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Quantitative multiphase determination using the Rietveld method with high accuracy
February 1996
ELSEVIER
Materials Letters 26 (1996) 171-175
Quantitative multiphase determination using the Rietveld method with high accuracy Hongchao Liu
*, Changlin
Kuo
Shanghai Insrituie of Ceramics, Chinese Academy of Sciences, Shanghui 200050, China
Received 4 July 1995; revised 28 August 1995; accepted 5 September 1995
Abstract Microabsorption correction was incorporated into the Rietveld program DBWS9411 for multiphase analysis. The results of quantitative phase analysis are much improved after this correction. Several factors that can influence the quantitative results were also discussed. Keywords: Rietveld method; Microabsorption; Quantitative phase analysis
1. Introduction
For more than 20 years, the Rietveld method [ 1,2] has been used as a powerful tool for crystal structure refinements based on powder diffraction data, especially in materials science. In the Rietveld method, the atom parameters in the unit cell are calculated fitting the entire powder diffraction diagram by the least-squares method. In quantitative phase analysis using the Rietveld method, relative weight fractions of crystalline phases in a multiphase sample can be calculated directly from scale factors of the respective calculated intensities as described by Hill and Howard [3]. Absolute quantities and total abundance of amorphous and minor phases can also be determined simultaneously by adding internal standards in known proportions. Using the Rietveld method, time consuming calibra-
* Corresponding author.
tion measurements can be avoided, and the phase abundance can be determined if all phases are identified and if their crystal structure parameters and chemical compositions are known. The Rietveld method also offers other advantages over the traditional integrated-intensity methods of quantitative phase analysis [4]. However, there are so many factors that can influence the accuracy of the results. Various corrections were introduced in the Rietveld method, such as texture, absorption contrast [5], sample transparency and displacement corrections. Less attention was paid to the microabsorption effect [6] in Rietveld analysis, especially in quantitative analysis. In the current program, only dose DBWS9411 takes into consideration microabsorption corrections. But it takes microabsorption parameters as global ones and regards that all phases in the diffraction pattern have the same microabsorption effect. This consideration is not consistent with real samples in which different phases have different microabsorption effect.
00167-577X/96/$12.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 00167-577X(95)00221-9
172
H. Liu, C. Kuo/Mutrriul.s
Microabsorption can cause angle-dependent intensity reductions. This effect can seriously influence the low angle intensities of the powder diffraction patterns measured in Bragg-Brentano geometry [7,8]. When the microabsorption effect is neglected, Rietveld refinement can yield unphysical, incorrect low thermal parameters, which may tend to negative values [9,10]; and the quantitative analysis will give unreliable results because the whole-pattern quantitative analysis is usually with intensity data of low-angle peak. It is needed to do microabsorption correction not only in structure analysis but also in quantitative determination. In this work, the DBWS9411 1111 program was modified so that the microabsorption correction can be applied to each phase in multiphase analysis. It is shown that the quantitative results are unreliable if the microabsorption in some samples is neglected. When the correction is included, these results are much improved.
2. Method of correction The diffraction data can be corrected for microabsorption effect before doing Rietveld analysis in two ways. One is by use of the measured angular dependence of the fluorescence intensity reduction as proposed by Sparks et al. [9,12]. Another one is by use of the theoretical model using parameters from physical measurements [ 131. It is practical to incorporate the microabsorption corrections into the Rietveld analysis procedure by making the microabsorption parameters refinable. The scattered intensity of one line of a phase j in a polycrystalline sample is given by Yci=SjCL,(F,12+(2@;-2@,)P,Aj+
Ybi,
where Sj is the scale factor of phase j, k represents the Miller indices, hkl, for a Bragg reflection, L, contains the Lorentz, polarization and multiplicity factors, @ is the reflection profile function, 20 is the Bragg angle, Pk is the preferred orientation function (March-Dollase model is often adopted), Fk is the structure factor for kth Bragg reflection, A, is the microabsorption correction function of phase j with certain size, and Ybi is the background intensity at the ith step. The summation is over all phases.
Letters 26 11996) 171-175
There are four models for correction of microabsorption. When they are incorporated in the DBWS9411 program, they were recast and normalized to 1.OOat 0 = 90”: (1) Combination of the Sparks and Sourtti model: Aj=r{l.O-p[exp(-q)]
+P[exp(-q/sin@)]]
+(I-,)[I-I(@-n/2)].
