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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 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
Jerusalem, Edmond J. Safra Campus, Givat Ram, Jerusalem, 91904, Israel. *Correspondence e-mail:
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 2� interval 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
The important parameters in the planning of PXRD
measurements are step size, counting time and angular range.
These parameters significantly affect the quality of the raw
data on one hand, and the duration and cost of the analysis on
the other. There are many recommendations regarding the
choice of step size and counting time in the literature on
PXRD and Rietveld refinement methods (Klug & Alexander,
1974; Cockcroft & Fitch, 2008; McCusker et al., 1999; Hill,
1993). However, there are no clear recommendations for
choosing the angular range. For example, Jenkins & Snyder
(1996) recommend recording the X-ray diffraction (XRD)
pattern in an angular range which ensures that at least 50
peaks are obtained. Guinebretiere (2010) believes that the
angular range should be as wide as possible because peaks at
high angles improve the refinement of the cell parameters and
of the atomic displacement parameters. In their article on the
results of a Rietveld refinement round robin test for mono-
clinic ZrO2, Hill & Cranswick (1994) wrote ‘As a general rule,
it is recommended that the widest possible range of d spacings
should be collected, in order to maximize the observations-to-
parameters ratio.’ However, Winburn (2002) reports on the
danger of using longer scans for complex mixtures due to
possible overloading of the Rietveld routines by data sets
containing excess information, thereby causing errors. It is
intuitively clear that the choice of the angular range should
depend on the features of the material under study (phase
composition, crystal structure, values of unit-cell parameters)
and the objectives to which the researchers aspire (a refine-
ment of the crystal structure including the atomic coordinates
and the atomic displacement parameters). However, a lack of
clear recommendations on this subject is probably one of the
reasons that different researchers while characterizing the
same material perform XRD pattern recording in sufficiently
different angular ranges. For example, participants of the
Rietveld refinement round robin test for monoclinic ZrO2
worked with XRD patterns acquired at 100–162� 2� upper
bounds (Hill & Cranswick, 1994). In the study of kaolinite,
Yan et al. (2016) recorded XRD patterns in the 5–130� 2�angular interval, Paz et al. (2018) used XRD patterns from 2 to
85� 2�, Zhu et al. (2017) acquired data in the angular interval
10–80� 2� and Zabala et al. (2007) chose the 3–70� 2� angular
range. When studying kidney stones, Pramanik et al. (2016)
recorded XRD patterns in the 5–120� 2� angular range, while
Uvarov et al. (2011) used the 6–66� 2� range for the same
materials. Jimenez et al. (2015) recorded XRD patterns in the
20–105� 2� angular range to quantify the crystalline phases of
alumina, which formed during the thermal decomposition of
boehmite, with the Rietveld refinement method. Perander et
al. (2009) studied the nature and impact of ‘fines’ (small
particles) in alumina with XRD patterns acquired at 20–80�
2�. In studying the mineralogy of Michelangelo’s fresco
plaster, Ballirano & Maras (2006) recorded XRD patterns of
calcite (for subsequent Rietveld refinement) in an angular
range of 5–100� 2�, while Aurelio et al. (2008) limited them-
selves to the angular range 15–85� 2� when studying selenium
and arsenic substitution in calcite with Rietveld refinement.
Tamer (2013) recorded XRD patterns of calcite in the 2–70� 2�
angular interval when studying carbonate rocks. For the
characterization of well known titanium dioxides (rutile and
anatase), some researchers (Bezerra et al., 2017) have
acquired XRD data in the 10–70� 2� range, while others
(Bessergenev et al., 2015; Al-Dhahir, 2013) used much wider
data ranges of 10–90 and 20–120� 2�, respectively. Braccini et
al. (2013) and Anupama et al. (2017) used the Rietveld method
when studying magnetite- and hematite-containing samples.
However, in the first case, XRD patterns were recorded in the
10–140� 2� range, and in the second case the 20–85� 2� interval
was used. Anyone interested in the choice of angular range
when planning XRD measurements can find many similar
examples for very different materials. Why is it believed that
increasing the angular range can improve the accuracy of
XRD results? Is this opinion supported by experimental data?
