PDF

1. Pair distribution
function analysis
The Analytical X-ray Company
Total scattering experiments
using high-energy X-rays on
a laboratory system
Summary
Analyzing powder diffraction data
of nanocrystalline and amorphous
materials using the atomic pair
distribution function (PDF) method
provides useful information about
the long- and short-range ordering
of the atoms in the materials. We
have developed the application
of PDF analysis on a standard
laboratory system employing an
X-ray tube with either a silver or a
molybdenum anode as X-ray source.
Data obtained from a variety of
samples are shown. Meaningful
results have been achieved, that
allowed extracting structural
information for comparison with
data reported in literature.
Introduction
Recent years have shown an increased
interest in the study of nanocrystalline
materials due to their specific properties
for application in e.g. semiconductors,
pharmaceuticals and polymers.
Structural information about these
materials is present as broad, not well
defined features in a diffractogram.
Analysis of nanomaterials therefore
requires a total scattering approach,
including both Bragg peaks and diffuse
scattering. One of the most promising
analytical methods used is atomic pair
distribution function (PDF) analysis.
Originally, this method was used to
study primarily amorphous and highly
disordered materials. More recently,
it has been used for the analysis of
nanostructured materials. Since the
method requires short wavelengths
to obtain high resolution in real space
(well defined interatomic distances),
often the measurements are performed
at synchrotron facilities, making use of
both the high photon energies and the
high photon flux that these facilities
offer.
We have investigated the possibility to
apply the pair distribution technique
on an in-house system, using Ag or
Mo Kα radiation. This application note
describes typical results on nanocrystals,
liquids and amorphous materials.
G[Å]-2
r [Å]
10 2 3 4 5 6
XRD APPLICATION NOTE
Schematic representation of the construction
of the atomic pair distribution function from a
square array of atoms [2]. The colored circles (a)
indicate the distance from a central atom where
a neighboring atom can be found. The arrows
indicate the corresponding peaks in the pair
distribution function.
(a)
(b)

2. Pair distribution function analysis
The pair distribution function G(r) describes the probability of finding two atoms separated by a distance r in the
material under investigation. The PDF method extracts structure-related information from powder diffraction data [3].
Since the technique takes both Bragg and diffuse scattering into account, it provides information not only about the
long-range (>10 nm) atomic ordering but also about the short-range ordering in materials. The method is performed in
the following steps:
(i) the diffraction pattern is corrected for background (using a separate diffraction measurement of an empty sample
container), Compton scattering, detector dead time, absorption, diffraction geometry and polarization;
(ii) the corrected X-ray diffraction data is scaled into electron units and the reduced structure function [4] is calculated;
(iii) the structure function is Fourier transformed to obtain the atomic pair distribution function:
G(r) = 4πr (ρ(r) - ρ0)
in which ρ(r) is the local atom number density, and ρ0 is the mean atom number density.
Since the method does not assume periodicity in the material, it is widely applied for the study of nanocrystalline and
amorphous materials. The data can be used for full-profile fitting to refine structural models [5].
Empyrean goniometer
with programmable
divergence slit and
incident beam anti-
scatter slit, capillary
spinner, programmable
receiving slit with
programmable
anti-scatter slit and
scintillation detector
PreFIX mounting
The optics and sample
stage are designed
according to the PreFIX
concept; they are
interchangeable with all
other available modules
for Empyrean without
the need for
realignment. The
capillary spinner stage
uses a goniometer head
to allow precise
alignment of the
capillary tube on the
goniometer axis.
Beam path of the used configuration
Instrumental configurations
X-ray diffraction measurements were
performed on a PANalytical Empyrean
system equipped with a programmable
divergence slit, a capillary spinner, a
dedicated anti-scatter device, and either
X’Celerator detector or a scintillation
detector with a programmable
receiving slit. The X-ray source was a
tube with either silver or molybdenum
anode delivering Kα radiation with a
wavelength of 0.0561 or 0.0709 nm,
respectively.
Additional shielding was applied to
the optical path in order to achieve a
feature-free background.
The samples were prepared in glass
capillaries with a diameter of 2 mm.
Scans along the 2θ axis were made up to
an angle of 160 degrees corresponding
to a scattering vector Q of 22 A-1 when
a silver anode is used. The scattering
vector is given by:
Q = 4π sinθ / λ
Initial data treatment, including
background subtraction and optional
Kα2 stripping, was done using X’Pert
HighScore.
For PDF analysis and fitting, we used the
software RAD [6] and PDFgui [5].
Experimental
setup Instrument Empyrean
X-ray tube Empyrean Tube with silver (Ag) or molybdenum (Mo)
anode, long fine focus
Incident beam optics Divergence slit with incident beam anti-scatter slit and
rhodium (Rh) or zirconium (Zr) beta-filters, focusing
beam X-ray mirrors
Sample stage Capillary spinner
Reflection-transmission spinner
Flat sample stage
Diffracted beam optics Programmable receiving slit with programmable
anti-scatter slit or a dedicated anti-scatter device
Detector X’Celerator, scintillation detector
Scan parameters Typically 2 - 160° 2θ, 0.06° step size
Total scan time 8-24 hours depending on the material and the
configuration used for the experiment

