Materials Characterization 58 (2007) 883 – 891
Metrological characterization of X-ray diffraction methods for
determination of crystallite size in nano-scale materials
V. Uvarov ⁎, I. Popov
The Hebrew University of Jerusalem, The Faculty of Natural Science, The Center for Nanoscience and Nanotechnology,
The Unit for Nanoscopic Characterization, E. Safra Campus, Givat Ram, Jerusalem, 91904, Israel
Received 30 May 2006; received in revised form 31 August 2006; accepted 1 September 2006
Abstract
Crystallite size values were determined by X-ray diffraction methods for 210 TiO2 (anatase) nanocrystalline powders with
crystallite size from 3 nm to 35 nm. Each X-ray diffraction pattern was processed using different free and commercial software. The
crystallite size calculations were performed using Scherrer equation and Warren–Averbach method. Statistical treatment and
comparative assessment of the obtained results were performed for the purpose of an ascertainment of statistical significance of the
obtained differences. The average absolute divergence between results obtained with using Scherrer equation does not exceed
0.36 nm for the crystallites smaller than 10 nm, 0.54 nm for the range 10–15 nm and 2.4 nm for the range N15 nm. We have also
found that increasing the analysis time improves statistics, however does not affect the calculated crystallite sizes. The values of
crystallite size determined from X-ray data were in good agreement with those obtained by imaging in a transmission electron
microscope.
© 2006 Elsevier Inc. All rights reserved.
Keywords: X-ray diffraction; Crystallite size determination; Anatase; Nanocrystallites; Statistical treatment
1. Introduction
The crystallite size is one of the important parameters
that influence physical properties of nanomaterials.
Fabrication of materials with specified properties
requires close control of crystallite size. Several techniques could be used for the estimation of this value.
The crystallite size determination can be based on direct
observation of particles by transmission electron
microscopy (TEM) or scanning electron microscopy
techniques. In this case we can also receive the im-
⁎ Corresponding author. Tel.: +972 2 6586761; fax: +972 2 6584809.
E-mail address: vladimiru@savion.huji.ac.il (V. Uvarov).
URL: http://www.nanoscience.huji.ac.il/unit (V. Uvarov).
1044-5803/$ - see front matter © 2006 Elsevier Inc. All rights reserved.
doi:10.1016/j.matchar.2006.09.002
portant information on the shape of particles. Data on
crystallite size can be obtained by X-ray diffraction
(XRD) technique as the crystallite size is related to the
diffraction peak broadening. It is important that TEM
and XRD methods allow not only to measure the
crystallite size, but also to identify crystalline phases.
The XRD methods for crystallite size determination
are applicable to crystallites in the range of 3–100 nm.
The diffraction peaks are very broad for crystallites
below 2–3 nm, while for particles with size above
100 nm the peak broadening is too small. Klug and
Alexander [1], Mittemeijer and Scardi [2] and Delhez et
al. [3] have described features of various XRD methods
for crystallite size determination. There are many
articles in which the results of the particle size determination obtained by various techniques are compared
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[4–8]. Usually the crystallite sizes determined by
various XRD methods differ, and it is not clear if the
observed divergences are significant or not. At the same
time there is no common opinion on advantages and
disadvantages of different XRD methods for the crystallite size determination. In our opinion for an estimation of applicability of any modification of XRD
method and for comparison of received results it is
necessary to have reliable statistics. We have not found
papers in which results of statistical analysis of the
crystallite size determination by various XRD methods
and their comparison with each other were presented.
Therefore performance rating of these methods is desirable and necessary.
In the present paper we report application of various
XRD methods for crystallite size determination of TiO2
powders with crystallite size from 3 nm to 35 nm. The
aim of this work was to compare the results of the
crystallite size obtained by various XRD methods and to
ascertain a statistical significance of the divergences
between these results.
