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research papers 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 490 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 research papers 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 491 research papers 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 492 A. Pramanick et al. Lattice parameters using a CPS detector in reflection geometry J. Appl. Cryst. (2009). 42, 490–495 research papers 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 493 research papers 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. References Figure 6 Lattice parameter of Smx/2Ndx/2Ce1xO2 as a function of dopant concentration (Omar et al., 2008). 494 A. Pramanick et al. Cullity, B. D. & Stock, S. R. (1978). Elements of X-ray Diffraction, 3rd ed. New Jersey: Prentice Hall. Deniard, P., Evain, M., Barbet, J. M. & Brec, R. (1991). Mater. Sci. Forum, 79–82, 363–370. Lattice parameters using a CPS detector in reflection geometry J. Appl. Cryst. (2009). 42, 490–495 research papers Evain, M., Deniard, P., Jouanneaux, A. & Brec, R. (1993). J. Appl. Cryst. 26, 563–569. Kim, D. J. (1989). J. Am. Ceram. Soc. 72, 1415–1421. 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