3. Results and
discussion
Intensity[counts]F(Q)[Å-1]
(Q) [Å-1]
2 theta [deg.] (Ag radiation)
62500
40000
22500
10000
2500
0
10 20 30 40 50
Empty container
Experiment data
60 70 80 90 100 110 120 130
-20
-10
0
10
20
30
40
0 5 10 15 20
Intensity[counts]F(Q)[Å-1]
(Q) [Å-1]
2 theta [deg.] (Ag radiation)
62500
40000
22500
10000
2500
0
10 20 30 40 50
Empty container
Experiment data
60 70 80 90 100 110 120 130
-20
-10
0
10
20
30
40
0 5 10 15 20
Radial distance [Å]
10
5
0
-5
10 20 30 40 50 60
Gdiff
Gobs
Gcalc
G(r)[Å-2]
Radial distance [Å]
Gdiff
Gobs
Gcalc
10
5
0
-5
5 10 15 20 25
a
b
c
d a: ¼<111>
b: ½<110>
c: <100>
d: ¼<331>
Gdiff
Gobs
Gcalc
G(r)[Å-2]
Samples of different nature – crystalline,
nanocrystalline, amorphous solid
and liquid – were selected to test
the applicability of PDF analysis on a
standard XRD system. The results of
these experiments are described below.
Silicon carbide
Figure 1a shows a diffraction pattern
of silicon carbide powder in a capillary,
together with a measurement of an
empty capillary. The reduced structure
function obtained from the corrected
intensity data is shown in Figure 1b.
After Fourier transformation the PDF as
shown in Figure 2 was obtained. Figure
2b shows the short distances of the PDF
in more detail. The maxima in this graph
could be identified as the interatomic
distances Si-C, Si-Si and C-C, derived
from the sphalerite crystal structure
of SiC [6]. The relation between the
orientation and the interatomic
distances in SiC is shown in Table 1.
Figure 1. (a) XRD
measurement and
(b) reduced structure
function of silicon
carbide
Figure 2. Experimental
(circles) and calculated
atomic PDF (red line)
of SiC
SiC crystal
structure
Table 1. Interatomic distances of SiC calculated
determined from the experimental PDF (Fig.2)
(a)
(a)
(b)
(b)
Atoms Orientation Interatomic
distance [Å]
Si - C ¼ <111> 1.89
Si - Si, C - C ½ <110> 3.08
Si - C ¼ <311> 3.61
Si - Si, C - C <100> 4.36
Si - C ¼ <331> 4.75
Si - Si, C - C ½ <211> 5.34

4. Variable counting time
Measurements performed for PDF
analysis typically require long-range
scans up to high 2θ angles, where the
diffracted intensities are low. Variable
counting time (VCT) methods can be
applied to spend longer counting times
at the high-angle, low-intensity region
of the diffractogram at the cost of
time spent on the low-angle region.
Schematically the redistribution of
measurement times is shown in Figure 3.
The total measurement time is the same
for both situations.
In order to investigate the improvement
of data quality at high Q-values,
measurements were performed on
nanocrystalline anatase (TiO2) with an
average particle size of 15 nm using
constant and variable measurement
times according to the scheme given in
Figure 3. The resulting diffractograms
and reduced structure functions are
shown in Figure 4 and 5 respectively.
Figure 3. Constant counting time (left) and variable counting time (right) as a function of 2θ angle
Figure 4. XRD
measurements on
nanocrystalline anatase
measured using (a)
constant counting
time and (b) variable
counting time
Results and
discussion ctd.
(b)
20 40 60 80 100 120 140
20 40 60 80 100 120 140
2theta [deg.]
0
50000
100000
150000
200000
Intensity[counts]
2theta [deg.]
0
10000
20000
30000
Intensity[counts]
(a)