2. Materials and experiment
2.1. Theoretical background
If analyzed crystals are free from microstrains and
defects, peak broadening depends only on the crystallite
size and diffractometer characteristics. In this case we
can use classical Scherrer [9] Eq. (1) for crystallite size
determination:
d¼
K k
b cosh
ð1Þ
where d is the crystallite size, λ is the X-ray wavelength, β is the width of the peak (full width at half
maximum (FWHM) or integral breadth) after correcting
for instrumental peak broadening (β expressed in radians), θ is the Bragg angle and K is the Scherrer
constant. According to [10] K value depends on the
crystal shape and the diffraction line indexes. Recently,
it has been found that Scherrer constant also depends on
the dispersion of crystallite sizes of the powder [11].
Usually, K is typically between 0.8 and 1.39 and for the
spherical particles K is nearly 1. The Scherrer equation
gives volume-weighted mean column length. In other
words, the d value calculated for (hkl) peak should be
understood as mean crystallite size in the direction that
is perpendicular to the (hkl) plane (hkl is Miller indices).
In contrast to this the Warren–Averbach analysis [12],
which is based on a Fourier deconvolution of the measured peaks, gives area-weighted mean column length.
The differences between the apparent domain size
obtained through volume and area weighting of column
lengths may be quite large [6].
In any case we should process a diffraction spectrum
to determine peak broadening (β). The modern software
for X-ray data processing allows choosing various
mathematical functions describing the peak shape for
the peak's width determination [13–17]. Pseudo-Voigt
or Pearson VII functions are usually used as profile
functions, since these functions provide the best fitting
for the usually observed peak shape. The newest available Fundamental Parameters approach (FP) allows
direct analysis of line broadening without a reference
sample when an instrument is well characterized [18].
Under such an approach an instrumental peak broadening is calculated based on real diffractometer characteristics (acquisition geometry, X-ray tube design, primary
and secondary optics, specimen size etc.).
2.2. Materials and methods
TiO2 powders prepared by hydrothermal synthesis
were obtained from Casali Institute for Applied Chemistry of the Hebrew University of Jerusalem. X-ray
powder diffraction measurements were performed on a
D8 Advance diffractometer (Bruker AXS, Karlsruhe,
Germany) with a goniometer radius 217.5 mm, Göbel
Mirror parallel-beam optics, 2° Sollers slits and 0.2 mm
receiving slit. The powder samples were placed on low
background quartz sample holders. XRD patterns from
20° to 60° 2θ were recorded at room temperature using
CuKα radiation (λ = 0.15418 nm) with the following
measurement conditions: tube voltage of 40 kV, tube
current of 40 mA, step scan mode with a step size of
0.02° 2θ and a counting time of 1 s per step. The
instrumental broadening was determined using LaB6
powder (NIST SRM 660).
TiO2 samples were examined by TEM imaging and
selected area electron diffraction (SAED). TEM observations were carried out with Tecnai F20 G2 (FEI
Company). TEM samples were prepared as follows: the
powder was ultrasonically dispersed for 5 min in absolute ethanol and then dropped on a 400 mesh copper grid
coated with amorphous carbon film.
Altogether 210 samples have been analyzed. The
statistical treatment and comparative assessment of the
obtained results were performed. An assessment of the
accuracy of the crystallite size determination by XRD
methods was carried out by comparison obtained results
with the results of TEM study of the same samples.
We used Powder Cell for Windows v.2.4 (PCW)
freeware [14], WinFit freeware [16] and TOPAS 2
V. Uvarov, I. Popov / Materials Characterization 58 (2007) 883–891
885
profile fitting: pseudo-Voigt (pV) and Pearson VII (P
VII) approximation functions and fundamental parameters approach (FP).
WinFit program calculates the crystallite size by
Warren–Averbach method. Pearson VII function was
used for profile fitting.
Statistical treatment of the obtained results was performed according to the recommendation published in
[19].
3. Results and discussion
Fig. 1. X-ray diffraction patterns acquired from samples with different
crystallite sizes (a — 21.6 nm, b — 14.8 nm, c — 9.2 nm, d — 4.6 nm).
(Bruker AXS) for profile fitting and crystallite size
calculations.
PCW calculates averaged value of the crystallite size
by Scherrer equation using all observable peaks.
Pseudo-Voigt function was used for profile fitting in
the Rietveld refinement procedure realized in PCW.