(1)
(2) The Sparks et al. model [ 121: A,=
l.O-t(@-n/2).
(2)
(3) The Sourtti et al. model [S]: Aj = 1.O - p[ exp( - q)] + p[ exp( -q/sin
o)] . (3)
(4) The Pitschke et al. model [14,15]: Aj=
l.O-
[pq(l.O-q)]
+(pq/sin@)(l.O-q/sin@).
(4)
Here p, q, r and t are refinable parameters. With the results obtained from the above procedures, the “so-called” ab initio quantitative phase analysis can be accomplished. Eq. (5) is used to calculated the relative abundance of phase j: Wj=Sj(ZMV)j/CSj(ZMV)j,
(5)
where Z and M are the number and mass of formula units, respectively; V is the volume of the unit cell. The summation is over all crystalline and identified components. If an internal standard phase s has been added to the mixture prior to data collection, then the amount of amorphous and minor phases may be estimated using the following equation: Wamorph
and minor phases
=
1 - cw,,
(6)
where Wj is determined by Wj= W,Sj(ZMV)j/S,(ZMV),.
(7)
The subscript s and j refer to standard phase and each crystalline phase in the mixture.
3. Experiment Al,03, CaCO,, TiO,, SiO, and Si were the selected phases. The X-ray powder diffraction pattern shows that there is no other than the needed phase appearing in above selected pure sample.
H. Liu, C. Kuo/Materials Table I Data collection
conditions
for diffraction
profiles
Diffractometer: Rig&u Dmax/IIIB
Generator:
40 kV, 25 mA
Wavelength: Cu Ka step: 0.04” (20) Count time: 2s
Diffracted beam monochromator: graphite Collimators (DS, RS, SS): l”, 0.6 mm, 1” Angular range: 20-90” (20)
Letters 26 (1996) 171-175
173
The X-ray diffraction data were collected on a Bragg-Brentano diffractometer under the conditions shown in Table 1.
4. Data treatment Al,O,, CaCO,, TiO, and Si were finely ground by ball grinding mill for two days while SiO, were manually ground from single crystals. SiO, was ground for one hour for the first two samples and half an hour for the other two samples. Then these components were weighed and mixed according to Table 2 (as labelled by 1). The mixtures for diffraction experiments were all prepared by shaking to homogenise and grinding for 10 min. This cycle was repeated several times. The samples were pressed into quartz holders for powder diffraction experiment.
The modified DBWS9411 program was used to treat the data. Pseudo-Vogit profile function was adopted to fit the experiment data. The range of calculated profile was 8 H,. Background was refined with a 3rd order polynomial in 20. Peaks below 35” were corrected for asymmetry. Run terminated when all calculated shifts were less than 10% standard deviation. There are 63 parameters refined altogether. The March-Dollase model was applied in preferred orientation correction and the Sourtti et al. model was chosen to do microabsorption correction.
380360340320300280260240220200180160140120lee80 60
;
* iL&hib
J_--
Ii_-___- ___ _ __ ____ __
Fig. 1. The diffraction pattern of sample No. 1. The dotted line is the experimental pattern and the straight line is the calculated differences are shown below them. The vertical bar indicates the reflection position.
pattern. The
174
H. Liu, C. Kuo/Mutrrials
The initial structure parameters of Al,O,, CaCO,, TiO,, SiO, and Si were taken from the Structure Report.
Letters 26 (19961 171-175 Table 3 Preferred Parameters No. 1 No.2
5. Results and discussion
No. 3
Table 2 Quantitative Sample No. 1
No. 2 No.3
No.4
1 2 3 1 3 1 3 4 1 3 4
phase analysis
results
At A
CaCO,
TiO,
SiO,
Si
29.78 31.50 30.89 34.81 36.77 45.30 48.98 47.70 23.63 29.3 1 25.02
15.42 17.87 14.88 20.54 19.45 20.40 19.36 19.23 32.20 32.92 31.89
30.14 29.13 30.33 20.16 20.54 24.66 26.30 24.33 4.89 4.85 4.99
19.66 17.30 19.33 14.85 13.44 5.21 1.78 5.15 29.70 24.27 29.11
4.99 4.19 4.57 9.62 9.80 4.43 3.58 3.60 9.57 8.66 8.98
Note: 1 - weighed phase composition; 2 - measured composition before preferred orientation correction; 3 - measured composition after preferred orientation correction; 4 - measured composition after all corrections.