How much do the refined parameters change when the
magnitude of the angular interval varies? Unfortunately, we
have not found any report on a systematic study relating to the
influence of the magnitude of the angular range of XRD
patterns on Rietveld refinement results. The purpose of the
present work was to clarify the quantitative aspects of this
issue, at least to some extent.
2. Materials and methods
We used well known commercial materials, natural crystalline
phases and their artificial mixtures as test samples in this study.
Powders for the preparation of artificial mixtures were
weighed in an AB104-S/ FACT analytical balance (Mettler
Toledo) with a precision of 1 mg. In total, 14 samples were
studied: seven artificial mixtures [iridium–iridium oxide (1:2),
calcite–gypsum (2:1), copper–cuprite (1:3), corundum–boeh-
mite (70:30), cristobalite–quartz (1:2), whewellite–uricite (4:1)
and hematite–magnetite (1:1)], natural kaolinite from
Gluhovets, P90 (Degussa) (about 95% anatase), corundum
(Kiocera), 100 nm copper nanoparticles, BaTiO3 (Sigma–
Aldrich), Mg2P2O7 powder and LaB6 (NIST SRM 660). Five
out of 21 studied phases had a cubic crystal structure, four
hexagonal, five tetragonal, one orthorhombic, five monoclinic
and one triclinic. The crystallite size values of the phases
present in the tested samples ranged from about 10 nm to
several micrometres. We believe that the chosen variety of
phase and chemical compositions, crystal structures, and
crystallite sizes covers a wide enough range of practical
applications of the PXRD method. Detailed information on
the crystalline phases used in this work is presented in
Table S1-1 (supporting information).
X-ray powder diffraction measurements were performed in
the Bragg–Brentano geometry on a conventional D8 Advance
diffractometer (Bruker AXS, Karlsruhe, Germany) with a
secondary graphite monochromator, 2� Sollers slits, a 2 mm
divergence slit and a 0.2 mm receiving slit, a reflectometer
sample stage, and an NaI(Tl) scintillation detector. Low-
background quartz sample holders (the diameter and the
depth of the cavity were 18–20 and 0.4–0.5 mm, respectively)
were carefully filled with the powder samples. The specimen
weight was approximately 0.2�0.3 g. XRD patterns from
research papers
J. Appl. Cryst. (2019). 52, 252–261 Vladimir Uvarov � Influence of XRD pattern range on Rietveld refinement 253
5–10� 2� small angles to 115–120� 2� were recorded using
Cu K� radiation (� = 1.5418 A) with the following measure-
ment conditions: tube voltage of 40 kV, tube current of 40 mA,
step-scan mode with a step size of 0.02� 2� and counting time
of 1 s per step. Rietveld refinements of the obtained data with
different angular ranges were performed using the TOPAS
V3.0 software (Bruker, 2006; Coelho, 2018). The fundamental
parameters approach (Cheary et al., 2004) was used for XRD
pattern processing. Strictly speaking, the radiation of Cu K� is
not monochromatic. Therefore the profile of the peaks was
rather asymmetrical, which was especially notable at large
angles. For that reason, in the Rietveld refinement we used the
Berger and Cu K�5 emission profiles (Bruker, 2006) based on
the phenomenological model proposed by Berger (1986) as
the most suitable in this case. In all cases, the value of the
angular range for the XRD pattern processing was gradually
increased from 60–65 to 115–120� 2� with step 5–10� 2�, and
thereby, step by step, the number of peaks used for Rietveld
refinement was also increased. The refined parameters were
scale factor, background (as a Chebyshev polynomial of sixth
order), specimen displacement, surface roughness, absorption,
lattice parameters, preferred orientation (using a spherical
harmonics function of order 2–6) and Beq (isotropic dis-
placement parameters). The starting values of the displace-
ment parameters can be accepted as equal to 1 (this is assumed
in TOPAS by default) or set to values reported in CIFs. The
Beq value was controlled during refinement, and we fixed its
value when it became negative or greater than four or when
the estimated standard deviation (e.s.d.) values were
comparable to the calculated Beq values. Once the results of
refinements performed with and without correction of
roughness do not differ significantly, there is no point in
applying a roughness correction. We emphasize that after
completion of each refinement the next was started from
scratch, i.e. each refinement was performed independently.