TOPAS 2 also calculates crystallite size by Scherrer
equation. But, we used three alternative techniques for
XRD study of the TiO2 powders has shown that in
most cases they were pure anatase with unit cell parameters a = 0.379 nm and c = 0.951 nm. Some samples
contained small amounts of rutile (TiO2), brookite
(TiO2) and hongquiite (Ti1 − xO1 − y). The unit cell parameters of examined anatase samples varied very
insignificantly.
Fig. 1 shows typical XRD patterns of the tested samples. Significant peak broadening is observed especially
for the smallest crystallite size values. Obviously the
(101) and (200) peaks are the most suitable for crystallite
size determination, because they do not overlap with
Fig. 2. Plot of crystallite size values calculated from (101) peak: a — Powder Cell vs TOPAS (FP), b — Powder Cell vs TOPAS (pV), c — Powder
Cell vs TOPAS (P VII), d — TOPAS (pV) vs TOPAS (P VII), e — TOPAS (FP) vs TOPAS (pV), f — TOPAS (FP) vs TOPAS (P VII).
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V. Uvarov, I. Popov / Materials Characterization 58 (2007) 883–891
Fig. 3. Plot of crystallite size values calculated from (200) peak: a — TOPAS (FP) vs TOPAS (pV), b — TOPAS (FP) vs TOPAS (P VII), c — TOPAS (pV)
vs TOPAS (P VII).
neighbors. We used these peaks for the calculation of
crystallite size. The microstrain value calculated by
PCW and TOPAS programs was in the range of 0.0001–
0.005. Therefore we considered microstrain as a minor
factor contributing to peak broadening [20].
Plots of the crystallite size values calculated by
different methods are shown on Figs. 2 and 3 for (101)
and (200) peaks, respectively. For comparison, Fig. 4
shows values of crystallite size calculated for the same
samples from (101) and (200) peaks separately. As is
seen from Figs. 2 and 3, all the presented datasets
exhibited good correlation. Thus, we conclude that
application of Scherrer equation to the crystallite size
calculation of nano-sized TiO2 provides very close
results through all tested mathematical routines.
Therefore, it was interesting to compare the above
results with those calculated from the same samples by
Warren–Averbach method (see Fig. 5). Although good
correlation was also observed for the plotted datasets,
the absolute value of crystallite size calculated by
Warren–Averbach method (WinFit program) was about
35% lower than those obtained with Scherrer equation.
This phenomenon is well known and was already reported in numerous publications earlier (for instance in
[6]). It is usually attributed to the intrinsic differences in
the physical meaning of coherent scattering regions
considered by these two methods.
For statistical treatment we divided our results onto 3
size groups as follows: less than 10 nm range, 10–15 nm
range and 15–35 nm range.
At the first step it was necessary to ascertain the
homogeneity of the variances in each group. Fisher's test
has been applied for this purpose. The results of
calculations of average values, sample variances and
standard deviation for all intervals and the applied
methods are presented in Tables 1 and 2. The sample
Fig. 4. Plot of crystallite sizes calculated with TOPAS from (101) peak vs that calculated from (200) peak with using: a — FP, b — pV and c —
Pearson VII.
V. Uvarov, I. Popov / Materials Characterization 58 (2007) 883–891
Fig. 5. Plot of crystallite size calculated by Scherrer equation (PCW) vs
that obtained by Warren–Averbach method (WinFit).
variance and the relative standard deviation are calculated as:
n
P
V ðdÞ ¼ s ðdÞ ¼
2
sr ðdÞ ¼
P 2
P
MD ¼
ð2Þ
n1
sðdÞ
ð3Þ
P
d
where V(d) is sampling variances, d is crystallite size, s
and sr are standard deviation and relative standard
deviation, n is number observations. According to Fisher's
test the variances are similar if the calculated value of
Fisher's criterion ξ is less than a tabulated value, i.e.
n¼
As is clearly seen in Tables 1 and 2, within each size
group the average values of crystallite size obtained by
all methods are very close. At the same time we have no
prior information on the accuracy of each applied
method. In such situation we are able to estimate only
the statistical significance of the observed differences.