Gl Gl Cl p Y Gl
The diffraction pattern, as well as the calculated
pattern, differences between these two patterns and all peak positions in the angular range of sample No. 1 are shown in Fig. 1. All the patterns show preferred orientation. The preferred orientation has a great influence on the quantitative results, as observed in this study. The phase abundance of sample No. 1 without preferred orientation correction is given in Table 2. Such a result cannot be accepted for further study. After structure refinement and preferred orientation correction, the quantitative phases analysis results became much better. R,, (weighed pattern R-factor) for sample No. 1 decreased from 20.06 to 15.26, goodness of fit from 2.30 to 1.75, D,, (Durbin-Watson statistic) increased from 0.72 to 1.04. The results also showed that the Rietveld analysis had great advantage over traditional single line or multiline X-ray diffraction quantitative analysis methods. The preferred orientation parameters are shown in Table 3. Preferred orientation direction of Al,O, and CaCO, is 104, TiO, 110, SiO, 011. The phase
orientation
No. 4
p Y
and microabsorption
parameters
A’2o3
CaCO,
TiO,
SiO,
0.9844 0.9802 0.9598
0.9088 0.8983 0.8770
0.9794 0.9693 1.OOoo
0.9874
0.9352
I .GQoo
l.OQOO 1.0000 0.4239 0.7892 0.1037 0.8952 0.7856 0.0747
which presents most serious preferred orientation in samples No. 3 and No. 4 is SiO,. This situation may be caused by its big particle size. The preferred orientation of other phases, which had the same particle size distribution in different samples, was not much varied. From the differences of the calculated and experimental patterns, the values of the R-factors, it can be seen that samples No. 1 and No. 2 show no apparent angular-dependent intensity reduction, so no further correction was applied. While the diffraction patterns of samples No. 3 and No. 4 show microabsorption of phase SiO,, the weight fraction of SiO, also greatly differs from the weighed one. Therefore, Eq. (3) was applied to correct microabsorption effect. The weight fractions after correction are shown in Table 2. It can be seen that much improvement is gained in the quantitative results. It can also be seen that microabsorption must be incorporated in the Rietveld quantitative analysis for such samples. Final values of p and q are listed in Table 3. It must be pointed out that the two microabsorption parameters of one phase are highly correlated with each other and also with other parameters such as temperature parameters (not refined in our study) and scale factors. Therefore, in the first several circles, all parameters but microabsorption parameters are refined. When there is a need to do microabsorption correction, parameters (with the exception of scale parameters) having been refined in former procedures are fixed. After the corrections mentioned above, the final weight fractions still have about 10% relative error compared to the weighed ones. There are still some factors which may be responsible for the discrepancy
H. Liu, C. Kuo/Materials
of the quantitative results by obtained the Rietveld method: (1) Sample preparation. The importance of sample preparation cannot be over-emphasized, although it can be remedied in the data treatment procedure by doing various corrections in Rietveld quantitative analysis. But in the case of an inhomogeneous sample, it is impossible to get correct results. (2) Structure model. The Rietveld method is structure-dependent. Large R values are a signal of incorrect structure or the presence of unknown phases. In our study, this problem occurs when we use the structure model of SiO, reported by Page et al. [ 161. (3) Data collection tragedy. According to Wang [17], both step size and scanning speed can cause errors in determination of X-ray diffraction peak positions, maximum intensity and full width at halfmaximum. The maximum relative error caused by step size is determined by the ratio of step size to FWHM, which was proposed as the shape-perfect coefficient of the XRD peak. A step size of 0.04” in our study can lead to 5% error in intensity. Intensity error caused by counting statistics will rise with increasing rate and decreasing measure time.
6. Conclusion The incorporation of microabsorption correction in the Rietveld program for multiphase refinement can greatly improve the results of quantitative analysis. After various corrections, the Rietveld method can give much higher accuracy quantitative results. Careful sample preparation, correct structure models and suitable data collection procedure can improve the results of Rietveld quantitative analysis.
