A Rietveld tutorial Mullite

Transcription

1 A Rietveld tutorial Mullite James A. Kaduk a Poly Crystallography Inc., 43 East Chicago Avenue, Naperville, Illinois Received 8 May 009; accepted September 009 The crystal structure of the mullite in a commercial material was refined by the Rietveld method using laboratory X-ray powder diffraction data. In this one refinement, most of the common challenges including variable stoichiometry partially occupied sites, multiple impurity phases, amorphous material, constraints, restraints, correlation, anisotropic profiles, microabsorption, and contamination during grinding are encountered and the thought processes during the refinement are described step-by-step. Interpretation of the refinements includes bulk chemical analysis, chemical composition of the mullite, assessment of the geometry, bond valence sums, the displacement coefficients, crystallite size and microstrain, comparison to similar structures to assess chemical reasonableness, and the nature of the amorphous phase. 009 International Centre for Diffraction Data. DOI: 0.54/ Key words: Rietveld, mullite, constraints, restraints, microabsorption, amorphous, quantitative phase analysis I. INTRODUCTION Newcomers to the Rietveld method often ask for a recipe a standard set of instructions for adding variables for carrying out a refinement. The reality is that each problem is different and that a one size fits all procedure is not possible. There are, however, many common features to real problems and some general principles. Refinement using a diffraction pattern from the common ceramic phase mullite illustrates many of the challenges faced in Rietveld refinements of inorganic phases and provides an opportunity to explain many of the thought processes involved in deciding what to do next. I encounter mullite most often in catalyst supports or inert materials used to contain the catalyst in a reactor, as well as in ceramic tableware and tile products. The structure is normally described in the orthorhombic space group Pbam No. 55. A search in the PDF for the mineral name mullite containing Al, Si, and O only and author s space group Pbam yields 3 entries, with average lattice parameters a= , b= , and c=.89 4 Å. The composition varies and averages Al 4.59 Si.4 O 9.7. There are correlations between lattice parameters and composition Deer et al., 98. The structure Figure consists of a three-dimensional network of corner-linked polyhedra. There is an octahedral site Al generally occupied only by Al. A tetrahedral site Al/Si3 is partially occupied by a mixture of Si and Al, and a second tetrahedral site Al4 generally has a low partial occupancy by Al. The Al/Si3 and Al4 sites are too close to each other to be occupied simultaneously. Charge balance requires the oxygen site occupancies to vary as the Al and Si occupancies change. Deriving such fine structural features is often easier using neutron diffraction data but can be done with care using good quality X-ray powder data. a Electronic mail: kaduk@polycrystallography.com II. EXPERIMENTAL The mullite used in this study was brought by Dilip Jang of the Kyanite Mining Co. Dillwyn, VA to an ICDD XRD clinic. Such mullite typically contains 0% to % glass, so we are faced with the prospect of quantitative phase analysis in the presence of amorphous material. As received, the sample was a polycrystalline aggregate of large grains Figure a. Mullite is hard and attempts to grind it in a mortar and pestle did not yield random powders. A portion was ground in a McCrone micronizing mill using corundum media and ethanol as the milling liquid and a fine powder was obtained Figure b. Examination of the unground mullite grains at higher magnification Figure c showed that the large grains were composed of fine needles, but that this fine structure was not present in the micronized material Figure d. One must always be aware that it is possible to change the sample while preparing the specimen. Even though it is possible to use the Rietveld method to model the nonideal features of real data sets, problems Figure. Color online The crystal structure of mullite viewed approximately down the c axis. The partially occupied Al4 site is designated P to permit plotting in a different color. 35 Powder Diffraction 4 4, December /009/4 4 /35//$ JCPDS-ICDD 35