Student's t-test for correlated samples has been applied
to ascertain a statistical significance of the differences
between the results of the crystallite size values calculated by different methods. Since we are interested
only in the differences between the methods, we should
consider only one variable Di = da,i - db,i (da,i and db,i are
crystallite sizes that were calculated by different
methods). The standard deviation σMD of the sampling
distribution MD and t-value for the Student's test were
calculated as
ðdi d Þ
i¼1
s21
V FðP; f 1; f 2Þ
s22
ð4Þ
where F(P, f 1, f 2) is tabulated value of Fisher coefficient,
P = 0.95 is confidence probability and f 1, f 2 are number
of degrees of freedom (i.e. number of the samples in each
size group). Value of F(P, f 1, f 2) is 1.45, 1.65 and 1.61 for
f = 106, 49 and 55, respectively. The Fisher's test reveals
homogeneity of variance for all size groups.
887
rMD
Di
n
ð5Þ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P 2
uP
ð
Di Þ
u
2
t Di n
¼
n ðn 1Þ
tcal: ¼
ð6Þ
j rM j
D
ð7Þ
MD
The calculations were performed only for (101)
anatase peak. The results of these calculations are summarized in Table 3.
According to Student's test the difference between the
methods is significant if tcal > ttab. The critical value ttab
for the 5% significance level and n > 40 is 2.00. As is
clearly seen, for all size groups only PCW vs TOPAS
(P VII) do not diverge at a statistically significant level.
All other compared pairs of calculated datasets exhibit
statistically significant differences in at least one size
group. However, the average absolute difference between
Table 1
Results of the preliminary statistical treatment of the experimental data for (101) anatase peak
Crystallite
size, nm
b10
10–15
N15
Number of
observations
106
49
55
Mean crystallite size, nm
Sampling variances
Powder
Cell
TOPAS 2
TOPAS 2
FP
pV
P VII
Powder
Cell
FP
pV
P VII
7.15
12.20
19.91
6.90
11.42
17.98
7.11
11.66
18.75
7.27
12.22
20.40
3.16 / 0.25
2.73 / 0.14
17.29 / 0.21
2.74 / 0.24
3.30 / 0.16
17.60 / 0.23
2.85 / 0.24
2.75 / 0.14
19.35 / 0.23
3.06 / 0.24
3.67 / 0.16
20.49 / 0.26
Relative standard deviation
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V. Uvarov, I. Popov / Materials Characterization 58 (2007) 883–891
Table 2
Results of the preliminary statistical treatment of the experimental data
for (200) anatase peak
Table 4
Statistical estimation of the divergence of the crystallite sizes obtained
from (101) and (200) diffraction peaks of anatase
Crystallite Number of Mean crystallite
size, nm observations size, nm
Compared
methods
Class of crystallite size, MD
nm
σMD
FP
b10
10–15
N15
b10
10–15
N15
b10
10–15
N15
0.087 0.87
0.173 3.24
0.216 3.66
0.109 1.90
0.214 3.09
0.238 7.18
0.085 2.00
0.244 3.67
0.253 10.08
FP
b10
106
10–15
49
N15
55
pV
Sampling variances
Relative standard
deviation
P VII FP
6.88 6.95 7.17
2.88 /
0.25
10.82 10.92 11.21 3.78 /
0.18
17.18 17.03 17.84 20.99 /
0.27
pV
P VII
2.44 /
0.22
3.69 /
0.17
20.53 /
0.27
2.58 /
0.22
4.97 /
0.20
24.07 /
0.28
methods does not exceed 0.36 nm for the b 10 nm size
group, 0.54 nm for 10–15 nm size group and 2.4 nm for
N15 nm range (see Table 3). Considering the mean
crystallite sizes presented in Table 1, we find that their
maximum relative deviations are 5.1%, 4.6% and 12.6%
respectively for the considered size groups. We presume
that comparative statistical treatment of the results
calculated by Warren–Averbach method and by Scherrer
equation should give very similar results.