5. Figure 5. Reduced
structure functions
of anatase, measured
using variable and
constant counting
times
The noise level at high Q-values of the
variable counting time measurement
is improved in comparison with the
constant counting time measurement
allowing the observation of additional
structure-related features. No reduction
in data quality has been observed in
the low Q-range. The experimental PDF
derived from the VCT experiment is in
good agreement with the calculated
PDF, as is shown in Figure 6.
Anatase - constant counting time
-5
0
5
10
0 5 10 15 20
Anatase - variable counting time
-5
0
5
10
0 5 10 15 20
F(Q)[Å-1]F(Q)[Å-1]G(r)[Å-2]
Radial distance [Å]
4
2
0
-1
0 10 20 30 40
Gdiff
Gobs
Gcalc
3
1
-2
-3
Figure 6.
Experimental
(circles) and
calculated atomic
PDF (red line) of
nanocrystalline
anatase

6. Focusing mirrors for high-energy X-rays
(Ag and Mo)
The graded multilayer focusing X-ray
mirror is a beam conditioner, which
is able to convert the divergent X-ray
beam from a tube in line focus position
to an intense monochromatic beam that
is focused onto the goniometer circle.
Experimental configurations for PDF
analysis using slit and mirror optics are
graphically compared in Figure 7 and 8.
Results and discussion ctd.
Figure 8: configuration with a focusing X-ray mirror using convergent X-ray beam
Sample in horizontal
orientation
Line- or point
detector - X’Celerator
or scintillation counter
Line- or point
detector - X’Celerator
or scintillation counter
Figure 7: standard configuration using divergent X-ray beam

7. The performance of the focusing X-ray
mirror for diffraction measurements in
transmission geometry (including PDF
analysis) was tested using silicon powder
(NIST SRM 640b) prepared in a 0.3 mm
glass capilary. Experiments showed that
the intensity and angular resolution
of the diffraction data collected with
the focusing mirror for Mo radiation
are suitable for PDF analysis as well as
traditional diffraction applications, such
as phase analysis.
Figure 9: silicon 640b: raw measurements. Data collected using a focusing mirror for Mo radiation
(λ = 0.7093 Å)
Figure 10: experimental PDF of silicon (blue dots) compared with a calculated PDF using the known
structure of Si (red line).
Figure 11: angular resolution is comparable to focusing mirror for Cu radiation.
10 20 30 40 50 60 70 80 90 100 110 120 130 140
0
10000
40000
90000
160000
Intensity[coutns]
Experimental data
Empty capillary 0.3 mm
2theta [deg]
6000
4000
2000
Intensity[counts]
21.6 21.7 21.8 21.9 22.0 22.1 22.2 22.3 22.4 22.5 22.6
21.9962 (°), 6800.6 (counts)
0.0474 (°)
2theta [deg]
10
5
0
-5
G(r)(Å-2
)
Radial distance [Å]
10 20 30 40 50 60
Gdiff
Gobs
Gcalc
-10
70
10 20 30 40 50 60 70 80 90 100 110 120 130 140
0
10000
40000
90000
Intensity[
2theta [deg]
6000
4000
2000
Intensity[counts]
21.6 21.7 21.8 21.9 22.0 22.1 22.2 22.3 22.4 22.5 22.6
21.9962 (°), 6800.6 (counts)
0.0474 (°)
2theta [deg]
10
5
0
-5
G(r)(Å-2
)
Radial distance [Å]
10 20 30 40 50 60
Gdiff
Gobs
Gcalc
-10
7010 20 30 40 50 60 70 80 90 100 110 120 130 140
0
10000
40000
90000
160000
Intensity[coutns]
Experimental data
Empty capillary 0.3 mm
2theta [deg]
6000
4000
2000
Intensity[counts]
21.6 21.7 21.8 21.9 22.0 22.1 22.2 22.3 22.4 22.5 22.6
21.9962 (°), 6800.6 (counts)
0.0474 (°)
2theta [deg]
10
5
0
-5
G(r)(Å-2
)
Radial distance [Å]
10 20 30 40 50 60
Gdiff
Gobs
Gcalc
-10
70

