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Journal of
Applied
Crystallography
ISSN 0021-8898
Received 30 July 2007
Accepted 18 March 2009
Lattice parameter determination using a curved
position-sensitive detector in reflection geometry
and application to Smx/2Ndx/2Ce1–xO2–d ceramics
A. Pramanick, S. Omar, J. C. Nino and J. L. Jones*
Department of Materials Science and Engineering, University of Florida, Gainesville, Florida, USA.
Correspondence e-mail: [email protected]
# 2009 International Union of Crystallography
Printed in Singapore – all rights reserved
X-ray diffractometers with curved position-sensitive (CPS) detectors have
become popular for their ability to perform fast data collection over a wide 2
range, enabling kinetics studies of chemical reactions and measurement of other
time-resolved solid-state phenomena. While the effect of sample displacement
on hkl-specific apparent lattice parameters has been explored for a transmission-mode Debye–Scherrer geometry, such effects for a reflection-mode Debye–
Scherrer geometry are not yet well understood. The reflection-mode Debye–
Scherrer geometry for CPS detectors is unique in the sense that the angle for the
incident X-ray beam is kept fixed with respect to the normal of a flat diffracting
sample, while the diffracted beams are measured at multiple angles with respect
to the sample normal. An efficient method for precise lattice parameter
determination using linear extrapolation of apparent lattice parameters
calculated from different hkl diffraction peaks is proposed for such geometries.
The accuracy involved with this method is investigated for an Si powder
standard. The extrapolation method is then applied to develop an empirical
relationship between composition (x) and the lattice parameter (ao) of
Smx/2Ndx/2Ce1xO2 ceramics for solid oxide fuel cell electrolytes. In this
system, the empirical relationship between x and ao is compared with a previous
theoretical prediction.
1. Introduction
X-ray diffraction is a widely accepted technique for the
identification and characterization of crystalline materials.
Most conventional X-ray diffractometers follow Bragg–
Brentano geometry for the collection of powder X-ray
diffraction patterns. In the Bragg–Brentano geometry, divergent X-rays from a fine-focus tube are incident on a flat
sample, and after diffraction, the convergent X-rays are
focused back on a detector with a receiving slit. In some
instruments, the sample is fixed, and the source and the
detector rotate in opposite directions along the arc of a circle
centered on the sample. In other instruments following the
same geometry, the source is kept fixed and the detector scans
along the focusing circle, while the sample is simultaneously
rotated in the same direction at an angular velocity half that of
the detector, thereby maintaining the diffraction vector
parallel to the sample normal. As an alternative to conventional Bragg–Brentano geometry, X-ray diffraction can also be
performed following a Debye–Scherrer arrangement. In a
conventional Debye–Scherrer camera, the X-ray beam is
incident on a capillary sample, and sections of diffraction rings
are recorded on a coaxial circular photographic film (Cullity &
Stock, 1978). In modern diffractometers, curved positionsensitive (CPS) detectors can be used instead of photographic
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doi:10.1107/S0021889809010085
films. The main benefits for using a Debye–Scherrer geometry
with a CPS detector include fast data collection over a wider
angular range with good counting statistics (Deniard et al.,
1991). Therefore, the effects of source instability on different
portions of the diffraction pattern can be reduced, and in situ
kinetic studies and other time-resolved phenomena can be
more easily measured. The diffraction geometry for CPS
detectors differs from that of a conventional Debye–Scherrer
camera in the following respect. In a conventional Debye–
Scherrer camera, diffraction is performed on a capillary
sample in transmission mode, and diffracted X-rays are
captured on a photographic film as concentric rings which are
symmetric with respect to the incident X-ray beam. However,
in many of the applications for laboratory X-ray diffractometers with CPS detectors, reflection geometry is followed,
with a fixed nonzero angle for the incident X-ray beam with
respect to the normal of a flat diffracting sample. In such a
case, the incident and the diffracted beam are not symmetric
with respect to the sample normal. Such a geometry may be
referred to as Debye–Scherrer geometry in reflection mode.