Letters 26 (1996) 171-175
175
Acknowledgements
The authors acknowledge Professor R.A. Young, Georgia Institute of Technology, USA, for providing us with the Rietveld Program DBWS9411 and Dr. W. Pitschke, Institute of Solid State and Materials Research, Dresden, Germany, for providing us with the modified DBWS9006-PC. Professor Lu Guanglie, Hangzhou University, P.R. China, is also acknowledged for his help in collecting X-ray diffraction data.
References Rietveld, Acta Cryst. 22 (1967) 15 I. Rietveld, J. Appl. Cryst. 3 (1969) 65. [31 R.J. Hill and C.J. Howard, J. Appl. Cryst. 20 (1987) 467. [41 R.J. Hill, Powder Diffraction 6 (1991) 74. I51 J.C. Taylor and C.E. Matulus, J. Appl. Cryst. 24 (1991) 14. [61 G.W. Brindly, Phil. Mag. 36 (1945) 347. 171P.M. de Wolff, Acta Cryst. 9 (1956) 682. RI R. Suortti, J. Appl. Cryst. 5 (1972) 325. Dl R. Kumar, C.J. Sparks, T. Shraishi, E.D. Specht, P. Zschack, G.E. Ice and K. Hisatsune, Mater. Res. Sot. Symp. Proc. 213 (1991) 369. [lOI N. Masciocchi, N. Toraya and W. Parrish, Mater. Sci. Forum 79-82 (1991) 245. [I II A. Sakhivel, R.A. Young, T.S. Moss and CO. Paiva-Santos, Program DBWS-9411 for Rietveld analysis of X-ray and neutron powder diffraction patterns (1995). 1121C.J. Sparks, R. Kumar, E.D. Specht, P. Zschack and G.E. Ice, Advan. X-Ray Anal. 35 (1991) 37. 1131 H. Hennann and M. Ermrich, Acta Cryst. A 43 (1987) 401. [I41 W. Pitschke, H. Hermann and N. Mattem, Powder Diffraction 8 (1993) 74. II51 W. Pitschke, N. Mattem and H. Hermann, Powder Diffraction 8 (1993) 223. 1161Y.L. Page and G. Donay, Acta Cryst. B 32 (1976) 2456. [I71 H. Wang, J. Appl. Cryst. 27 (1994) 716.
111H.M. 121H.M.
ELSEVIER
Materials Letters 26 (1996) 171-175
Quantitative multiphase determination using the Rietveld method with high accuracy Hongchao Liu
*, Changlin
Kuo
Shanghai Insrituie of Ceramics, Chinese Academy of Sciences, Shanghui 200050, China
Received 4 July 1995; revised 28 August 1995; accepted 5 September 1995
Abstract Microabsorption correction was incorporated into the Rietveld program DBWS9411 for multiphase analysis. The results of quantitative phase analysis are much improved after this correction. Several factors that can influence the quantitative results were also discussed. Keywords: Rietveld method; Microabsorption; Quantitative phase analysis
1. Introduction
For more than 20 years, the Rietveld method [ 1,2] has been used as a powerful tool for crystal structure refinements based on powder diffraction data, especially in materials science. In the Rietveld method, the atom parameters in the unit cell are calculated fitting the entire powder diffraction diagram by the least-squares method. In quantitative phase analysis using the Rietveld method, relative weight fractions of crystalline phases in a multiphase sample can be calculated directly from scale factors of the respective calculated intensities as described by Hill and Howard [3]. Absolute quantities and total abundance of amorphous and minor phases can also be determined simultaneously by adding internal standards in known proportions. Using the Rietveld method, time consuming calibra-
* Corresponding author.
tion measurements can be avoided, and the phase abundance can be determined if all phases are identified and if their crystal structure parameters and chemical compositions are known. The Rietveld method also offers other advantages over the traditional integrated-intensity methods of quantitative phase analysis [4]. However, there are so many factors that can influence the accuracy of the results. Various corrections were introduced in the Rietveld method, such as texture, absorption contrast [5], sample transparency and displacement corrections. Less attention was paid to the microabsorption effect [6] in Rietveld analysis, especially in quantitative analysis. In the current program, only dose DBWS9411 takes into consideration microabsorption corrections. But it takes microabsorption parameters as global ones and regards that all phases in the diffraction pattern have the same microabsorption effect. This consideration is not consistent with real samples in which different phases have different microabsorption effect.