Comparison of the calculated crystallite size for (101)
peak with that for (200) peak shows that the latter value is
usually slightly lower: 1.3% lower for b 10 nm size group,
7.2% — for 10–15 nm range and on 9.7% — for N15 nm
range. Similar results were reported in [4]. Apparently,
pV
P VII
0.076
0.561
0.792
0.207
0.663
1.710
0.170
0.895
2.550
j j
t ¼ MD
rMD
such deviations could be caused by non-equiaxial morphology of the tested crystallites.
We estimated statistical significance of these divergences (see Table 4). The divergences are not significant
only for the smallest crystallites of less than 10 nm. We
suppose that the smallest crystallites have the highest
probability of isotropic morphology.
It is commonly accepted that the quality of fitting
affects the final value of parameters extracted from XRD
data. In order to check how sensitive to fitting quality
crystallite size value is we performed the following test.
XRD pattern was acquired from the same sample
initially as a single scan with counting time 1 s/per step
and then with using auto-repeating mode we recorded
patterns resulted from the accumulation of 2 to 10
succeeding scans. Ten XRD patterns obtained this way
Table 3
Statistical estimation of the divergence of the crystallite sizes obtained
by various XRD methods
j j
Compared
methods
Class of crystallite
size, nm
MD
σMD
PCW–FP
b10
10–15
N15
b10
10–15
N15
b10
10–15
N15
b10
10–15
N15
b10
10–15
N15
b10
10–15
N15
0.249
0.461
1.934
0.042
0.291
1.167
− 0.114
− 0.110
− 0.435
− 0.207
− 0.170
− 0.767
− 0.360
0.541
− 2.419
− 0.150
− 0.159
− 1.652
0.081 3.05
0.162 2.84
0.248 7.79
0.044 0.49
0.122 2.39
0.211 5.53
0.088 1.30
0.142 0.84
0.250 1.74
0.042 4.92
0.092 1.95
0.108 7.10
0.030 12.24
0.113 4.81
0.212 11.41
0.030 4.43
0.237 0.67
0.143 11.55
PCW–pV
PCW–P VII
FP–pV
Fp–P VII
PV–P VII
t ¼ MD
rMD
Crystallite sizes were calculated from (101) peak for TOPAS and as
mean for all observed peaks for PCW.
Fig. 6. Values of crystallite sizes and Rwp factor calculated for different
counting time with all four techniques for the same sample.
V. Uvarov, I. Popov / Materials Characterization 58 (2007) 883–891
889
Fig. 7. TEM images and SAEDs (insets) obtained from the samples for which calculated crystallite size (PCW) was: a — 26.1 nm, b — 14.2 nm, c —
9.5 nm, d — 4.4 nm.
were processed with four tested techniques in order to
calculate the value of crystallite size and estimate the
quality of the fitting procedure.
Usually fitting quality for the whole diffraction pattern is estimated by the value of Rwp factor [15] that is
calculated by equation:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
uP
u wi jyi yc;i j2
u
i1;n
P
ð8Þ
Rwp ¼ 100u
t
wi y2i
i¼1;n
where yi and yc,i are measured and calculated profile
intensities, wi is weight of the observation, wi = 1 / σi2
and σi2 is the variance of the profile intensity yi.
Therefore, the lower the value of Rwp the higher the
fitting quality is. Values of the Rwp factor and crystallite
size calculated from data with increased counting time
are presented in Fig. 6.
For all the patterns acquired as a single scan with 1 s/
step counting time we typically got Rwp values of 10–
13%. As is seen in Fig. 6 accumulating of raw data over a
longer counting time really resulted in decreasing the
value of the Rwp factor from 10–13% for 1 s to less than
4% for 10 s, i.e. fitting quality was improved. At the same
time, the calculated value of the crystallite size remained
practically the same for each counting time. For practical
considerations, this result confirms that the crystallite size
value calculated from the data acquired at the time limiting
conditions (i.e. 1 s/step counting time) could be used.
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V. Uvarov, I. Popov / Materials Characterization 58 (2007) 883–891
Statistical treatment has revealed that, as a rule, the
divergences between the crystallite sizes values calculated by different techniques are significant. However
obtained statistics does not give the information on the
accuracy of the crystallite sizes determination. In our
opinion the obtained results are comparable on accuracy
so that there is no basis to prefer one or another method.