The effect of this asymmetry in the diffraction geometry on
the peak profile has been explored before (Evain et al., 1993;
Masson et al., 1996). The possible error in peak position due to
sample displacement and specific methods for correction have
also been discussed (Masson et al., 1996). It has been shown
J. Appl. Cryst. (2009). 42, 490–495
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that, with proper sample positioning, X-ray diffraction
patterns from a Debye–Scherrer diffraction setting in reflection mode can be suitable for use in Rietveld refinement of
crystal structure (Shishiguchi et al., 1986). However, exact
positioning of the sample surface at a predetermined position
is usually not guaranteed for every diffraction pattern
obtained. This can be attributed to the variations in sample
size, the exact mechanical positioning of specimen holders or
the instability of the radiation source. Such errors in sample
displacement may lead to errors in peak positions and
consequently affect the calculated lattice parameter of a
crystalline phase. Rietveld refinement can be used to
compensate for these effects, though this involves refinement
of the entire diffraction pattern by modifying a large number
of parameters (Young, 1995).
To calculate the lattice parameter without the use of the
Rietveld method, and to compensate for the effect of sample
positioning on the final calculated lattice parameter, an
extrapolation method can be used in either Bragg–Brentano
geometry or conventional Debye–Scherrer cameras. In this
method, a geometrical relationship between the sample
displacement and the various hkl peak positions is established.
This geometrical relationship is used to calculate the actual
value of the lattice parameter from a linear extrapolation of
apparent lattice parameters calculated from different hkl
diffraction peaks (Cullity & Stock, 1978). This method
provides an easy and efficient way for calculating lattice
parameters with a known accuracy. In order to apply the
extrapolation method for a Debye–Scherrer geometry in
reflection mode, as is applicable for diffractometers with CPS
detectors, modification of the already existing relations for
lattice parameter extrapolation is necessary. This paper aims
to provide a suitable relation for lattice parameter extrapolation for Debye–Scherrer geometry in reflection mode.
The method is applied to determine a relationship between
the lattice parameter and the material composition for
Smx/2Ndx/2Ce1xO2 ceramics with application in solid oxide
fuel cell (SOFC) electrolytes.
sample and an angle 2 with the incident X-ray beam. For this
geometry, we have the following relations:
’=’ ¼ S=S
ð1Þ
’=’ ¼ 2=ð2 !Þ;
ð2Þ
and
where S is the deviation in the measured position of the
peak along the circumference of the detector and 2 is the
error in the measured Bragg angle 2. For a vertical
displacement of the sample surface by y, S is given by
S ¼ yðsin 2Þ=sin !;
ð3Þ
as is illustrated in Fig. 1(b). Since, for small-angle geometry,
’ = S/R and ’ = , the following relation can be arrived
at using equations (1)–(3):
2 ¼ yðsin 2Þ=ðR sin !Þ:
ð4Þ
By differentiating Bragg’s law with respect to , the error in
the calculated value for a specific hkl interplanar spacing d
can be related to the error in the measured value for through
d=d ¼ ðcos =sin Þ:
ð5Þ
From equations (4) and (5), we obtain
2. Effect of sample displacement on peak positions and
determination of the lattice parameter
2.1. Theoretical derivation
In this section, a geometrical relationship between the
sample position and the final estimated lattice parameter is
developed for a reflection-mode Debye–Scherrer geometry
with a parallel incident X-ray beam.
The geometry for the diffraction setup is illustrated in
Fig. 1(a). The detector is in the form of a circular arc of radius
R with its center coincident with the center of the diffraction
volume. The angle of the incident X-ray beam is ! with respect
to the sample surface. Let us consider that the diffracted
beam, corresponding to a particular hkl diffraction peak, is
recorded at a distance S along the circumference of the
detector, and it makes an angle ’ with the surface of the
J. Appl. Cryst. (2009). 42, 490–495
A. Pramanick et al.
Figure 1
(a) Debye–Scherrer diffraction geometry in reflection mode with parallel
incident X-ray beam. (b) Estimation of S for a vertical sample
displacement of y.
Lattice parameters using a CPS detector in reflection geometry
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Table 1
Standard Si powder (Standard Reference Material 640b)
was obtained from NIST. An Inel CPS 120 detector was used
to obtain the diffraction pattern of Si using Cu K radiation.
