THE RIGAKU JOURNAL VOl. 9 / NO.2 / Ceramics Research Laboratory, Nagoya Institute of Technology, Asahigaoka, Tajimi 507, Japan

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1 THE RIGAKU JOURNAL VOl. 9 / NO.2 / 1992 ACCURATE MEASUREMENT OF UNIT-CELL PARAMETERS BY THE POWDER DIFFRACTION METHOD: THE USE OF SYMMETRIC EXPERIMENTAL PROFILE AND A NEW ALGORITHM FOR SYSTEMATIC ERROR CORRECTION HIDEO TORAYA Ceramics Research Laboratory, Nagoya Institute of Technology, Asahigaoka, Tajimi 507, Japan A procedure for the accurate determination of unit-cell parameters using the conventional (para-focusing geometry and 0-20 scanning) powder diffractometer and internal standard technique is described. It has been developed on the two bases: one is the use of symmetric diffraction profile, which was demonstrated to be very effective in reducing the error accompanied with the determination of peak maximum positions. Nearly symmetric diffraction profiles were obtained by using a high-resolution type diffractometer with a larger goniometer radius and a narrower Soller slit aperture compared to those in standard-type diffractometer. The other is a new algorithm for correcting systematic errors in peak position and wavelength. In this technique, observation equations for both sample to be investigated and internal standard reference material are solved simultaneously, and the unit-cell parameters and a form of polynomial error function are determined in the course of the least-squares. An example using Si +W, W + Ce0 2, Ce0 2 + Si mixtures is given, where the unit-cell parameters of W, Ce0 2, and Si in respective mixtures were measured successively, showing that the unit-cell parameters of these materials could be measured routinely in the accuracy of the order of 10 p.p.m. with respect to the unit-cell parameter of standard reference material. 1. Introduction It is just one parameter, in the case of cubic crystals, that must be determined by the least-squares in the measurement of unit-cell parameters. Most of currently used powder diffractometers can put out digitized profile intensity data obtained by step scanning, and some of them are equipped with commercially avairable computer programs for peakposition determinations such as by using second derivative or profile fitting techniques. Then it seem to be an easy task to determine the unit-cell parameters even for the people, who are not acquainted with the profound knowledge of X-ray diffraction technique. If they are all experts in the field of X-ray powder diffraction, then it may be very probable that all of them will derive the same value, within the experimental errors, in the unit-cell-parameter determination of the same samples using the same internal standard reference materials. According to the 1960 report for Round Robin of the unit-cell parameter determination, the sixteen laboratories from Australia, the United States, and European countries particapated in the contest of measuring the unit-cell parameters of the same samples of diamond, silicon, and tungsten powders by using the same values of wavelengths, thermal expansion coefficients, and refraction coefficients. They declaired the accuracy of 0.001% (lop.p.m.) in their measurements. The results were, however, reported that the agreement among the sixteen data sets was 0.01%, which was less by one order in magnitude than those declaired by individual experiments [1]. Most data were of the film data that used for the intensity data collection in that time. Nowadays, the diffractometer data are generally believed to be more precise than the film data. In materials science reports, we find many discussions on the unit-cell parameters of materials. However, we also find large discrepancies, which exceed the estimated standard deviations (e.s.d.'s), in the measured values of unit-cell parameter, and their causes are attributed to reported and unreported factors, such as wavelength, instrumental factors, thermal expantion, crystallite size, chemical compositions, and sometimes to inpurity. After all, is the accurate measurement of unit-cell parameters not so an easy task, and is there no established way for the routine measurement by using powder diffraction technique? In the present paper, a procedure for the accurate measurement of unit-cell parameters using the internal 17