We consider the practical meaning of our results as a
recommendation to use more than one physical method
for the evaluation of the crystallite size of a nanodimensional material. Namely, each one of the tested
XRD techniques may be used with practically equal
accuracy, although additional information (for instance,
from direct imaging) will improve the accuracy and
extend the understanding of materials structure.
We applied such approach to the analysis of several
samples. First of all, we analyzed a well-known sample of
commercial P-25 powder manufactured by Degussa AG.
According to the reported data, the crystallite size of
anatase in P-25 powder is in the range 20.7–32 nm [21–
24]. In our study the crystallite size of P-25 averaged on
four XRD methods is 28.8 nm (PCW — 26.1 nm, FP —
28.2 nm, pV — 29.4 nm and P VII — 31.8 nm) and
approximately 27 nm as found by TEM study (see Fig. 7a).
We have also performed a TEM study on additional 15
samples prepared by hydrothermal synthesis. Typical
TEM images and electron diffraction patterns of the examined samples are shown in Fig. 7. Direct TEM imaging
revealed that all the tested samples have practically equiaxial morphology and narrow distribution of crystallite
size. The latter is usually attributed to the properties of the
hydrothermal synthesis itself. It is also clearly seen that size
and morphology of the particle strongly affects the appearance of an electron diffraction pattern. A number of diffraction rings and intensity distribution within the ring are
highly sensitive for such small changes in crystallite size
as, for example, increasing from 4 to 9 nm (see Fig. 7d and
c). Thus, additional semi-quantitative information source is
allowed through SAEDs which being acquired from a
relatively large area, provides information about huge
amounts of crystallites. Finally, we have to stress that in
each case very good agreement was found between XRD
and TEM results.
4. Conclusions
Experimental results of crystallite size estimation for
nano-dimensional TiO2 powders presented above allow
us to conclude:
1. When crystallite size of a nano-scale material (up to
35 nm size) is calculated from XRD data with Scherrer
2.
3.
4.
5.
equation, all the tested routines available within PCW
freeware and commercial TOPAS provide very close
results. Absolute value of crystallite in nano-sized
TiO2 as-determined from X-ray data practically
coincides with that obtained by direct imaging in TEM.
For the first time the statistical estimation of the
revealed divergences was performed and numerical
characteristics for these divergences were received.
The least and the not significant divergence has been
revealed between the results obtained by PCW and
TOPAS with P VII approximation function for profile
fitting. In other cases divergences were statistically
significant.
Calculation of crystallite size by Warren–Averbach
method gives lower values than that obtained from
Scherrer equation. However, close correlation exists
between the crystallite sizes calculated by both methods.
Using (200) anatase peak gives slightly lower
crystallite size (6% on average) in comparison with
that for (101) peak. At the same time (200) peak can be
used for crystallite size calculation in difficult cases,
for example when brookite is present in the sample.
Increasing the counting time from 1 to 10 s/step does
not affect the final value of crystallite size calculated
from XRD data, although it improved the fitting
quality of the whole diffraction pattern.
We believe that the presented results could serve as
practical recommendations for the application of the
XRD technique for the estimation of crystallite size in
nano-scale materials.
References
[1] Klug HP, Alexander LE. X-ray diffraction procedures for
polycrystalline and amorphous materials. 2nd ed. New York:
J. Wiley & Sons; 1974.
[2] Mittemeijer EJ, Scardi P. Diffraction analysis of the microstructure of materials. Berlin Heidelberg: Springer-Verlag; 2004.
[3] Delhez R, de Keijser TH, Mittemeijer EJ. Determination of
crystallite size and lattice distortions through X-ray diffraction
line profile analysis. Recipes, methods and comments. J Anal
Bioanal Chem 1982;312(1):1–16.
[4] Weibel A, Bouchet R, Boulc'h F, Knauth P. The big problem of
small particles: a comparison of methods for determination of
particle size in nanocrystalline anatase powders. Chem Mater
2005;17:2378–85.
[5] Tian HH, Atzmon M. Comparison of X-ray analysis methods
used to determine the grain size and strain in nanocrystalline
materials. Philos Mag, A 1999;79(8):1769–86.