00167-577X/96/$12.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 00167-577X(95)00221-9
172
H. Liu, C. Kuo/Mutrriul.s
Microabsorption can cause angle-dependent intensity reductions. This effect can seriously influence the low angle intensities of the powder diffraction patterns measured in Bragg-Brentano geometry [7,8]. When the microabsorption effect is neglected, Rietveld refinement can yield unphysical, incorrect low thermal parameters, which may tend to negative values [9,10]; and the quantitative analysis will give unreliable results because the whole-pattern quantitative analysis is usually with intensity data of low-angle peak. It is needed to do microabsorption correction not only in structure analysis but also in quantitative determination. In this work, the DBWS9411 1111 program was modified so that the microabsorption correction can be applied to each phase in multiphase analysis. It is shown that the quantitative results are unreliable if the microabsorption in some samples is neglected. When the correction is included, these results are much improved.
2. Method of correction The diffraction data can be corrected for microabsorption effect before doing Rietveld analysis in two ways. One is by use of the measured angular dependence of the fluorescence intensity reduction as proposed by Sparks et al. [9,12]. Another one is by use of the theoretical model using parameters from physical measurements [ 131. It is practical to incorporate the microabsorption corrections into the Rietveld analysis procedure by making the microabsorption parameters refinable. The scattered intensity of one line of a phase j in a polycrystalline sample is given by Yci=SjCL,(F,12+(2@;-2@,)P,Aj+
Ybi,
where Sj is the scale factor of phase j, k represents the Miller indices, hkl, for a Bragg reflection, L, contains the Lorentz, polarization and multiplicity factors, @ is the reflection profile function, 20 is the Bragg angle, Pk is the preferred orientation function (March-Dollase model is often adopted), Fk is the structure factor for kth Bragg reflection, A, is the microabsorption correction function of phase j with certain size, and Ybi is the background intensity at the ith step. The summation is over all phases.
Letters 26 11996) 171-175
There are four models for correction of microabsorption. When they are incorporated in the DBWS9411 program, they were recast and normalized to 1.OOat 0 = 90”: (1) Combination of the Sparks and Sourtti model: Aj=r{l.O-p[exp(-q)]
+P[exp(-q/sin@)]]
+(I-,)[I-I(@-n/2)].
(1)
(2) The Sparks et al. model [ 121: A,=
l.O-t(@-n/2).
(2)
(3) The Sourtti et al. model [S]: Aj = 1.O - p[ exp( - q)] + p[ exp( -q/sin
o)] . (3)
(4) The Pitschke et al. model [14,15]: Aj=
l.O-
[pq(l.O-q)]
+(pq/sin@)(l.O-q/sin@).
(4)
Here p, q, r and t are refinable parameters. With the results obtained from the above procedures, the “so-called” ab initio quantitative phase analysis can be accomplished. Eq. (5) is used to calculated the relative abundance of phase j: Wj=Sj(ZMV)j/CSj(ZMV)j,
(5)
where Z and M are the number and mass of formula units, respectively; V is the volume of the unit cell. The summation is over all crystalline and identified components. If an internal standard phase s has been added to the mixture prior to data collection, then the amount of amorphous and minor phases may be estimated using the following equation: Wamorph
and minor phases
=
1 - cw,,
(6)
where Wj is determined by Wj= W,Sj(ZMV)j/S,(ZMV),.
(7)
The subscript s and j refer to standard phase and each crystalline phase in the mixture.
3. Experiment Al,03, CaCO,, TiO,, SiO, and Si were the selected phases. The X-ray powder diffraction pattern shows that there is no other than the needed phase appearing in above selected pure sample.
H. Liu, C. Kuo/Materials Table I Data collection
conditions
for diffraction
profiles
Diffractometer: Rig&u Dmax/IIIB
Generator:
40 kV, 25 mA
Wavelength: Cu Ka step: 0.04” (20) Count time: 2s
Diffracted beam monochromator: graphite Collimators (DS, RS, SS): l”, 0.6 mm, 1” Angular range: 20-90” (20)
Letters 26 (1996) 171-175
173
The X-ray diffraction data were collected on a Bragg-Brentano diffractometer under the conditions shown in Table 1.