The ‘short’ interval (less than 60� 2�) was not examined,
because ‘short’ XRD patterns are often insufficient for
unambiguous identification of phases.
3. Results and discussion
3.1. XRD measurements
The results obtained for four samples are shown in Tables 1–
4 and Figs. 1–4. The results of the analysis of the remaining ten
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254 Vladimir Uvarov � Influence of XRD pattern range on Rietveld refinement
Table 1Results of Rietveld refinement of LaB6 (NIST SRM 660).
Angular range in Rietveld refinement, 2��
Parameter 10–65 10–70 10–75 10–85 10–90 10–101 10–110 10–115
Rwp† 34.94 34.52 34.7 35.17 35.40 36.13 36.54 36.8a = b = c (A ) 4.156220 4.156188 4.156299 4.156352 4.156365 4.156364 4.156318 4.156333e.s.d.‡ 0.000100 0.000089 0.000082 0.000071 0.000062 0.000054 0.000048 0.000046d (nm) 1040 1020 1040 1020 1000 1020 998 972e.s.d.§ 130 120 120 110 100 100 96 90
† Rwp is the residual-weighted parameter characterizing the refinement quality. Rwp = {P
wi[yi(obs) � yi(calc)]2/P
wi[yi(obs)]2}1/2 [where yi(obs) is the observed intensity at step i, yi(calc) is the
calculated intensity and wi is the weight]. ‡ The Rietveld e.s.d. calculated by the TOPAS software. § d is the calculated crystallite size.
Table 3Results of Rietveld refinement of the calcite–gypsum (2:1) mixture.
Angular range in Rietveld refinement, 2��
Parameters 10–70 10–80 10–90 10–100 10–120
Rwp 34.52 35.85 36.79 37.65 38.87
Calcite (wt%) 64.2 65.0 66.1 67.2 66.17e.s.d. 4.7 4.7 4.2 4.6 4.2a = b (A) 4.98556 4.98532 4.98514 4.98498 4.98500e.s.d. 0.00025 0.00024 0.00021 0.00019 0.00016c (A) 17.0500 17.0491 17.0483 17.04819 17.04825e.s.d. 0.0012 0.0011 0.0011 0.00096 0.00084d (nm) 164 165 163 162 159e.s.d. 12 12 12 12 11
Gypsum (wt%) 35.8 35.0 33.9 32.8 33.83e.s.d. 4.7 4.7 4.2 4.6 4.2a (A) 6.28633 6.28625 6.28595 6.28599 6.28614e.s.d. 0.00073 0.00071 0.00071 0.00068 0.00066b (A) 15.1947 15.1936 15.1935 15.1938 15.1926e.s.d. 0.0018 0.0018 0.0017 0.0017 0.0015c (A) 5.67779 5.67813 5.67761 5.67716 5.67759e.s.d. 0.00084 0.00084 0.00082 0.00079 0.00076� (�) 114.157 114.171 114.172 114.173 114.172e.s.d. 0.012 0.012 0.012 0.012 0.011d (nm) 117.0 118.1 117.6 117.8 116.8e.s.d. 8.1 8.2 8.1 8.0 7.7
Table 2Results of Rietveld refinement of the iridium–iridium oxide mixture (1:2).