2 Figure. a SEM image 00 of the as-received mullite. b SEM image 00 of the micronized mullite. c SEM image 5000 of the as-received mullite. d SEM image 5000 of the micronized mullite. caused by poor data never go away and the best practice is to collect the best possible data from well-prepared specimens Buhrke et al., 998. Because I wanted to carry out an absolute quantitative phase analysis in the presence of amorphous material, a portion of the micronized mullite was blended with 8.36 wt % NIST 640b silicon internal standard in a Spex 8000 mixer/ mill. My experience is that 0 to 5 wt % is generally a convenient concentration of the internal standard. In this case mullite scatters relatively weakly compared to silicon I/I c =0.77 and 4.7, respectively, so I chose a lower Si concentration to avoid having the internal standard peaks dominate the pattern. The powder pattern was measured 5 to 50, steps, s/step from a front-packed rotating flat plate specimen on a Bruker D8 Advance diffractometer equipped with a VÅNTEC- position-sensitive detector. JADE 8.5 Materials Data, Inc., 007 was used to identify mullite, quartz, and silicon as the major phases Figure 3. Several PDF entries for mullite match the observed pattern about equally well. A common source of consternation is having to choose the best one among these or at least the one to use as the initial structural model in the Rietveld refinement. The PDF editors have expended a great deal of effort extending the quality mark system used for experimental patterns to patterns calculated from structural data, as well as designating primary and alternate patterns. Entry is an old indexed-quality pattern NBS, 964 but does not point directly to a crystal structure. Entry Ban and Okada, 99 has a blank quality mark and is Figure 3. Color online Identification of the major phases. 35 Powder Diffr., Vol. 4, No. 4, December 009 James Kaduk 35

3 ICDD mullite micronized w/ 8.36% Si (KADU86) Hist Lambda.5406 A, L-S cycle Obsd. and Diff. Profiles ICDD mullite micronized w/ 8.36% Si (KADU86) Hist Lambda.5406 A, L-S cycle 7 Obsd. and Diff. Profiles X0u X0u Figure 4. Color online A portion of the Rietveld plot after refinement of three background terms and three phase fractions. The peaks are not calculated at the observed positions, so add a specimen displacement coefficient and the mullite and quartz lattice parameters. Keep the silicon lattice parameter fixed at the certified value. designated alternate. Entry comes from another structure in the same paper and is designated blank and primary. Entry Li et al., 004 is a starquality primary pattern and I chose to use these coordinates ICSD collection code 9937 for the initial mullite model. III. THE RIETVELD REFINEMENT The binary raw data file was converted to an ASCII UXD-format file using Bruker tools and then to a GSASformat data file using POWDER4 Dragoe, 004. Manual editing was required to add the title and to ensure that the step width was correct. The refinement was begun with mullite, quartz, and silicon as phases,, and 3 using the GSAS refinement package Larson and Von Dreele, 004. For mullite, I used the model of Li et al. and set U iso =0.0 Å for Al and Si, and 0.0 Å for O. For quartz, I used my own structural model Kaduk, 998. For silicon, I fixed the cubic lattice parameter at the certified value of Å, and used the well-known structure Fd-3m; Si at /8,/8,/8 and U iso =0.0 Å. Since the low-angle portion of the pattern contained no peaks and thus no information needed for the fit, I used only the 5 to 50 portion of the pattern as histogram. The instrument parameter file was derived from a refinement using data on NIST SRM 660a LaB 6. With only one data set, I turned off the default refinement of the histogram scale factor and refined one phase fraction for each phase. A good initial background function for this instrument is a three-term shifted Chebyshev polynomial. The initial refinement of these six variables using observations yielded the residuals Rwp= , Rp= 0.5, and = A look at the strong peaks Figure 4 showed that the peaks were not calculated at the observed positions. A good general principle is to deal with the largest discrepancies first and the refinement cannot progress until I get the peaks in the right positions. The major variables that affect peak positions are the specimen displacement since the peaks are observed too low in angle, the effective surface of Figure 5. Color online A portion of the Rietveld plot after adding the specimen displacement and lattice parameters. The peaks are not the right shape, so add profile Y for mullite, quartz, and silicon. the specimen is lower than the focusing circle of the diffractometer