2 standard technique is described. The diffractometer data are now widely used in laboratories. Then the extrapolation technique, which correct systematic error by extrapolating the plot of back-reflection film data to 28 = [2], is unpopular, while the internal (or external) standard technique becomes important for systematic error correction. The reader will know that NIST (National Institute of Standards and Technology) standard reference material (SRM) 640x Si powders, distributed as SRM all over the world, were determined by using the internal standard technique [3]. This technique has a limitation that its accuracy cannot surpass that of the unit-cell parameter of SRM itself. However, this technique has some advantages more than the simplicity in experiment. For examples, the error accompanied with the thermal expantion can be eliminated when the thermal expantion coefficient of the sample to be measured is almost the same as that of SRM [3]. It will be demonstrated, in this paper, that this technique can correct the error in wavelength used in calculating the unit-cell parameters [4,5] (section 6). This is a great advantage when we use the synchrotron radation as an X-ray source, of which wavelength will easily be changed by a subtle variation in monochromator angle. Furthermore, all experimentalists should be able to deduce the same unit-cell parameter values within the limit of e.s.d. if they use the same sample and the same SRM in their experiments. A well characterized SRM can be used just as a "prototype meter", and it may be very useful to discuss the problem in unit-cell parameters. Hart et al. described that I p.p.m. (,1a/a) may be set as a goal of the accuracy in the unit-cell parameter measured by the powder diffraction method [6]. The present procedure aims to achieve the 10 p.p.m. accuracy in a routine measurement by using the diffractometer in laboratory system (para-focusing, 8-28 scanning). The present procedure has been developed on the two bases: one is the improvement of the accuracy in determining the peak-maximum positions. The other is a new algorithm for the systematic error correction [4]. Probable sources of error in measuring the peak positions are the profile asymmetry and KIXI-KIX2 doublet. These two sources can completely be removed by the use of parallel-beam synchrotron radiation [7]. A recent synchrotron radiation experiment has shown that the unit-cell parameter of tungsten powder could be measured in the accuracy of 3 to 6 p.p.m. with respect to the unit-cell parameter of NIST SRM 640a Si powder [6]. The laboratory system is, however, used in the present paper. The problem of profile asymmetry will be resolved by the use of high-resolution type diffractometer, and the problem of KIXI-KIX2 doublet partly by the use of profile fitting technique [8,9] (sections 2 and 3). The present technique has been applied to the sample of high crystallographic symmetry as cubic. One method of expanding the range of application to the samples with low crystilographic symmetry is the whole-powder-pattern fitting technique [10,11]. The measurement of unit-cell parameters by using the whole-powder-pattern fitting will be described elsewhere [12]. 2. Problems in Analyzing Asymmetric Profile In the first, an example is given to show how largely the peak-maximum positions are affected by profile model in profile fittings when the observed profile shape is asymmetric. A reflection (111) from IX-Si0 2 at 28 = was used as a test profile. The intensity data were collected by using the "standardtype" powder diffractometer (see section 3) and analyzed with a computer program PRO-FIT for individual profile fitting [10] by using a split-type Pearson VII function to represent a profile shape*. Table 1 gives, together" with a list of symbols, three results of refinements, a to c, which differ in the constraint applied to profile shape parameters during the least-squares. A fitting result in the case of refinement c is shown in Fig. 1. The refinement a in Table 1 was obtained by assuming the symmetric profile function (A = 1 and m L = m H ), and the Rwp factor had the highest value among the three refinements. If we made the profile width asymmetric by varying the asymmetry parameter A during the least-squares, the Rwp factor rapidly descended to 5.54% (refinement b). This change accompanied the shift of peak-maximum position by (28) to the high-angle side. Then if we made the tails of profile shape asymmetric by varying exponent parameters, m L and m H, independently during the last-squares, the Rwp factor was further reduced by about 1% and the peakmaximum position was shifted by (28) in the same direction (refinement c). Is the difference in peak-maximum position between the refinements b and c of just 5/1000 C in 28) too small to be taken into account? This magnitude is just in question in the present paper. Several methods have been devised to model the profile asymmetry [14]. The result presented in Table 1, however, shows how largely the form of profile model, assumed in profile fittings, affects the peak-maximum position, even if we make a sophisticated model for profile asymmetry. 3. Symmetric Experimental Profile For the accurate determination of peak-maximum * Mathematical forms of split-type functions of both Pearson VII and pseudo-voigt are given in references 10 and The Rigaku Journal