The incident X-ray beam was 100 mm 5 mm in dimensions,
with the larger dimension parallel to the detector width. The
incident angle ! was kept constant at 6 . The sample holder
was rotated about its vertical axis during the acquisition of a
diffraction pattern. The top reflecting surface of the powder
specimen was systematically displaced vertically downward by
known distances for the collection of additional diffraction
patterns. The vertical displacement of the sample was
accomplished by inserting spacers of known thicknesses
between the sample holder and the stage on the diffractometer. The phrase ‘no displacement’ refers to the fact that no
spacer was introduced in this case. It can be observed from
Fig. 2 that a vertical downward displacement of the top surface
of the powder specimen causes a shift of the diffraction peaks
towards lower 2 values.
For each diffraction pattern, positions of the various hkl
diffraction peaks were determined using a commercial software package (Igor Pro 5.05A; http://www.wavemetrics.com/).
For all peaks except the 111 diffraction peaks, two symmetric
Gaussian profiles were fitted to each peak to account for
Cu K1 and Cu K2 components of the incident X-ray beam.
The peak positions of the Cu K1 component ( =
1.540598 Å) were used for all subsequent calculations. A
single Gaussian peak profile was used for fitting the 111
diffraction peaks, since in this case the Cu K1 and Cu K2
components were highly overlapped. Bragg’s law was applied
to each hkl diffraction peak for calculating hkl interplanar
spacings and subsequently lattice parameter a. The calculated
lattice parameters from the different hkl peaks are shown with
respect to their corresponding cos2 values in Fig. 3. A linear
fit is obtained following equation (7). The value of ao obtained
at cos2 = 0 provides the correct value of the lattice parameter
of Si, after accounting for sample displacement effects. The
effect of sample displacement on the calculated value of the
lattice parameter is listed in Table 1, and comparison with the
standard value provided by NIST for Si lattice parameter is
shown in Fig. 4. The errors reported in Table 1 and in Fig. 4 are
derived from the linear fit in Fig. 3. It can be seen that the
Figure 2
Figure 3
The 111 diffraction peak of Si for different vertical displacements of the
sample. The plus signs show the actual data and the solid lines show a
Gaussian fit to the peak profile. A downward displacement of the sample
causes the diffraction spectrum to shift to lower 2 values.
A plot of the apparent lattice parameter a with respect to cos2,
calculated from different hkl diffraction peaks, for various sample
displacements. The straight lines representing equation (12) converge
near the actual lattice parameter of Si.
Calculated values of lattice parameter of Si for various Bragg plane
displacements.
Vertical sample displacement (mm)
Calculated value of ao (nm)
0
150
300
450
0.54296 (22)
0.54305 (26)
0.54290 (27)
0.54290 (27)
d=d ¼ K0 cos2 ;
ð6Þ
where K 0 ¼ y=ðR sin !Þ is a constant. Rearranging equation
(6) and expressing interplanar spacing in terms of the lattice
parameter, we can write
a ¼ ao þ ao K 0 cos2 ;
ð7Þ
where ao is the true estimation of the lattice parameter and a is
the apparent lattice parameter calculated from the angular
position of a particular hkl diffraction peak. The implication of
equation (7) is that the actual lattice parameter ao can be
obtained by plotting a with respect to cos2 and extrapolating
it to cos2 = 0. Equation (7) is equivalent to the relation
obtained using Debye–Scherrer geometry in transmission
mode (Cullity & Stock, 1978). However, for a Bragg–Brentano
geometry the corresponding lattice parameter extrapolation
relation is given by a ¼ ao þ ao K 0 ðcos2 = sin Þ, which has an
additional factor of sin (Cullity & Stock, 1978).
2.2. Experimental verification
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Lattice parameters using a CPS detector in reflection geometry
J. Appl. Cryst. (2009). 42, 490–495
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extrapolation method correctly determines the lattice parameter of Si to within an error of less than one picometre for
displacements up to 450 mm. For larger sample displacements,
the small-angle approximation, as is adopted in equation (4),
may not be applicable. Furthermore, peak distortions will
increase with large asymmetric displacements of the sample
surface with respect to the detector arc. These factors can limit
the application of this method for larger sample displacements.