8. Figure 12. XRD
measurement of
vanadium oxide
xerogel using Mo
radiation. Background
intensity has been
subtracted from the
data.
Figure 13. Reduced
structure function
of vanadium oxide
xerogel
Figure 14. Atomic PDF
of vanadium oxide
xerogel. Experimental
(circles) and calculated
(red line)
Vanadium oxide xerogel
Vanadium oxide xerogel (V2O5 nH2O)
does not form crystals, that can be
analyzed with the use of traditional
crystallographic methods. The
diffraction pattern (see Figure 12) only
shows a combination of Bragg-like
peaks and broad diffuse features. The
reduced structure function is shown in
Figure 13.
The PDF, derived from the measurement
was compared with the PDF obtained
from a structure model described by
Petkov et al. [8]. This model describes
the crystallites consisting of bilayers of
V2O5, made of square pyramidal VO5
units and separated by water molecules.
PDF analysis in Figure 14 shows a good
fit at distances in the intralayer region
(r < 11 Å) and a not so good agreement
in the interlayer region
(r > 11 Å). The same observation by
Petkov et al. [8] was attributed
to the fact that the bilayer slabs
are not perfectly stacked, but are
turbostratically disordered.
Results and
discussion ctd.
-2
2 6 10 14 184 8 12 16 20
0
2
4
6
-5
0 2 4 6 8 10 12 14 16
0
5
10
100
0 20 40 60 80 100 120 140
1000
10000
Intensity[arb.units]F(Q)[Å-1
]G(r)[Å-2
]
2theta [deg]
Scattering vector Q [Å-1
]
Interatomic distance r [Å-1
]
G(r)[Å
-2
]
Interatomic distance r [Å-1
]
8
6
4
2
0
-2
-4
0 5 10 15 20
Gtrunc
Gdiff
Gcalc
-2
2 6 10 14 184 8 12 16 20
0
2
4
6
-5
0 2 4 6 8 10 12 14 16
0
5
10
100
0 20 40 60 80 100 120 140
1000
10000
Intensity[arb.units]F(Q)[Å-1
]G(r)[Å
-2
]
2theta [deg]
Scattering vector Q [Å-1
]
Interatomic distance r [Å-1
]
G(r)[Å-2
]
Interatomic distance r [Å-1
]
8
6
4
2
0
-2
-4
0 5 10 15 20
Gtrunc
Gdiff
Gcalc
-2
2 6 10 14 184 8 12 16 20
0
2
4
6
-5
0 2 4 6 8 10 12 14 16
0
5
10
100
0 20 40 60 80 100 120 140
1000
10000
Intensity[arb.units]F(Q)[Å-1
]G(r)[Å
-2
]
2theta [deg]
Scattering vector Q [Å-1
]
Interatomic distance r [Å-1
]
G(r)[Å
-2
]
Interatomic distance r [Å-1
]
8
6
4
2
0
-2
-4
0 5 10 15 20
Gtrunc
Gdiff
Gcalc

9. Figure 16. Atomic PDF
of fumed silica
Figure 1. Fumed silica
powder prepared in a
glass capillary
Amorphous solids
Fumed silica powder was used as an
example of applying PDF analysis
to amorphous materials. Traditional
structure analysis does not give much
information; only a few ’humps‘ can
be seen in the scan in Figure 15a.
After calculating the reduced structure
function in Figure 15b more structure
can be observed.
PDF analysis of these data helps
to reveal the short range order by
determining average distances between
the nearest neighbouring atoms.
The PDF in Figure 16 shows five clear
peaks that could be determined as first
and second order Si-Si, O-O or Si-O
interatomic distances in silica as given by
Mozzi and Warren [9].
Figure 15b. Reduced
structure function
calculated from the
experimental data
from Figure 15a.
Figure 15a. XRD
measurement of
fumed silica (red
line) and empty glass
capillary (blue line)
performed with Mo
radiation
-2
1
1
2
3 4
5
2 3 4 5 6 7 8
-1
0
1
2
3
4
5
-2
0 2 4 6 8 10 12 14 16
0
2
-1
100
0 20 40 60 80 100 120 140
1000
10000
Intensity[arb.units]F(Q)[Å-1
]G(r)[Å-2
]
2theta [deg]
Scattering vector Q [Å-1
]
Interatomic distance r [Å]
-2
1
1
2
3 4
5
2 3 4 5 6 7 8
-1
0
1
2
3
4
5
-2
0 2 4 6 8 10 12 14 16
0
2
-1
100
0 20 40 60 80 100 120 140
1000
10000
Intensity[arb.units]F(Q)[Å-1
]G(r)[Å-2
]
2theta [deg]
Scattering vector Q [Å-1
]
Interatomic distance r [Å]
-2
1
1
2
3 4
5
2 3 4 5 6 7 8
-1
0
1
2
3
4
5
-2
0 2 4 6 8 10 12 14 16
0
2
-1
100
0 20 40 60 80 100 120 140
1000
10000
Intensity[arb.units]F(Q)[Å-1
]G(r)[Å-2
]
2theta [deg]
Scattering vector Q [Å-1
]
Interatomic distance r [Å]
(a)
(b)