3 l(counts) 4X1~r I I I~ 1\ J)l I I o - ~~ - 35 I I (deg) 38 Fig. I Profile fitting result for 110 reflection from a-si0 2 (standard-type diffractometer data). Observed profile intensities are represented with solid squares, calculated intensities with solid lines, and their difference at the bottom of the diagram. Two short vertical bars represent Bragg reflection positions for Kai and Ka2 peaks. Table 1. Refined parameters of split-type Pearson VII function I under different constraints a to c in profile fitting for 110 reflection from (X-Si0 2. a b c A I 1.70(4) 2.13(9) ml 1.48(10) 1.44(4) 1.59(4) mh (4) 26 max (1) (1) (1) Rwp Notations: A: a ratio of FWHM on the low-angle side to that on the high-angle side, ml: exponent of Pearson VII function on the low-angle side, mho exponent of Pearson VII function on the high-angle side, 26 max : peak maximum position (0), Rwp: weighted R factor for profile intensity (%). positions, it may be a better strategy to improve the experimental profile shape rather than to make a sophisticated model for profile asymmetry. Major factors, which make the observed profile shape asymmetric, are the axial divergence, the specimen transparency, and the flat specimen effect [15]. The axial divergence can be suppressed by elongating the radius of Debye-Scheller ring (to adopt a larger I'i! Soller slit Fig. 2 Comparison between a) high-resolution type diffractometer and b) standard-type diffractometer, showing differences in goniometer radius and Soller slit aperture. goniometer radius) and/or by slicing vertically Debye- Scheller rings (to use Soller slits with a narrow aperture) [16]. These two factors reduce the diffracted beam intensity. The following example, however, will show how effective these factors are in reducing the profile asymmetry and, as a result, in 19

4 A a b Two-theta Fig. 3 Variation of asymmetric parameter A with 20 for reflections from Ce0 2, observed by using high-resolution type diffractometer (a and b) and standard-type diffractometer (c and d). Receiving slit widths, used in experiments, were 0.15 mm for a and c, and 0.3 mm for band d. improving the accuracy in the determination of peak-maximum positions. The two diffractometers used in the present study are schematically presented in Fig. 2. One is the diffractometer of a standard-type setup (Rigaku PMG-A2), and hereafter called as the standard-type (S-type) diffractometer. The other is the diffractometer of high-resolution setup (Rigaku PMG-VH [l6j) and hereafter called as the high-resolution-type (HRtype) diffractometer. The goniometer radius was 250 mm and the aperture of Soller slits was 2 in HR-type diffractometer, while they were 185mm and 5 in S-type. The diffractometers, which are currently used in many laboratories, may be of S-type. The entire diffraction patterns of NIST SRM 674 Ce0 2 powder were collected with both HR- and S-type diffractometers using CuKa radiation, and they were analyzed by PRO-FIT. The asymmetry parameters A, obtained by profile fittings, are plotted against 28 in Fig. 3. Observed profile shapes become largely asymmetric in the low-angle region when the S-type diffractometer is used for the data collection. On the other hand, they are almost symmetric (A = I) in the entire 28 region, if we use the HR-type diffractometer with a receiving slit of 0.3 mm. The HR-type diffractometer with larger goniometer radius and narrow Soller slits has been shown to be very effective in suppresing the axial divergence. Then it was used for testing the reproducibility in determining peak-maximum positions. The test was performed by measuring repeatedly ten times the L Table 2 Results of the reproducibility test in positioning the peak maximum (0 in 20) (only average values are presented for 220 and 531 reflections) [5].. ASWmodel SWmodel No. 26 max Deviation 26 max Deviation (16) (5) (17) (6) (16) (5) (15) (5) (16) (5) (16) (5) (17) (6) (15) (5) (15) (5) (15) (5) avo (16) (5) avo (8) (3) avo (19) (5) three reflections of Ce0 2, 220 (28=47.47 ), 421 (88.42 ), and 531 ( ), from the low-angle side. The split-type pseudo-voigt function was used in two different ways to represent the profile shape in PRO-FIT analysis. One is a model with symmetric profile width (A = I but Y/L o;6y/h' and hereafter called as a SW model), while the other is a model with asymmetric profile width (A 0;6 1 and Y/L o;6y/h' ASW model). Test results are given in Table 2. Individual 20 The Rigaku Journal