The value for the constant K0 in equations (6) and (7) is
related to the vertical sample displacements according to
K0 ¼ y=ðR sin !Þ. For example, for a vertical displacement of
150 mm, the best linear fit for equation (7) yields K 0 equal to
0.0050 (7). For comparison, the calculated value for y =
150 mm is K 0 = 0.0052. The values for K0 calculated using the
two different methods are consistent.
3. Application
The knowledge of precise lattice parameters and their
evolution with temperature, chemistry and other environmental conditions can provide insight into the origin of solid
state phenomena, material properties and behavior. For
example, in many solid solutions, the lattice parameter and
dopant concentration are strongly correlated (Omar et al.,
2007). In the present section, the variation in the lattice
parameters of Smx/2Ndx/2Ce1xO2 solid solutions as a function of composition (x) is investigated using the technique
developed in x2.
In the past few decades, ceria-based materials have received
significant attention as potential candidates for electrolyte
application in SOFCs (Omar et al., 2006). Oxygen ion
conductivity in these materials takes place via oxygen vacancies. These oxygen vacancies are incorporated into the cubic
fluorite lattice of pure ceria by replacement of Ce4+ by an
acceptor dopant cation. This can be represented by the
following defect equation using Kröger–Vink notation:
Figure 4
Comparison of the estimated lattice parameter of Si with the NIST
standard for various sample displacements.
J. Appl. Cryst. (2009). 42, 490–495
A. Pramanick et al.
D2 O3 ! 2D0Ce þ 3O
O þ VO :
2CeO2
ð8Þ
Depending on the ionic radius of the dopant and the host
cation, the cubic fluorite lattice can either expand or contract.
Moreover, the lattice parameter of the material is also
dependent on the concentration of the dopant cations. Typically, in the low dopant concentration regime, solid solutions
obey Vegard’s law, i.e. the lattice parameter is linearly
dependent on the dopant concentration (Vegard & Dale,
1928). As the dopant concentration increases, negatively
charged dopant cations tend to interact with positively
charged oxygen vacancies. The attractive interactions between
these two defects lead to the formation of local defect structures, which as a result contract the unit cell and lower the
mobile oxygen vacancy concentration (Stephens & Kilner,
2006; Parks & Bevan, 1973). Since ionic conductivity is
dependent on mobile oxygen vacancies, in this regime it
significantly decreases with dopant content (Omar et al., 2008).
In terms of crystal structure, these oxide materials start to
differ from Vegard’s law at higher dopant concentration
(Parks & Bevan, 1973). This difference was interpreted as the
amount of local defect ordering present in the system. From
the lattice expansion and compositional studies, the extent of
defect interactions and their effect on the ionic conductivity in
these materials can be correlated. For the present study, Sm3+
and Nd3+ were selected as dopant cations. As the ionic radii of
3+
(r3+
both Sm3+ (r3+
Sm,VIII = 0.1079 nm) and Nd
Nd,VIII =
4+
4+
0.1109 nm) are higher than that of Ce (rCe,VIII = 0.097 nm),
expansion in the cubic fluorite lattice is expected (Shannon,
1976). Powders of Smx/2Ndx/2Ce1xO2 (with x = 0.01, 0.03,
0.05, 0.08, 0.10, 0.12, 0.15, 0.18 and 0.20) were processed using
a conventional solid oxide reaction method, as reported
elsewhere (Omar et al., 2008). To confirm complete dissolution
of the dopants in the ceria lattice, X-ray diffraction (XRD)
patterns were measured for each composition using the same
instrumental conditions as described in x2.
Fig. 5 shows the XRD patterns measured at room
temperature for all the compositions of Smx/2Ndx/2Ce1xO2.
Tungsten was used as an internal standard in the powder
sample. It can be observed that, with an increase in the total
dopant concentration (x), the 200 peak of the material shifts to
lower 2 angles, while the 110 peak positions of the tungsten
standard remain stable in all the compositions. This clearly
indicates that lattice expansion occurs in doped ceria with
increasing dopant concentration.