and the lattice parameters. I know the diffractometer zero from the refinement to develop the instrument parameter file and do not need to refine it until and unless the diffractometer configuration has changed. When I added the mullite and quartz lattice parameters and a common constrained to be the same for all phases specimen displacement coefficient shft in GSAS terminology, the residuals for refinement of variables dropped to Rwp=0.89, Rp=0.760, and =3.33 Figure 5. The largest errors were clearly now in the shapes/widths of the peaks. After micronizing, the profile shapes are generally dominated by microstrain broadening Stephens, 999, so I added the profile Y terms for mullite and quartz. Experience tells me that the silicon peaks also contain a strain broadening component, so I added profile Y for silicon also. Refinement of 5 variables yielded the residuals Rwp= 0.49, Rp=0.73, and =3.74. The error plot Figure 6 showed that the description of the background needed improvement. ICDD mullite micronized w/ 8.36% Si (KADU86) Hist Lambda.5406 A, L-S cycle 9 Normalized Error Distr Figure 6. Color online This error plot shows that the description of the background needs improvement. 353 Powder Diffr., Vol. 4, No. 4, December 009 A Rietveld tutorial Mullite 353

4 ICDD mullite micronized w/ 8.36% Si (KADU86) Hist Lambda.5406 A, L-S cycle 9 Radial dist. function ICDD mullite micronized w/ 8.36% Si (KADU86) Hist Lambda.5406 A, L-S cycle Obsd. and Diff. Profiles D(r), arb. units X0u X0u radius, A Figure 7. Color online The reduced pair correlation function generated POWPLOT/R by Fourier transforming the difference plot corresponding to Figure 6. The characteristic distances to be used in a type diffuse scattering function can be picked off the plot. While I could have added more terms to the shifted Chebyshev background function, my knowledge of the sample suggested that there should be an amorphous component to the background. The R option in the GSAS program POWPLOT carries out a Fourier transform of the difference plot and generates a reduced pair correlation function Figure 7, from which we can pick off characteristic distances 0.63,.39,.53, and 4.9 Å to be used in a diffuse scattering function. Calculation of a proper diffuse scattering function requires sampling much higher in reciprocal space than can be achieved using Cu radiation, so such functions calculated from laboratory data sometimes contain artifacts. Since my interest was in a good model for the background, I could tolerate a philosophically questionable choice here and included the unphysical 0.63 Å distance in the diffuse scattering function. I added these four distances as type diffuse scattering terms with amplitudes of 0, +0, 0, and +0 initial guesses, respectively, and designated the 0.63 and.39 Å distances Si-O, the.53 Å distance O-O, and the 4.9 Å ICDD mullite micronized w/ 8.36% Si (KADU86) Hist Lambda.5406 A, L-S cycle Normalized Error Distr. Figure 9. Color online A portion of the difference plot showing the unaccounted for peaks of the impurity phases. distance Si-Si. I chose a somewhat arbitrary value derived from a few attempts at refining such terms of 0.05 Å for the U values in the diffuse scattering function. These terms in a Debye function are large the amorphous bumps result from small differences among these terms, and are not orthogonal, so I applied a large damping factor of 8 to the refinement of the diffuse scattering amplitudes. The error plot was now much flatter Figure 8 and the residuals for refinement of 9 variables dropped considerably: Rwp =0.0878, Rp=0.0658, and =4.76. Although the largest errors were now in the mullite peak intensities and there was more work to do on the background, a significant problem was there were weak peaks that are not accounted for by the three major phases Figure 9. Picking these peaks off of the difference plot Table I and entering them into Sieve+ an add-on to the PDF-4+ database or another phase identification program did not yield plausible results, so another strategy was needed. I used the Boolean logic capabilities of the PDF-4+. In such a ceramic, it is reasonable to assume that all of the phases will be mineral related. It is also reasonable to assume that the peaks I observed in the difference plot will be among the three strongest peaks of the impurity phase. So using Sieve+ to find entries that are mineral related and have one of their three strongest peaks near one of the observed difference peaks, and starting with the strongest at Å, I Figure 8. Color online The error plot after addition of diffuse scattering to the refinement. TABLE I. Peaks in the difference plot. d Å I counts Identity Cristobalite Quartz Cu K Zircon Rutile Kyanite Zircon Rutile Corundum Kyanite Kyanite Kyanite 354 Powder Diffr., Vol. 4, No. 4, December 009 James Kaduk 354