5 peak-maximum positions and their deviations from the mean value are presented for reflection 421, while only mean values for 220 and 531. The gradual change in peak-maximum position to the low-angle side in the first three measurements are considered to be due to the subtle change of the instrument until reaching the electrical and thermal equilibrium, and excluded from calculating mean values. The e.s.d.'s and deviations in the case of SW model are about one third of those in ASW model. This result shows that we can assume the symmetric profile width model if the experimental profile shape is virtually symmetric, and this constraint (A = I) is very effective in reducing the error in the measurement of peakmaximum positions below 1/1000 CO in 2e). The symmetric experimental profile can not only reduce the random errors but also eliminate the problem of profile asymmetry. The importance of using symmetric experimental profile will be shown again in section New Algorithm for Systematic Error Correction In the conventional technique for systematic peakshift correction using the internal (or external) SRM, two steps are required in deriving the unit-cell parameters. First step is to draw a calibration curve by plotting against 2e the amounts of peak-shift, which is defined as the difference between the observed and calculated peak positions, for the reflections from SRM. The second step is the determination of unit-cell parameters by the least-squares after applying the peak-shift correction for the observed peak-positions of the sample to be measured, and the Cohen's method used for the least-squares determination [17] is a well known procedure [18,19]. The two sets of reflection data from SRM and sample are processed separately in conventional technique. In new algorithm for systematic error correction here described, these two steps are excuted. simultaneously by using both data sets as a single data set [4]. From Bragg's equation,. [2e?bS-L12ei-Ci] A.as-L1A. * sm = d: 2 2', where 2e?bSis the observed scattering angle for the ith reflection, L12ei is the error in 2e arising from systematic peak-shift, e, is the random error in observation, A.as is the wavelength assumed in the calculation, L1A. is the difference between the A.asand the true wavelength, and d i * is the magnitude of reciprocal vector. Then the e, CO) is given by c.=2eobs_2 180 Sin-1[A.as-L1A. d.*]-l12e., 'n 2', ~2e?bS-2 1:0 sin-{ A.;s L1A tan e~s-l12e n Aas 1 ' " where efs is the Bragg angle calculated by (2) from d i * given where au is the coefficient, Xj is the least-squares parameter, and the number of x/s, m, changes from m = 1 in cubic system to m = 6 in triclinic system. In the case of triclinic system as a general case, ai/s are given by (3) a i1 =h 2, ai2=k 2, ai3=12, ai4 = 2kl, ais =2Ih, ai6 =2hk, (4) where hkl are the indices of reflection, and x/s by Xl =a*2, X2 =b*2, X3 =C*2, X4 = b*c* cos 0(*, Xs = c*a* cos [3*, x6=a*b*cosy*, (5) where a*, b*, c*, 0(*, [3*, and y* are recirpocal lattice constants. The third term on the right hand side of Eq. (2) depends on 2e and the fourth term do so in general. The sum of these two terms is the systematic error, and we introduce the following linear function L1(2ei) to represent the systematic error correction. m+n L1(2ei) = L t(2ejjxj, j=m+ I where t(2ei)j is the jth coefficient and Xj is the least-squares parameter. In the present study, the following two equations were used to represent L1(2eJ [4]. L1(2eJ=x m + 1 +x m +22ei+ +x m + n 2e7-1, (7) (6) L1(2eJ=x m + 1 +x m + 2 tanei+ X m + n tan n - l e i, (8) (1) Please note that the second term in Eq. (8) has the same form as that for error in wavelength [third term on the right hand side of Eq. (2)]. Then we obtain the two kinds of observation equations. One is for the sample to be measured, having m + n least-squares parameters, and the other for the SRM with n least-squares parameters (m = 0 because we have the unit-cell parameters of SRM). It is explicitly assumed that the peak-positions of both SRM and sample to be measured are on the same error curve. Then in the least-squares calculation, these two kinds of observation equations are solved 21