Angular range in Rietveld refinement, 2��
Parameter 10–60 10–70 10–80 10–90 10–100 10–110 10–120
Rwp 25.48 25.71 27.04 26.87 27.53 28.60 29.39
Iridium (wt%) 35.1 34.5 34.11 33.0 32.8 33.4 33.0e.s.d. 1.9 1.6 0.83 1.5 1.2 1.1 1.0a = b = c (A) 3.8346 3.8372 3.8367 3.83648 3.83668 3.83652 3.83658e.s.d. 0.0014 0.0017 0.0017 0.00094 0.00057 0.00077 0.00069d (nm) 22.95 22.45 21.80 21.61 21.46 21.50 21.46e.s.d. 0.88 0.77 0.69 0.56 0.54 0.52 0.54
Iridium oxide(wt%)
64.9 65.5 65.89 67.0 67.2 66.6 67.0
e.s.d. 1.9 1.6 0.83 1.5 1.2 1.1 1.0a = b (A) 4.4921 4.4951 4.4939 4.4940 4.4948 4.49503 4.49505e.s.d. 0.0017 0.00020 0.0017 0.0012 0.0010 0.00093 0.00087c (A) 3.1507 3.1520 3.1511 3.1510 3.15089 3.15063 3.15068e.s.d. 0.0014 0.0016 0.0013 0.0011 0.00095 0.00087 0.00069d (nm) 12.22 12.12 11.99 11.98 11.87 11.87 11.84e.s.d. 0.24 0.20 0.20 0.18 0.17 0.17 0.17
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.
Angular range in Rietveld refinement, 2��
Parameters 10–75 10–80 10–91 10–100 10–110 10–120
Rwp 9.56 9.55 9.60 9.62 9.62 9.60
Copper (wt%) 23.8 23.3 23.2 23.2 24.5 25.0e.s.d. 2.1 1.9 1.4 1.3 1.3 1.3a = b = c (A) 3.61345 3.61339 3.61247 3.61242 3.61249 3.61249e.s.d. 0.00034 0.00032 0.00020 0.00019 0.00019 0.00018d (nm) 92.5 91.5 91.4 91.5 91.9 92.2e.s.d. 5.6 5.5 5.2 5.2 5.2 5.2
Cuprite (wt%) 76.2 76.7 76.8 76.8 75.5 75.0e.s.d. 2.1 1.9 1.4 1.3 1.3 1.3a = b = c (A) 4.26812 4.26806 4.26730 4.26695 4.26708 4.26708e.s.d. 0.00047 0.00044 0.00033 0.00032 0.00031 0.00030d (nm) 38.43 38.68 38.68 38.61 38.41 38.28e.s.d. 0.76 0.77 0.80 0.80 0.78 0.77
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.
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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-
gence, flat sample, specimen transparency, sample displace-
ment and zero shift affects the peak position and does not
depend linearly on the angle. Unfortunately, the systematic
errors, which are in fact significant, cannot be estimated within
the procedure of Rietveld refinement (McCusker et al., 1999;
Scott, 1983).
To estimate the value of the random error, we verified the
reproducibility of the method. With this purpose the XRD
patterns from P90 Degussa (a material that contains more
than 90% anatase with a crystallite size of about 15 nm),
BaTiO3 (about 100 nm crystallite size) and corundum
(submicrometre crystallite size) specimens were recorded five
or six times and processed in the same way as described above.
All powder samples were taken out and repacked between
each data collection. The main results of the statistical esti-
mation of reproducibility are given in Tables 5 (for anatase),
S11 (for BaTiO3) and S12 (for corundum).
Since the number of compared intervals is sufficiently large
(from seven to 11), we used the ANOVA (analysis of variance)
method (https://www.itl.nist.gov/div898/handbook/prc/section4/
prc431.htm) to estimate the statistical significance of the
difference between mean values and variances simultaneously
for all angular intervals. The null hypothesis for an ANOVA is
that there is no significant difference between the results
obtained with Rietveld refinement of different data sets. Thus,
the method is in fact a Student’s test for a great number of
results obtained from data sets that have different sizes. As we
have results for which only one factor (the number of
measured intensity points, which is determined by the width of
the angular range) varies, the one-way ANOVA method is
well suited to our task. We used the ANOVA online calculator
(http://astatsa.com/OneWay_Anova_with_TukeyHSD/) to
perform all the calculations. In this case, the ANOVA calcu-
lator allowed us to compare up to ten intervals simultaneously.