5 ICDD mullite micronized w/ 8.36% Si (KADU86) Hist Lambda.5406 A, L-S cycle 40 Normalized Error Distr. ICDD mullite micronized w/ 8.36% Si (KADU86) Hist Lambda.5406 A, L-S cycle 73 Obsd. and Diff. Profiles X0u Figure 0. Color online A portion of the error plot after the addition of rutile and cristobalite. Figure. Color online A small portion of the Rietveld plot after refinement of the mullite structure. The effects of the % peak tail cutoff are visible as steps near the strongest peaks, so change the peak cutoff to 0.%. obtained a hit list with many entries for rutile, which is certainly a reasonable phase in such a sample. Rutile also accounts for the.49 Å peak in the difference plot, so the identification is confident. Adding rutile refining the phase fraction only to the refinement as phase 4 decreased the reduced to The next most prominent peak was the one at d=4.05 Å. Repeating the Boolean search yielded cristobalite, another quite reasonable phase. Adding cristobalite as phase 5 caused the reduced to drop to only At this point, the approximate concentrations of rutile and cristobalite were 0.7 and 0. wt% respectively and the largest errors were in the mullite peak intensities Figure 0, soit was time to refine the mullite structure. Before refining the mullite structure, I wanted to look at the geometry using the GSAS program DISAGL and decide how to incorporate chemical knowledge into restraints and constraints. The octahedral Al is bonded to two O5 at.90 Å and four O6 at.04 Å. The tetrahedral Al/Si3 is bonded to one O5 at.66 Å, two O6 at.67 Å, and one O7 at.67 Å. The tetrahedral Al4 is bonded to one O5 at.69 Å, two O6 at.86 Å, and one O8 at.48 Å. Based on experience or the bond valence formalism Brown, 00, I know that.90 Å is a reasonable value for an octahedral Al-O bond distance,.6 is reasonable for a tetrahedral Si-O bond, and.74 Å for a tetrahedral Al-O bond. Since I do not yet know the composition of the Al/Si3 site, I used the current value of.67 Å for the restraints. In GSAS, the atoms participating in an angle restraint must be in the same asymmetric unit. In mullite, all of the heavy atoms lie on special positions, so their coordination spheres span multiple asymmetric units and I must use nonbonded distance restraints to incorporate knowledge of the angles in octahedral and tetrahedral coordination spheres. A little solid geometry shows that a cis O O distance in an AlO 6 coordination sphere is.69 Å and that the O O distance in a tetrahedral coordination sphere is.7 Å. A restraint of.70 4 Å on nonbonded O O distances will be adequate to restrain the angles. I want to constrain the coordinates of Al and Si3 to have the same values. The occupancies of Al, Si3, Al4, O7, and O8 will be considered at a later stage of refinement. A laboratory X-ray pattern does not usually contain enough resolved reflections to permit refinement of individual U iso, so I used constraints to refine a single displacement coefficient for the Al and Si, and another for the O. I also refined the quartz structure with bond restraints and the silicon U iso since its initial value was only a reasonable guess. A point that can cause controversy is the relative weighting of the data and the restraints. When creating restraints, there is a weighting factor FACTOR which can change the relative weights of the restraints and the data. I chose to use a FACTOR= 0 for these distance restraints, with the result is that they make only a small contribution to the final reduced. The relative weighting will depend on the problem, the quality of the data, and your confidence in the quality of the restraints. Refining the structures of the three major phases yielded the residuals Rwp=0.0638, Rp=0.050, and =.68 for the 39 variables, and the structural parameters were chemically reasonable. I could now consider adding the mullite site occupancies. The average Al/Si3-O bond distance is.696 Å. The ideal Si-O distance is.6 Å and the corresponding ideal Al-O distance is.74 Å. Interpolating between the two yields a Al/Si ratio of 0.66/0.33. Beginning the refinement ICDD mullite micronized w/ 8.36% Si (KADU86) Hist Lambda.5406 A, L-S cycle 3 Normalized Error Distr Figure. Color online An error plot before the addition of a fourth shifted Chebyshev background term. 355 Powder Diffr., Vol. 4, No. 4, December 009 A Rietveld tutorial Mullite 355