4. Data treatment Al,O,, CaCO,, TiO, and Si were finely ground by ball grinding mill for two days while SiO, were manually ground from single crystals. SiO, was ground for one hour for the first two samples and half an hour for the other two samples. Then these components were weighed and mixed according to Table 2 (as labelled by 1). The mixtures for diffraction experiments were all prepared by shaking to homogenise and grinding for 10 min. This cycle was repeated several times. The samples were pressed into quartz holders for powder diffraction experiment.
The modified DBWS9411 program was used to treat the data. Pseudo-Vogit profile function was adopted to fit the experiment data. The range of calculated profile was 8 H,. Background was refined with a 3rd order polynomial in 20. Peaks below 35” were corrected for asymmetry. Run terminated when all calculated shifts were less than 10% standard deviation. There are 63 parameters refined altogether. The March-Dollase model was applied in preferred orientation correction and the Sourtti et al. model was chosen to do microabsorption correction.
380360340320300280260240220200180160140120lee80 60
;
* iL&hib
J_--
Ii_-___- ___ _ __ ____ __
Fig. 1. The diffraction pattern of sample No. 1. The dotted line is the experimental pattern and the straight line is the calculated differences are shown below them. The vertical bar indicates the reflection position.
pattern. The
174
H. Liu, C. Kuo/Mutrrials
The initial structure parameters of Al,O,, CaCO,, TiO,, SiO, and Si were taken from the Structure Report.
Letters 26 (19961 171-175 Table 3 Preferred Parameters No. 1 No.2
5. Results and discussion
No. 3
Table 2 Quantitative Sample No. 1
No. 2 No.3
No.4
1 2 3 1 3 1 3 4 1 3 4
phase analysis
results
At A
CaCO,
TiO,
SiO,
Si
29.78 31.50 30.89 34.81 36.77 45.30 48.98 47.70 23.63 29.3 1 25.02
15.42 17.87 14.88 20.54 19.45 20.40 19.36 19.23 32.20 32.92 31.89
30.14 29.13 30.33 20.16 20.54 24.66 26.30 24.33 4.89 4.85 4.99
19.66 17.30 19.33 14.85 13.44 5.21 1.78 5.15 29.70 24.27 29.11
4.99 4.19 4.57 9.62 9.80 4.43 3.58 3.60 9.57 8.66 8.98
Note: 1 - weighed phase composition; 2 - measured composition before preferred orientation correction; 3 - measured composition after preferred orientation correction; 4 - measured composition after all corrections.
Gl Gl Cl p Y Gl
The diffraction pattern, as well as the calculated
pattern, differences between these two patterns and all peak positions in the angular range of sample No. 1 are shown in Fig. 1. All the patterns show preferred orientation. The preferred orientation has a great influence on the quantitative results, as observed in this study. The phase abundance of sample No. 1 without preferred orientation correction is given in Table 2. Such a result cannot be accepted for further study. After structure refinement and preferred orientation correction, the quantitative phases analysis results became much better. R,, (weighed pattern R-factor) for sample No. 1 decreased from 20.06 to 15.26, goodness of fit from 2.30 to 1.75, D,, (Durbin-Watson statistic) increased from 0.72 to 1.04. The results also showed that the Rietveld analysis had great advantage over traditional single line or multiline X-ray diffraction quantitative analysis methods. The preferred orientation parameters are shown in Table 3. Preferred orientation direction of Al,O, and CaCO, is 104, TiO, 110, SiO, 011. The phase
orientation
No. 4
p Y
and microabsorption
parameters
A’2o3
CaCO,
TiO,
SiO,
0.9844 0.9802 0.9598
0.9088 0.8983 0.8770
0.9794 0.9693 1.OOoo
0.9874
0.9352
I .GQoo
l.OQOO 1.0000 0.4239 0.7892 0.1037 0.8952 0.7856 0.0747
which presents most serious preferred orientation in samples No. 3 and No. 4 is SiO,. This situation may be caused by its big particle size. The preferred orientation of other phases, which had the same particle size distribution in different samples, was not much varied. From the differences of the calculated and experimental patterns, the values of the R-factors, it can be seen that samples No. 1 and No. 2 show no apparent angular-dependent intensity reduction, so no further correction was applied. While the diffraction patterns of samples No. 3 and No. 4 show microabsorption of phase SiO,, the weight fraction of SiO, also greatly differs from the weighed