An example of application of the ANOVA test is given in the
supporting information. The results of this test showed that
the differences between the mean values and variances
obtained for different angular intervals during estimation of
the reproducibility are statistically insignificant. On the basis
of the obtained results, it can be concluded that, from a
statistical point of view, the data sets (Rietveld refinement
results) for different angular intervals belong to the same
statistical population.
As a rule, the standard deviations of unit-cell parameters
calculated in reproducibility tests (Tables 5, S11 and S12) were
an order of magnitude larger than the values of errors of unit-
cell parameters calculated by TOPAS in the Rietveld refine-
ment (Tables 2, 3, 4 and S1–S10). The standard deviations and
TOPAS errors were roughly the same only for the phases with
a crystallite size of less than 40 nm, and also in cases when the
phase content was small (uricite in the whewellite–uricite
mixture, muscovite and quartz in kaolin).
It is very difficult to say anything specific about the values of
e.s.d. calculated by the TOPAS software. Here we note a
curious observation we made on application of Rietveld
refinement to our experimental data. It is well known and
specified in the guides to Rietveld refinement that one cannot
perform the ‘zero error’ and ‘specimen displacement’ correc-
tions simultaneously. It is implied that both corrections lead to
the same result. In this work we checked this implication and
found that for all our samples and for all the tested angular
intervals the ‘zero error’ correction always gave larger unit-
cell parameter values than the ‘specimen displacement’
correction. Although the absolute difference was only 0.0001–
0.0009 A, it was always observed (see Table S13). Perhaps this
fact should be taken into account in cases of precise refine-
ment of the unit-cell parameters and atomic coordinates.
In order to assess the situation when the systematic error is
absent, we simulated XRD patterns based on CIFs. We used
CIFs from the Inorganic Crystal Structure Database (ICSD;
Hellenbrandt, 2004) and the Mercury software (Macrae et al.,
2006) to simulate ‘ideal’, i.e. free of systematic errors, patterns
for anatase (ICSD-154604), hematite (ICSD-415251) and
corundum (ICSD-92628). It is assumed that such ideal XRD
patterns will have no systematic errors. These calculated XRD
patterns were processed according to the same scheme as the
real XRD patterns. The results of Rietveld refinement are
presented in Tables S14–S16 and Figs. S18–S20. The results
obtained by processing of simulated XRD patterns effectively
do not differ from those of real experimental data. In other
words, the observed tendency was the same: for the extended
angular interval the value of Rwp increases, the error value
decreases, the calculated values of the unit-cell parameters
change slightly, but the calculated crystallite sizes essentially
do not change. The errors calculated by the TOPAS software
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J. Appl. Cryst. (2019). 52, 252–261 Vladimir Uvarov � Influence of XRD pattern range on Rietveld refinement 259
Table 5Means and standard deviations calculated for anatase (P90 sample) in thefivefold reproducibility test.
Mean and standard deviations for different angular intervals, 2��
Parameter 10–60 10–65 10–80 10–90 10–100 10–110 10–120
a = b (A) 3.78554 3.7849 3.7850 3.7852 3.7851 3.7851 3.7830e.s.d. 0.00079 0.00141 0.00142 0.00146 0.00154 0.00159 0.00317c (A) 9.49738 9.4970 9.4966 9.4970 9.4970 9.4947 9.4931e.s.d. 0.00605 0.00563 0.00423 0.00415 0.00396 0.00243 0.00315d (nm) 16.23 16.20 16.43 16.42 16.44 16.56 16.66e.s.d. 0.40 0.34 0.23 0.24 0.23 0.17 0.34Anatase
(wt%)94.41 94.70 94.28 94.26 94.22 94.22 94.16
e.s.d. 0.66 0.64 0.60 0.62 0.61 0.62 0.58
have practically the same value for both real and ideal XRD
patterns.