10. Liquids
As in amorphous materials, liquids
do not have a periodic arrangement
of the atoms, and therefore no sharp
diffraction maxima are observed in the
diffractogram. Figure 18a shows the
XRD measurement of tap water as an
example. PDF analysis helps to observe
the average distance betweens atoms.
Calculation of the reduced structure
function from this diffractogram reveals
the structural information, as can be
seen in Figure 18b.
The PDF in Figure 19 shows a relatively
narrow peak for the first O-O distance
(ca. 2.8 Å). The maxima for the second
and third coordination spheres are less
sharp. These results are in accordance
with the synchrotron data reported by
Hura et al. [10].
Figure 19. Atomic PDF
of liquid water
Results and
discussion ctd.
Figure 18b. Reduced
structure function
calculated from the
experimental data
from Figure 17a.
Figure 18a. XRD
measurement of liquid
water (red line) and
empty glass capillary
(blue line) performed
with Mo radiation
-1
Intensity[arb.units]F(Q)[Å-1
]G(r)[Å
-2
]
2theta [deg]
Scattering vector Q [Å-1
]
Interatomic distance r [Å]
-3
1
1
2 3
2 3 4 5 6 7 8 9 10
-2
-1
0
1
2
3
4
-2
0 2 4 6 8 10 12 14 16
0
2
4
100
0 20 40 60 80 100 120 140
1000
10000
-1
Intensity[arb.units]F(Q)[Å-1
]G(r)[Å
-2
]
2theta [deg]
Scattering vector Q [Å-1
]
Interatomic distance r [Å]
-3
1
1
2 3
2 3 4 5 6 7 8 9 10
-2
-1
0
1
2
3
4
-2
0 2 4 6 8 10 12 14 16
0
2
4
100
0 20 40 60 80 100 120 140
1000
10000
-1
Intensity[arb.units]F(Q)[Å-1
]G(r)[Å
-2
]
2theta [deg]
Scattering vector Q [Å-1
]
Interatomic distance r [Å]
-3
1
1
2 3
2 3 4 5 6 7 8 9 10
-2
-1
0
1
2
3
4
-2
0 2 4 6 8 10 12 14 16
0
2
4
100
0 20 40 60 80 100 120 140
1000
10000
(a)
(b)
Figure 20. Water
sample prepared in a
glass capillary

11. Conclusion
Results of PDF analysis on a range of samples measured on a standard
laboratory XRD system, equipped with an X-ray tube with a silver or
molybdenum anode, were shown. Meaningful results were achieved, that
allowed for comparison with data reported in literature. Empyrean allows
performing experiments with both high-energy X-ray radiation and a wide 2θ
measurement range and this permitted to obtain data with good quality up to
scattering vectors of 17Å-1 (Mo anode) or 22Å-1 (Ag anode). Larger Q-vectors
can be obtained by using X-rays with higher energy at synchrotron facilities
although in practice values higher than 30 A-1 are rarely used.
The flexibility and the accessibility of the Empyrean system make it the ideal
tool for preparation and pre-screening for valuable synchrotron beam time.
References
1. J. te Nijenhuis, M. Gateshki, M. J. Fransen, Z. Kristallogr. Suppl. 2009, 30, 163.
2. Billlinge, S.J.L., 2007, Z. Kristallog. Suppl., 26, 17.
3. Egami, T & Billinge, S.J.L, 2003, Underneath the Bragg peaks: Structural
Analysis of Complex Materials (Amsterdam, The Netherlands: Elsevier
Science B.V.).
4. Klug, H. P. & Alexander, L. E., 1974, X-ray Diffraction Procedures for
Polycrystalline Materials (New York, NY, USA: Wiley).
5. Farrow, C. L., Juhas, P., Liu, J. W., Bryndin, D., Bozin, E. S., Bloch, J.,
Proffen, Th. & Billinge, S. J. L., 2007, J. Phys.: Condens. Matter, 19,
335219.
6. Petkov, V., 1989, J. Appl. Cryst., 22, 387.
7. Braekken, H., 1930, Z. Kristallog, 75, 572-573.
8. Petkov, V., Bozin, E., Billinge, S.J.L., Vogt, T., Trikalitis, P. & Kanatzidis, M.,
2002, J. Am. Chem. Soc., 124, 10157.
9. Mozzi, R.L. & Warren, B.E., 1969, J. Appl. Cryst., 2, 164.
10. Hura, G., Sorenson, J., Glaeser, R.M. & Head-Gordon, T., 2000, J. Chem.
Phys., 113, 9140.
Acknowledgements
The authors gratefully acknowledge
Prof. V. Petkov, Central Michigan
University, Mt. Pleasant, MI, USA
and Prof. B. Palosz, Institute of High
Pressure Physics, Polish Academy of
Sciences, Warsaw, Poland respec-
tively for providing the vanadium
oxide xerogel and silicon carbide
samples described in this paper.

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