one. Therefore, Eq. (3) was applied to correct microabsorption effect. The weight fractions after correction are shown in Table 2. It can be seen that much improvement is gained in the quantitative results. It can also be seen that microabsorption must be incorporated in the Rietveld quantitative analysis for such samples. Final values of p and q are listed in Table 3. It must be pointed out that the two microabsorption parameters of one phase are highly correlated with each other and also with other parameters such as temperature parameters (not refined in our study) and scale factors. Therefore, in the first several circles, all parameters but microabsorption parameters are refined. When there is a need to do microabsorption correction, parameters (with the exception of scale parameters) having been refined in former procedures are fixed. After the corrections mentioned above, the final weight fractions still have about 10% relative error compared to the weighed ones. There are still some factors which may be responsible for the discrepancy
H. Liu, C. Kuo/Materials
of the quantitative results by obtained the Rietveld method: (1) Sample preparation. The importance of sample preparation cannot be over-emphasized, although it can be remedied in the data treatment procedure by doing various corrections in Rietveld quantitative analysis. But in the case of an inhomogeneous sample, it is impossible to get correct results. (2) Structure model. The Rietveld method is structure-dependent. Large R values are a signal of incorrect structure or the presence of unknown phases. In our study, this problem occurs when we use the structure model of SiO, reported by Page et al. [ 161. (3) Data collection tragedy. According to Wang [17], both step size and scanning speed can cause errors in determination of X-ray diffraction peak positions, maximum intensity and full width at halfmaximum. The maximum relative error caused by step size is determined by the ratio of step size to FWHM, which was proposed as the shape-perfect coefficient of the XRD peak. A step size of 0.04” in our study can lead to 5% error in intensity. Intensity error caused by counting statistics will rise with increasing rate and decreasing measure time.
6. Conclusion The incorporation of microabsorption correction in the Rietveld program for multiphase refinement can greatly improve the results of quantitative analysis. After various corrections, the Rietveld method can give much higher accuracy quantitative results. Careful sample preparation, correct structure models and suitable data collection procedure can improve the results of Rietveld quantitative analysis.
Letters 26 (1996) 171-175
175
Acknowledgements
The authors acknowledge Professor R.A. Young, Georgia Institute of Technology, USA, for providing us with the Rietveld Program DBWS9411 and Dr. W. Pitschke, Institute of Solid State and Materials Research, Dresden, Germany, for providing us with the modified DBWS9006-PC. Professor Lu Guanglie, Hangzhou University, P.R. China, is also acknowledged for his help in collecting X-ray diffraction data.
References Rietveld, Acta Cryst. 22 (1967) 15 I. Rietveld, J. Appl. Cryst. 3 (1969) 65. [31 R.J. Hill and C.J. Howard, J. Appl. Cryst. 20 (1987) 467. [41 R.J. Hill, Powder Diffraction 6 (1991) 74. I51 J.C. Taylor and C.E. Matulus, J. Appl. Cryst. 24 (1991) 14. [61 G.W. Brindly, Phil. Mag. 36 (1945) 347. 171P.M. de Wolff, Acta Cryst. 9 (1956) 682. RI R. Suortti, J. Appl. Cryst. 5 (1972) 325. Dl R. Kumar, C.J. Sparks, T. Shraishi, E.D. Specht, P. Zschack, G.E. Ice and K. Hisatsune, Mater. Res. Sot. Symp. Proc. 213 (1991) 369. [lOI N. Masciocchi, N. Toraya and W. Parrish, Mater. Sci. Forum 79-82 (1991) 245. [I II A. Sakhivel, R.A. Young, T.S. Moss and CO. Paiva-Santos, Program DBWS-9411 for Rietveld analysis of X-ray and neutron powder diffraction patterns (1995). 1121C.J. Sparks, R. Kumar, E.D. Specht, P. Zschack and G.E. Ice, Advan. X-Ray Anal. 35 (1991) 37. 1131 H. Hennann and M. Ermrich, Acta Cryst. A 43 (1987) 401. [I41 W. Pitschke, H. Hermann and N. Mattem, Powder Diffraction 8 (1993) 74. II51 W. Pitschke, N. Mattem and H. Hermann, Powder Diffraction 8 (1993) 223. 1161Y.L. Page and G. Donay, Acta Cryst. B 32 (1976) 2456. [I71 H. Wang, J. Appl. Cryst. 27 (1994) 716.
111H.M. 121H.M.