We can calculate the value of the systematic error only for
the LaB6 sample because the value of its unit-cell parameter is
certified. Then we can calculate the total error as
�total ¼ ð�2r þ �
2s þ �
2TÞ
1=2; ð1Þ
�r ¼
Pðai � �aaÞ2
n� 1
� �1=2
; ð2Þ
�s ¼ �aa� aref; ð3Þ
where �total, �r, �s and �T are the total, random, systematic and
TOPAS errors (Rietveld e.s.d.), n is the number of observa-
tions, and ai, �aa and aref are the calculated, mean and reference
values of the LaB6 unit-cell parameter.
The results of these calculations are shown in Table 6 and
Fig. 6. The results obtained confirm the well known assertion
that, in the case when the random and program errors (i.e. the
Rietveld e.s.d. values calculated by the applied software) are
several times less than the value of the systematic error, the
effect of the random and program errors on the absolute error
of measurement can be neglected. These results also demon-
strate that the e.s.d. values calculated by the used software are
less than the values of the random errors for all calculated
parameters. For this reason, unfortunately, it should be
recognized that when performing Rietveld refinement of
routine XRD patterns one cannot expect that the accuracy of
the unit-cell parameters will exceed the fourth decimal place.
This is true for the data obtained for any angular interval.
4. Conclusion
The results obtained show that the values of the parameters
calculated in a Rietveld refinement change only slightly when
increasing the angular interval in which the XRD pattern was
recorded. The most significant difference was observed when
the XRD pattern was ‘interrupted’ at about 60–70� 2�.
However, in most cases, this clearly observed difference was
not statistically significant. In addition, these observed
differences are comparable to the values of errors for para-
meters that are calculated by the TOPAS software. Our data
show that the differences between the values of parameters
calculated for different angular intervals are statistically
insignificant. This means that if a specific XRD experiment is
not aimed at refinement of atomic positions, interatomic
distances, site occupancies and displacement parameters the
XRD pattern could be acquired up to 75–85� 2�. Since
systematic error can only be determined and taken into
account with an internal standard, it probably does not make
sense to extend the angular range even for precise measure-
ments of the unit-cell parameters. Our results indicate that
changes in the interval in Rietveld refinement in practice do
not affect the calculated phase content and crystallite sizes.
This is because peak positions do not affect the calculated
values of these parameters. Here, the mean difference
between the calculated and real phase percentage in artificial
mixtures was 1.2% and did not exceed 5%. The average
differences for the crystallite size values were 0.33, 2.3 and
4.2 nm at crystallite sizes smaller than 20, 50 and 120 nm,
respectively. In the present study we did observe some
increase in precision (i.e. decreasing estimated standard
deviation) of Rietveld refinement results with the increased
angular range of the processed pattern. However, it is clear
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260 Vladimir Uvarov � Influence of XRD pattern range on Rietveld refinement J. Appl. Cryst. (2019). 52, 252–261
Figure 6Graphical representation of results of calculations of random, systematic,TOPAS and total errors for different angular intervals in the LaB6
analysis.
Table 6Results of calculations of random, systematic and TOPAS errors for different angular intervals in LaB6 analysis.
Angular range in Rietveld refinement, 2��
Parameters 10–70 10–75 10–85 10–90 10–102 10–110 10–115
a1 (A)† 4.156188 4.156299 4.156352 4.156365 4.156364 4.156318 4.156333a2 (A) 4.155970 4.156018 4.156067 4.156069 4.156063 4.156032 4.156075a3 (A) 4.155929 4.156025 4.156063 4.156075 4.156067 4.156035 4.156075Mean 4.156029 4.156114 4.156161 4.156170 4.156165 4.156128 4.156161
Random error (A) 0.000139 0.000160 0.000166 0.000169 0.000173 0.000164 0.000149Systematic error (A) �0.000887 �0.000802 �0.000755 �0.000746 �0.000751 �0.000788 �0.000755TOPAS error (A)‡ 0.000089 0.000082 0.000071 0.000062 0.000054 0.000048 0.000046Total error (A) 0.000469 0.000397 0.000356 0.000348 0.000353 0.000381 0.000344
† 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.
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J. Appl. Cryst. (2019). 52, 252–261 Vladimir Uvarov � Influence of XRD pattern range on Rietveld refinement 261