying in the selected angular interval. As the
angular interval increases, we increase the size of the sample
(i.e. the number of diffraction peaks) to be processed.
Therefore, from this point of view, the results obtained after
performing Rietveld refinement of the data obtained from
different angular intervals of the same investigated material
cannot be regarded as independent.
On the basis of the foregoing argument, we stress that the
unambiguous choice of the angular range in the planning of an
XRD experiment is not a trivial task. However, if we prove
that the results obtained for different angular ranges have the
same accuracy from a statistical point of view, then it could
simplify the problem.
3.2. Estimation of precision and accuracy of the results
At this point it is appropriate to recall that the precision and
accuracy of measurements are two different things. The
precision is associated with random errors and characterizes
the variability of the method from a statistical point of view.
research papers
258 Vladimir Uvarov � Influence of XRD pattern range on Rietveld refinement J. Appl. Cryst. (2019). 52, 252–261
Figure 5The calculated phase percentages for some artificial mixtures as a function of the magnitude of the angular ranges (the actual values of phase percentageare indicated in brackets).
The accuracy is associated with systematic errors and char-
acterizes the obtained result, the difference between the
obtained and ‘true’ value.
In the present work for each tested sample we have a
number of ‘independent’ measurements, from which after
their processing the parameters characterizing the sample
were obtained. It is known that any measurement result, in
fact, contains errors (random and systematic) and therefore
the true value of a measurand can never be established.
For Rietveld refinement the error of the results consists of
three components: a systematic error (related to the diffract-
ometer alignment and some physical factors), a random error
and an error in the calculations (related to the features of the
software). The systematic error arising from the axial diver-
† a1, a2 and a3 are the calculated values of unit-cell parameter for the three XRD patterns of LaB6. ‡ The Rietveld e.s.d. calculated by the TOPAS software.
that the absolute values of the estimated standard deviations
depend on the complexity of the material being analyzed
(phase and chemical composition, crystal structure and crys-
tallite size of the phases contained in the sample, etc.). Addi-
tionally, it was demonstrated that errors calculated by the
Rietveld software for crystallite size, percentage and unit-cell
parameters were smaller than the random errors obtained for
different angular intervals in the reproducibility tests. But
because of the possible presence of systematic error, this has
little effect on the accuracy of the obtained results. In fact, our
work demonstrates the stability of the Rietveld refinement
results obtained from the TOPAS software when the angular
interval changes.
We hope that the results of the present study will give
readers pause for thought and will help researchers in plan-
ning XRD measurements of a wide range of materials aimed
at structural characterization, quantitative analysis, QA/QC
etc. Note that all measurements were performed in Bragg–
Brentano reflection geometry and the conclusions are related
solely to this geometry.
Acknowledgements
The author would like to acknowledge Dr Inna Popov, leader
of The Unit for Nanoscopic Characterization of the Harvery
M. Krueger Center for Nanoscience and Nanotechnology at
the Hebrew University of Jerusalem, for her valuable
comments and editing, which significantly improved this
paper.
References
Al-Dhahir, T. A. (2013). Diyala J. Pure Sci. 9, 108–119.Anupama, A. V., Keune, W. & Sahoo, B. (2017). J. Magn. Magn.
Mater. 439, 156–166.Aurelio, G., Fernandez-Martinez, A., Cuello, G. J., Roman-Ross, G.,
Alliot, I. & Charlet, L. (2008). Rev. Soc. Esp. Mineral., 9, 39–40.Ballirano, P. & Maras, A. (2006). Am. Mineral. 91, 997–1005.Berger, H. (1986). X-ray Spectrom. 15, 241–243.Bessergenev, V. G., Mateus, M. C., do Rego, A. M. B., Hantusch, M. &
Burkel, E. (2015). Appl. Catal. Gen. 500, 40–50.Bezerra, P. C. S., Cavalcante, R. P., Garcia, A., Wender, H., Martines,
M. A. U., Casagrande, G. A., Gimenez, J., Marco, P., Oliveira, S. C.& Machulek, A. Jr (2017). J. Braz. Chem. Soc. 28, 1788–1802.
Bish, D. L. & Howard, S. A. (1988). J. Appl. Cryst. 21, 86–91.Braccini, S., Pellegrinelli, O. & Kramer, K. (2013). World J. Nucl. Sci.
Germany.Chantler, C. T., Rae, N. A. & Tran, C. Q. (2007). J. Appl. Cryst. 40,
232–240.Chauhan, A. & Chauhan, P. (2014). J. Anal. Bioanal. Tech. 5, 212.Cheary, R. W., Coelho, A. A. & Cline, J. P. (2004). J. Res. Natl Inst.
Stand. Technol. 109, 1–25.
