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 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:

vladimiru@savion.huji.ac.il

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

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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.

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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.

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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

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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