6 ICDD mullite micronized w/ 8.36% Si (KADU86) Hist Lambda.5406 A, L-S cycle 348 Obsd. and Diff. Profiles X0u Figure 3. Color online The Rietveld plot at the conclusion of the first refinement. The vertical scale has been multiplied by a factor of 4 from 5 to 8 and by a factor of 0 at angles 8. with the Al and Si occupancies at these values, I set up a constraint so that the shift in the Al occupancy was always that of the shift in the Si3 occupancy; the result is that the Al and Si occupancies will stay in the ratio deduced from the bond distances. O8 is only occupied when Al4 is present, so I constrained the occupancies of Al4 and O8 to the same value. I let the occupancy of O7 refine freely. Refinement of 4 variables decreased the residuals to only Rwp=0.0638, Rp=0.050, and =.65. At this point I could consider fine details. The mullite peaks were dominated by microstrain broadening; the profile Y was 44.7 compared to the instrumental value of.03. The current treatment of the mullite microstrain broadening was isotropic, but it is reasonable to expect some peak width anisotropy. I changed to profile function no. 4, which incorporates the Stephens 999 anisotropic strain broadening model. On high magnification of the Rietveld plot Figure, I could see the effects of the default % peak wing cutoff which is set in the instrument parameter file on the peaks of the three major phases. The calculated intensity suddenly drops to zero at points on the peak tails. One of the many things that the program POWPREF does is to determine which peaks from ICDD mullite micronized w/ 8.36% Si (KADU86) Hist Lambda.5406 A, L-S cycle 348 Normalized Error Distr Figure 4. Color online The error plot at the conclusion of the first refinement. Figure 5. Color online Particle size distributions in the micronzed mullite, NIST 640b silicon, and the micronized blend. which phases contribute to which points in the pattern based on the lattice parameters and the profile function. Once an observed data point lies outside the window, the calculated intensity from that peak is set to zero. Changing the cutoff to 0.% for these three phases slows the refinement but extends the profile computation until the pattern noise level is reached. The refinement of 47 variables yielded the residuals Rwp=0.0598, Rp=0.0467, and =.33. After thinking about the consequences of shape anisotropy in mullite, I added second-order spherical harmonic terms to describe any preferred orientation of this phase. The residuals for refinement of 49 variables decreased only slightly, and the texture index refined to.00, so preferred orientation was slight. The difference/error plot Figure, or the equivalent LIVE- PLOT view suggested that some improvement in the background was needed particularly to describe the curvature at low angles. I could have chosen to add more diffuse scattering terms or another background coefficient. The background and diffuse scattering functions interact strongly with each other the coefficients are highly correlated, so this step needs to be taken with caution. I chose to add one more term to the background function. The refinement of 50 variables almost diverged actually increased for a few cycles before settling down but eventually converged with residuals Rwp=0.058, Rp=0.045, and =.99. TABLE II. Potential microabsorption effects in a 9.64% mullite/8.36% silicon mixture. Phase Mullite Silicon / cm /g g/cm cm avg cm D m peak D Size Coarse powder Coarse powder 356 Powder Diffr., Vol. 4, No. 4, December 009 James Kaduk 356

7 Figure 6. Color online The increase in corundum contamination after the second micronization. At this point, the description of mullite microstrain broadening was completely Lorentzian Cauchy. My experience is that this is almost always the case, but in a few cases there is a Gaussian component to the strain broadening. I tested this possibility by adding the profile U and coefficients, but GENLES refused to refine away from and the value of U remained very small. Apparently there is no Gaussian component to the strain broadening, so I fixed these values at their instrumental values. Adding refinement of the lattice parameters and a profile X for phases 4 and 5 caused the residuals for the refinement of 56 variables to decrease to Rwp=0.0567, Rp=0.0440, and =.094. The fit to