6 I simultaneously by minimizing the quantity D given by N D= L Wier, t= 1 where Wi is a weight assigned to the ith observation, and N is the number of observations. In the least-squares, we can minimize the following function instead of Eq. (9). N D= L w;[d*(obs)r-d*(calc)l]2, i= 1 where d(obs)[ is the observed d i *, and d(calc)[ is the corresponding calculated value. The advantage of using Eq. (10) is that we can solve the problem by using a linear least-squares [4]. On the other hand, non-linear least-squares is used to minimize the function by Eq. (9). There is little difference in determined unit-cell parameters between the two minimization functions (9) and (10). Several computer programs use Eq. (9), while several others use Eq. (10), and approximate unit-cell parameters are required to start the non-linear least-squares in the former. In the computer program UNITCELL written by the present author [I], the unit-cell parameters are first determined by minimizing function by (10), and then they are refined by minimizing Eq. (9). Thus starting parameters are not required, even though non-linear least-squares is used for deriving final parameters. At this point, the reader of this paper will have a question what is the advantage of this new algorithm? Fig. 4 explaines it schematically. In general, materials of high crystallographic symmetry, L\28 L\28?.SRM o Sample Fig. 4 Angle calibration curves for systematic-peak-shift correction used in a new algorithm (upper diagram) and the conventional two-stage analysis (lower one) (9) (10) such as of cubic crystal, are used as SRM. One reason is that the cubic crystal gives a proper number of reflections in the entire 2() range to make a calibration curve without having so many peak overlappings. The other reason may be that materials, of which unit-cell parameters have been measured accurately, are very few, and just cubic crystals with one unit-cell parameter might meet this requirement. In practice, however, just a few reflections can be used to draw a calibration curve (for example, the diffraction pattern of Si has just one reflection, in a verage, per every 12 2() in the range up to 2()= 140 for CuKa). This number is not enough to draw an accurate calibration curve, in particular, when the peak-shift has a large 2() dependency. Moreover there is uncertainty in extrapolating the calibration curve beyond the plot (Fig. 4 lower one). Thus, in some cases, multi-measurements were required to increase the statistical precision. On the other hand, the new algorithm uses not only the data from SRM but also the data from sample to be measured in order to determine a' form of error function (Fig. 4 upper diagram). Thus this technique can determine the error function much more precisely than the conventional technique, and moreover, it does not accompany the uncertainty in extrapolation. In practical applications of internal standard technique, it often happens that just one reflection from SRM is avairable because of severe overlappings with reflections from sample. The reader will find, in Table I of reference 4, that, in an ultimate case, just one reflection from SRM is enough to correct the systematic error. In section 6, it is described that this new technique can also correct the error in wavelength. 5. Testing the New Procedure Then a next question is how to test the accuracy of the present procedure? A test method here proposed is the self-consistensy test by making the most of the internal standard technique [5]. The procedure is schematically represented in Fig. 5. This test consists of three steps, and use, in each step, one of the three kinds of mixtures. In the present example, the mixtures were Si + W, W + Ce0 2, and CeO 2, + Si prepared from NIST SRM 640b Si powder [20], W powder used for 1960 IUCr Round Robin [I], and NIST SRM 674 Ce0 2 powder. In the first, the unitcell parameter of W powder in Si + W mixture is determined by assuming that the Si is SRM. In the second, the unit-cell parameter of Ce0 2 powder in W + Ce0 2 mixture is determined by assuming that the W is the SRM, where the unit-cell parameter of W powder, determined one step before, is used as a standard value. Then finally, the unit-cell parameter of Si powder in Ce0 2 + Si mixture is determined 22 The Rigaku Journal