Cockcroft, J. K. & Fitch, A. N. (2008). Powder Diffraction: Theoryand Practice, edited by R. E. Dinnebeir & S. J. L. Billinge, p. 44.Cambridge: RSC Publishing.
Coelho, A. A. (2018). J. Appl. Cryst. 51, 210–218.Feret, F. R. (2013). Powder Diffr. 28, 112–123.Guinebretiere, R. (2010). X-ray Diffraction by Polycrystalline
Materials, p. 192. New York: Wiley.Hellenbrandt, M. (2004). Crystallogr. Rev. 10, 17–22.Hill, R. J. (1992). J. Appl. Cryst. 25, 589–610.Hill, R. J. (1993). The Rietveld Method, edited by R. A. Young, pp. 61–
101. New York: Oxford University Press.Hill, R. J. & Cranswick, L. M. D. (1994). J. Appl. Cryst. 27, 802–844.Jenkins, R. & Snyder, R. L. (1996). Introduction to X-ray Powder
Diffractometry, pp. 265–266. New York: Wiley.Jimenez, J. A., Padilla, I., Lopez-Delgado, A., Fillali, L. & Lopez-
Andres, S. (2015). Int. J. Appl. Ceram. Technol., 12, E178–E186.Klug, H. P. & Alexander, L. E. (1974). X-ray Diffraction Procedures
New York; Wiley.Langford, J. I. & Louer, D. (1996). Rep. Prog. Phys. 59, 131–234.Macrae, C. F., Edgington, P. R., McCabe, P., Pidcock, E., Shields, G. P.,
Taylor, R., Towler, M. & van de Streek, J. (2006). J. Appl. Cryst. 39,453–457.
Madsen, I. C., Scarlett, N. V. Y., Cranswick, L. M. D. & Lwin, T.(2001). J. Appl. Cryst. 34, 409–426.
McCusker, L. B., Von Dreele, R. B., Cox, D. E., Louer, D. & Scardi, P.(1999). J. Appl. Cryst. 32, 36–50.
Paz, S. P. A., Kahn, H. & Angelica, R. S. (2018). Miner. Eng. 118, 52–61.
Perander, L. M., Zujovic, Z. D., Kemp, T. F., Smith, M. E. & Metson,J. B. (2009). JOM, 61, 33–39.
Pramanik, S., Ghosh, S., Roy, A. & Mukherjee, A. K. (2016). J. Mater.Res. 31, 328–336.
Rietveld, H. M. (1967). Acta Cryst. 22, 151–152.Rietveld, H. M. (1969). J. Appl. Cryst. 2, 65–71.Scarlett, N. V. Y., Madsen, I. C., Cranswick, L. M. D., Lwin, T.,
Groleau, E., Stephenson, G., Aylmore, M. & Agron-Olshina, N.(2002). J. Appl. Cryst. 35, 383–400.
Scott, H. G. (1983). J. Appl. Cryst. 16, 159–163.SRM 660 (1989). Instrument Line Position and Profile Shape
Standard for X-ray Powder Diffraction, National Institute ofStandards and Technology, US Department of Commerce,Gaithersburg, MD, USA. https://www-s.nist.gov/srmors/certificates/archives/660.pdf.
Tamer, M. (2013). J. Modern Phys. 4, 1149–1157.Toby, B. H. (2006). Powder Diffr. 21, 67–70.Tsubota, M. & Kitagawa, J. (2017). Sci. Rep. 7, 15381.Ufer, K. & Raven, M. D. (2017). Clays Clay Miner. 65, 286–297.Uvarov, V. & Popov, I. (2008). J. Pharm. Biomed. Anal. 46, 676–682.Uvarov, V. & Popov, I. (2013). Mater. Charact. 85, 111–123.Uvarov, V., Popov, I., Shapur, N., Abdin, T., Gofrit, O. N., Pode, D. &
Duvdevani, M. (2011). Environ. Geochem. Health, 33, 613–622.Winburn, R. S. (2002). Adv. X-ray Anal. 46, 210–218.Yan, K., Guo, Y., Wu, X. & Cheng, F. (2016). Powder Diffr. 31, 185–
191.Zabala, S. M., Conconi, M. S., Alconada, M. & Sanchez, R. M. T.
(2007). Cienc. Suelo 25, 65–73.Zhu, X., Zhu, Z., Lei, X., Yan, C. & Chen, J. (2017). J. Wuhan Univ.
Technol.-Mater. Sci. Edit. 32, 373–377.Zunic, T. B., Katerinopoulou, A. & Edsberg, A. (2011). J. Mineral.
Geochem. 188, 31–47.
research papers
J. Appl. Cryst. (2019). 52, 252–261 Vladimir Uvarov � Influence of XRD pattern range on Rietveld refinement 261