the mullite peak shapes at low angles was not perfect, so I added the profile X and ptec unique axis 00 to test if there was any size broadening. These coefficients refined to X= and ptec=.58 46, so any size broadening was small. The residuals decreased slightly to Rwp =0.0554, Rp=0.0430, and =.004 for refinement of 58 variables. The portion of the pattern between 5 and 6 degrees contained only background which was not fitted well. Rather than adding more background terms to fit it, I changed excluded region to 0 6 degrees. The residuals dropped to Rwp=0.0545, Rp=0.049, and =.90. Now I examined the remaining impurity peaks. A Hanawalt search in Sieve+ readily identified kyanite Yang et al., 997. A search for mineral-related phases having one of their three strongest lines at Å suggested zircon mullite/8.35% Si micronized blend (KADU38) Hist Lambda.5406 A, L-S cycle 37 Obsd. and Diff. Profiles mullite/8.35% Si micronized blend (KADU38) Hist Lambda.5406 A, L-S cycle 37 Normalized Error Distr. X0u Figure 7. Color online The final Rietveld plot. The vertical scale has been multiplied by a factor of 4 from 5 to 8, and by a factor of Figure 8. Color online The final error plot. 357 Powder Diffr., Vol. 4, No. 4, December 009 A Rietveld tutorial Mullite 357

8 TABLE III. Quantitative phase analysis. Phase Raw wt % Scaled wt % True wt % Mullite Al 4.85 Si.8 O Quartz SiO Silicon Si Rutile TiO Cristobalite SiO Kyanite Al SiO Zircon ZrSiO Corundum Al O Amorphous or perhaps a feldspar. The best explanation for the.0870 Å peak is probably corundum from the grinding elements in the micronizing mill. Adding kyanite, zircon Siggel and Jansen, 990, and corundum as phases 6, 7, and 8 and refining their phase fractions decreased the residuals to Rwp= , Rp =0.0397, and =.673. Refining the lattice parameters and profile X coefficients for these three phases 74 variables decreased the residuals to Rwp= , Rp= , and =.64. Reviewing the refinement at this stage, I noted that the restraints soft constraints contributed 0.7% of the final reduced. I prefer to have the refinement dominated by the data rather than the restraints, so the relative weighting was acceptable. Both the Rietveld plot Figure 3 and the error plot Figure 4 suggested that the possibilities of this model have been exhausted. The R F =0.084 and R F = The largest peak and hole in the difference Fourier map were and 0.39 e/å 3. The slope and intercept of the normal probability plot were.63 and showing that there were only very small systematic errors in the fit. At this point, the silicon concentration despite the appearance of the background and our expectations was lower than the expected value 8. 3 wt % versus 8.36 wt % suggesting that the concentration of amorphous material was negative. Despite the quality of the fit, something was wrong. The lower-than-expected Si concentration is consistent with the effect of microabsorption Brindley, 945 ; the concentrations of the less-absorbing phases were higher than they should be. I therefore needed to test the possibility that microabsorption effects were present. Once the particle not crystallite size distributions of the micronized sample and NIST 640b silicon were measured by scattering of laser light Figure 5, the potential microabsorption effects could be estimated Table II. Caution is required when considering the possibility of microabsorption; one of the largest sources of error by participants in the IUCr quantitative phase analysis round robin Madsen, et al., 00 was making a microabsorption correction when none was warranted. The main factors in determining whether microabsorption effects are present are significant differences in both absorption coefficients and the particle size among the phases. There is a significant difference between the linear absorption coefficients of mullite and silicon, and thus in the absorption contrast. For the purpose of this discussion, I ignored the quartz and other minor phases. In choosing in internal standard, it is wise to make the absorption contrast with the sample phases as small as possible; perhaps silicon was not the best choice of standard for this problem. The peak in the mullite size distribution is m the largest particles are more important, while the peak in the narrow size distribution of SRM 640b is 7.8 m. For each phase, the product of the linear absorption coefficient and the particle diameter D is 0. showing that despite the relatively small particle size both of these phases are coarse particles according