7 Table 4 Unit-cell parameters (A) of W, Ce0 2, and Si obtained under various analysis conditions (Ll is the difference between the measured unit-cell parameters of Si and the starting value of (35) A) [5]. Run No. W Ce02 Si t:. Fig. 5 A scheme for testing the accuracy of the procedure for the determination of unit-cell parameters with internal standard reference materials. ITable 3 Some parameters in step scanning (S: Si, C: Ce0 2 ) [5]. Run No. I 2 3 Sample S+W,W+C,C+S s-w.w-c.c-s S+W,W+C,C+S RS width (mm) range (') peak regions Step width (') Counting rime (s) 20,20,20 10,12, ,10-40,10-50 by assuming that CeOz is the SRM. Thus we can measure the unit-cell parameter of Si powder, of which unit-cell parameter (standard value) is used as the starting value in this cyclic test. As shown in Fig. 5, this test will be carried out in both clockwise and counter-clockwise. The difference L1 between the measured unit-cell parameter of Si and the starting value will be used as a measure of accuracy of the present procedure. The unit-cell parameters of Si + W, W + Ce0 2, and Ce0 2 + Si were measured according to the scheme shown in Fig. 5. Profile intensity data were collected by using CuKex radiation and HR-type powder. diffractometer under the three conditions, which differ in step width and receiving slit width as given in Table 3. The profile intensity data were analyzed by using PRO-FIT [10], and the split-type pseudo- Voigt function was assumed as the profile function. The computer program UNITCELL [4] mentioned above was used for the least-squares determination of unit-cell parameters by assuming Eq. (8) for the error function, A for the unit-cell parameter of Si powder [20], and A for A of CuKex I radiation [21]. No correction was given for refraction and thermal expantion. Data analysis is described in more detail in reference 5. The refined unit-cell parameters, obtained by using both SW and ASW models, are compared in Table 4. The unit-cell parameters obtained by using synchrotron radiation data collected at Photon Factory in Tsukuba are si~w~ce02-7si ASWmodel I (4) (6) (7) (4) (7) (5) (4) (7) (5) Si~Ce02~W~Si I (4) (6) (6) (4) (5) (6) (7) (5) (6) Si~W~Ce02~Si SWmodel (4) (6) (5) (4) (6) (5) (3) (7) (5) Si~Ce02~W~Si Synchrorron (4) (5) (6) (3) ~(5) (6) (4) (5) (6) SWmodel radiation data (1) (1) (1) given for a comparison purpose [22]. The advantage of using experimentally symmetric profile and SW model has already been mentioned in section 3. When the ASW model is used, the L1 amounts to 29", 33 p.p.m. for three measurements (Run nos. I to 3). Signs of L1 in clockwise test were all reversed in counter-clockwise test. On the other hand, L1 was reduced to less than 6 p.p.m..when the SW model is used. There is a little deviation in the unit-cell parameters of Ce0 2 However, the average deviation is just 9 p.p.m. in clockwise test and 10 p.p.m. in counter-clockwise test. From the comparison of these two results, the cause of relatively large L1 in the case of ASW model will be ascribed to the systematic error arising from the use of asymmetric profile width model. On the other hand, we can see the importance of using the symmetric experimental profile, coupled with SW model, in determing the peak-position. The mean deviations of peak-maximum positions for three measurements were to when the SW model is used. This magnitude is comparable with those obtained by synchrotron radiation experiment at Stanford Synchrotron Radiation Laboratory [23]. The unit-cell parameters measured with the para-focusing diffractometer are well in accordance with those measured with synchrotron radiation data within the e.s.d. However, the e.d.s.'s 23

8 are less by 1/3 to 1/6 when the synchrotron radiation data were used. This result indicates that synchrotron radiation data using strictly monochromatic beam is more advantageous in determining the peakmaximum position than that by using Krxl-Krx2 doublet. 6. Wavelength Error Free Which wavelength does the reader use in calculating the unit-cell parameters. For example, the unitcell parameter of NIST SRM 640b Si powder will change from A to A, if we replace the wavelength value of A, which was used for deriving it, with A, which was cited from International Tables for X-ray Crystallography Vol. IV [21]. The difference in wavelength value of 23 p.p.m. between the two sources directly influences the derived value of unit-cell parameters by the same amount. The next example shows that the second (tangent) term on the right hand side in Eq. (8), which corresponds to the third term in Eq. (2), can correct completely the error in wavelength. Two results for the unit-cell parameter determination of W powder in Si + W mixture (Run no. 2 in Table 3) are compared in Table 5. The Eq. (8) was used again as the error function. One result (Iefthand side in Table 5) was obtained by using A of A (for CuKrx 1), while the other (right-hand side) by using A of A (for CuKrx2). Since all peak-positions, determined by using the PRO-FIT, were those of CuKrx I peak, and thus we should use the A of A in the least-squares calculation. Otherwise the large error will arise from the wavelength. Although the two wavelengths used in the calculation have a