[6] Balzar D, Audebrand N, Daymond MK, Fitch A, Hewat A,
Langford J, et al. Size–strain line-broadening analysis of the ceria
round-robin sample. J Appl Crystallogr 2004;37:911–24.
[7] Marinkovic B, de Avellez RR, Saavedra A, Assuncao FCR. A
comparison between the Warren–Averbach method and alternate
V. Uvarov, I. Popov / Materials Characterization 58 (2007) 883–891
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
methods for X-ray diffraction microstructure analysis of
polycrystalline specimens. Mater Res 2001;4(2):71–6.
Santra K, Chatterjee P, Sen Gupta SP. Voigt modeling of size–
strain analysis: application to α-Al2O3 prepared by combustion
technique. Bull Mater Sci 2002;25(3):251–7.
Scherrer P. Estimation of the size and internal structure of
colloidal particles by means of r.ovrddot.ontgen rays. Nachr. Ges.
Wiss. Gottingen; 1918. p. 96–100.
Shull CG. The determination of X-ray diffraction line widths.
Phys Rev 1946;70(9–10):679–84.
Pielaszek, R. Diffraction studies of microstructure of nanocrystals exposed to high pressure, Ph.D. thesis, Warsaw University,
Department of Physics, Warsaw, Poland; 2003.
Warren BE, Averbach BL. The effect of cold-work distortion on
X-ray patterns. J Appl Phys 1950;21:595–9.
Cheary, RW, Coelho, AA. Software: XFIT-FOURYA, deposited in
CCP14 Powder Diffraction Library, Engineering and Physical
Sciences Research Council, Daresbury Laboratory, Warrington,
England; 1996 (http://www.ccp14.ac.uk/tutorial/xfit-95/xfit.htm).
Kraus W, Nolze G. POWDER CELL — a program for the
representation and manipulation of crystal structures and calculation of the resulting X-ray powder patterns. J Appl Crystallogr
1996;29:301–3.
Roisnel T, Rodríguez-Carvajal J. WinPLOTR: a Windows tool
for powder diffraction patterns analysis. In: Delhez R, Mittenmeijer EJ, editors. Mat Sci Forum, Proc Seventh European
Powder Diffraction Conference (EPDIC 7); 2000. p. 118–23.
891
[16] Krumm S. WINFIT 1.0 — a computer program for X-ray diffraction
line profile analysis. Acta Univ Carol, Geol 1994;38:253–6.
[17] Stewart JM, Zhang Y, Hubbard CR, Morosin B, Venturini EL.
XRAYL: a new powder diffraction profile fitting program. J Appl
Crystallogr 1989;22(6):640–1.
[18] Cheary RW, Coelho AA, Cline JP. Fundamental parameters line
profile fitting in laboratory diffractometers. J Res Natl Stand
Technol 2004;109(1):1–25.
[19] Lowry, R., Concepts & Applications of inferential statistics,
(http://faculty.vassar.edu/lowry/VassarStats.html).
[20] Lee Penn R, Banfield JF. Imperfect oriented attachment:
dislocation generation in defect-free nanocrystals. Science
1998;281:969–71.
[21] Nagaveni K, Sivalingam G, Hegde MS, Giridhar Madras. Solar
photocatalytic degradation of dyes: high activity of combustion
synthesized nano TiO2. Appl Catal B: Environ 2004;48(2):83–93.
[22] Tayade Rajesh J, Kulkarni Ramchandra G, Jasra Raksh V.
Photocatalytic degradation of aqueous nitrobenzene by nanocrystalline TiO2. Ind Eng Chem Res 2006;45(3):922–7.
[23] Sreethawong T, Suzuki Y, Yoshikawa S. Synthesis, characterization,
and photocatalytic activity for hydrogen evolution of nanocrystalline mesoporous titania prepared by surfactant-assisted templating
sol–gel process. J Solid State Chem 2005;178(1):329–38.
[24] Nagaveni K, Hegde MS, Ravishankar N, Subbanna GN,
Girindhar Madras. Synthesis and structure of nanocrystalline
TiO2 with lower band gap showing high photocatalytic activity.
Langmuir 2004;20:2900–7.