Two symmetric Pearson VII functions were fitted to each
peak to account for the Cu K1 and Cu K2 components of
the incident X-ray beam. Precise lattice parameters (ao) of all
the compositions of Smx/2Ndx/2Ce1xO2 were calculated
from the hkl peak positions corresponding to Cu K1 radiation. The lattice parameter of pure CeO2 is determined to be
0.54113 (23) nm at room temperature, which is consistent with
the previously reported value of 0.541134 (12) nm (Joint
Committee on Powder Diffraction Standards card 34-0394).
Fig. 6 shows the lattice parameter of Smx/2Ndx/2Ce1xO2
solid solutions as a function of total dopant concentration (x).
It can be seen that the lattice parameter increases linearly with
Lattice parameters using a CPS detector in reflection geometry
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increasing dopant concentration following Vegard’s law. Using
a least-squares fitting algorithm, a linear relationship was
obtained between ao and x. This can be represented as
ao ðxÞ ¼ 0:5412 þ 0:0154x:
ð9Þ
It is essential to note that the crystal lattice of Smx/2Ndx/2Ce1xO2 continues to expand linearly even at higher dopant
concentration (> 10 mol%). This points to the fact that the
local oxygen vacancy ordering in the Smx/2Ndx/2Ce1xO2
system is not as pronounced as that in other ceria-based
systems (Omar et al., 2007).
In addition to the dopant concentration, the following
relationship has been previously reported for the lattice
parameter of doped CeO2 as a function of ionic radius and
Coulombic charge of the dopant cation (Kim, 1989):
ao ðxÞ ¼ 0:5413 þ ½2:20ðrd rh Þ þ 0:015ðzd zh Þx;
ð10Þ
where rd and rh are the ionic radii of the dopant cation and the
host cation in nanometres, respectively, and zd and zh are the
valence charges of the dopant cation and the host cation,
respectively. The above-mentioned relationship was obtained
by multiple regression analysis, for the dilute regime of dopant
concentration. In these regimes, defect interactions are
assumed to be negligible. The following relationship is
obtained when equation (10) is extended to co-doped ceria:
ao ðxÞ ¼ 0:5413 þ ½2:20ðrd1 þ rd2 2rh Þ
þ 0:015ðzd1 þ zd2 2zh Þx;
ð11Þ
where rd1, rd2 and rh are the ionic radii of the dopant cation 1,
the dopant cation 2 and the host cation in nanometres,
respectively, and zd1, zd2 and zh are the valence charges of the
dopant cation 1, the dopant cation 2 and the host cation,
respectively. Here, it is assumed that the concentration of each
dopant cation in the host lattice is the same. If the ionic radii
and Coulombic charges of Sm3+ and Nd3+ are input in the
empirical relationship [equation (10)] then the equation
transforms to
ao ðxÞ ¼ 0:5413 þ 0:0123x:
ð12Þ
It can be seen that the slope in the experimentally obtained
lattice parameter–dopant concentration relationship differs
from that of the theoretical relationship. This may be associated with the formation of local defect structures in the
system which are not taken into account in the theoretical
relationship. The presence of two different dopant cations in
ceria may also lead to deviation of the experimental relationship from the theoretical model.
4. Conclusion
Figure 5
XRD profiles of different composition Smx/2Ndx/2Ce1xO2 . Tungsten is
used as an internal standard.
The procedure for precise lattice parameter determination
from a linear extrapolation of apparent lattice parameters,
calculated from different hkl diffraction peaks, is provided for
a reflection-mode Debye–Scherrer diffraction geometry. It is
shown that this procedure can be used to calculate the lattice
parameter of a material within picometre resolution. As an
example for practical material applications, the extrapolation
method is used to obtain the composition dependence of the
lattice parameter of Smx/2Ndx/2Ce1xO2. These compounds
are shown to follow Vegard’s law for increasing Sm3+ and Nd3+
content. The experimentally derived Vegard law for
Smx/2Ndx/2Ce1xO2 compounds is compared with previous
theoretical calculations. It was observed that for Smx/2Ndx/2Ce1xO2 the experimentally determined lattice parameter–
dopant concentration relationship deviates from the theoretical model based on multiple regression analysis.
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Figure 6
Lattice parameter of Smx/2Ndx/2Ce1xO2 as a function of dopant
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