difference amounting to 0.25% in magnitude, the refined unit-cell parameters and parameters x m + j in Eq. (8) are entirely the same for both wavelengths except for the value for X m + 2. The difference of in X m + 2 between the two results is identical to (180/n)(LlA/ A), showing the perfect correction for the error in wavelength by the present procedure. It is well known that the internal standard Table 5 Refined unit-cell parameters of Wand parameters in error function obtained on the assumption of different wavelengths. Parameter A= A A = A a (4) (4) xm (5) 0.012(5) xm (16) 0.286(16) xm (14) 0.003(14) xm (3) (3) technique can be applied without the knowledge of wavelength. The example of determining the unit-cell parameter of W powder in Si + W mixture in the accuracy of 3 to 6 p.p.m. by using the internal standard technique without the knowledge of wavelength is given in reference 6. The present algorithm requires the wavelength in calculation, and its error is corrected automatically by tangent term in Eq. (8). The reader will not be bothered anymore by the error in wavelength. 7. Conclusion A new procedure for the determination of unit-cell parameters based on the internal standard technique has been described. It aims the accuracy of the order of 10p.p.m. with respect to the unit-cell parameter of SRM in a routine measurement using the conventional para-focusing powder diffractometry. In the experiment, the use of symmetric experimental profile has been shown to be very important in reducing the error accompanied with the determination of peakmaximum positions. The nearly symmetric profile was obtained by using a high-resolution type powder diffractometer with a larger goniometer radius and a narrower Soller sits compared to those in standardtype setup. In the theory, the new technique for systematic error correction has been introduced. In this algorithm, all Bragg reflection data from both sample and SRM are used in the least-squares determination of the unit-cell parameters and a form of error function. This algorithm has several advantages that I) the correction is much more precise than the conventional two-stage analysis, 2) there is no uncertainty in extrapolating the curve of error function, and 3) it is free from the error in wavelength. The self-consistency test for examining the accuracy of the procedure by using three kinds of mixtures, A + B, B + C, and C + A was proposed. In this test, the unit-cell parameters of sample A, which is used as an internal SRM in the mixture A + B, is measured in the third mixture C + A, and the difference in the unit-cell parameters of sample A between the measured value in C + A and the starting value in A + B gives a measure of the accuracy. The unit-cell parameters of W, Ce0 2, and Si were determined, showing that the accuracy of the order of 10 p.p.m. was obtained in the routine measurement of unit-cell parameters of cubic materials. References [ I] W. Parrish: Acta Cryst, 13 (1960), 838. [2] N. Morimoto: X-ray Crystallography, Vol. II, Ed. by. I. Nitta, Maruzen, Tokyo, (1961), 57 (in Japanese). [3] C. R. Hubbard: J. Appl. Cryst., 16 (1983), 285. [4] H. Toraya and M. Kitamura: J. Appl. Cryst., 23 (1990), The Rigaku Journal

9 [5] H. Toraya and W. Parrish: Adv. X-ray Anal., 35 (1992) (in press). [6] M. Hart, R. J. Cernik, W. Parrish and H. Toraya: J. Appl. Cryst., 23 (1990), 286. [7] J. B. Hastings, W. Thomlinson and D. E. Cox: J. Appl. Cryst., l7 (1984), 85. [8] D. Taupin: J. Appl. Cryst., 6 (1973), 266. [9] W. Parrish, T. C. Huang and G. L. Ayers: Trans. Am. Cryst. Assoc., 12 (1976), 55. [10] H. Toraya: J. Appl. Cryst., 19 (1986), 440. [I I] H. Toraya: J. Am. Ceram. Soc., 72, (1989), 662. [12] H. Toraya: Abstract for Int. Conf. Accuracy in Powder Diffraction ll, Gaithersburg, May [13] H. Toraya: J. Appl. Cryst., 23 (1990), 485. [14] H. Toraya: J. Cryst. Soc. Jpn., 34 (1992),86. (in Japanese) [15] H. P. Klug and L. E. Alexander: X-ray Diffraction Procedure for Polycrystalline and Amorphous Materials, John Wiley, New York, (1974), 271. [16] The Rigaku J., 5, No. I (1988),39. [17] M. U. Cohen: Rev. Sci. Instrum., 6 (1935), 68. [18] L. V. Azaroff and M. J. Buerger: The Powder Method in X-ray Crystallography, McGraw-Hill, New York, (1958), 239. [19] J. I. Langford: J. Appl. Cryst., 6 (1973),190. [20] S. D. Rasberry: NIST Certificate for SRM 640b Si powder (1987). [21] International Tablesfor X-ray Crystallography, Vol. IV, The Kynoch Press, Birmingham, (1974), 3. [22] H. Toraya, K. Ohno and K. Ohsumi: Photon Factory Activity Report, 9, (1991), 56. [23] W. Parrish, M. Hart, T. C. Huang and M. Bellotto: Adv. X-ray Anal., 30 (1987),