to Brindley and that microabsorption effects might be significant. Microabsorption affects the expected or classical ratio of phase concentrations by multiplying it by a factor, K = M = V V S M 0 e M avg x dv V S V M S 0 e S avg x dv, where the integrations are carried out over the volumes of the particles of the two phases mullite and silicon. Brindley supplies tables for evaluating these integrals for spherical particles, so we can estimate the microabsorption effect here, K = M =.066 S =., so microabsorption effects though not large should be significant. The way to overcome these effects is to micronize the mullite/silicon blend. The particles in the micronized blend are smaller Figure 5 and I assume the size distribution for all phases is the same. The powder pattern was remeasured, the old experiment file was copied to the new name, the histogram replaced using EXPEDT, and refinement continued. The micronized blend contained slightly more corundum than the first sample Figure 6. My experience is TABLE IV. Refined structural parameters of mullite. Space group Pbam, a=5.5478, b= , c = Å, and V= Å 3. Atom x y z frac U iso Å Al Al Si Al O O O O Powder Diffr., Vol. 4, No. 4, December 009 James Kaduk 358

9 that corundum from the grinding elements is highly strain broadened, so the profile X was fixed at the instrumental value and the profile Y was allowed to refine starting from a typical value of 80. The refinement was actually slightly better than the first specimen Figures 7 and 8 and refinement of the same 74 variables yielded the residuals Rwp =0.048, Rp=0.0377, and =.533. The restraints soft constraints again contributed 0.7% of the final reduced. The R F =0.083 and R F = The largest peak and hole in the difference Fourier map were Å from two Al and 0.38 e/å Å from Al, so the map was acceptably flat. The slope and intercept of the normal probability plot were.4 and showing that there were only very small systematic errors in the fit. IV. INTERPRETATION OF THE RESULTS The quantitative phase analysis, including the amorphous material, is calculated in Table III. GSAS, like other contemporary Rietveld programs, calculates a quantitative phase analysis from the phase fractions under the assumption that no amorphous material is present. This is basically Chung s matrix flushing method Chung, 974. Strictly speaking, the Rietveld refinement gives us the ratios of the concentrations of the crystalline phases, which are the true concentrations if no amorphous material is present. In this sample, I added silicon internal standard with a concentration of 8.36 wt % before the contamination with corundum during the grinding. It is therefore legitimate to ignore the corundum and rescale the raw concentrations by multiplying each of them by /8.57. The amorphous content is then calculated by subtraction the sum of the concentrations of the crystalline phases from 00%. Since one of the goals of the study was to determine the amorphous content of the initial sample and not the specimen to which the internal standard had been added, the concentrations thus need to be renormalized by dividing by to obtain the true concentrations in the original sample. The standard uncertainties on the true concentrations were calculated by propagating the relative standard uncertainties on the raw concentrations. The standard uncertainty on the amorphous concentration was estimated by adding the standard uncertainties on the crystalline concentrations in quadrature. The analysis resulted in 7.6 wt % amorphous material, which is consistent with the appearance of the pattern and the information supplied with the sample. The sample is 77.6 wt % mullite. In the absence of amorphous material, the bulk chemical analysis calculated from the phase concentrations can be compared to the measured bulk analysis for a valuable internal consistency check. In this case, such analysis would have to be done phase by phase in the electron microscope. From the refined structural parameters Table IV we can calculate the composition of the mullite as Al 4.85 Si.8 O 9.77, and thus carry out quantitative phase analysis on the atomic scale as well. It is important to check the chemical reasonableness of the refined results Kaduk, 007. The unconstrained sum of the positive charges in the unit cell is +9.8 and the sum of the negative charges is The agreement is to % of stoichiometric, which I consider acceptable. It is possible to restrain the cation and anion TABLE V. Bond distances in mullite. Atom O5 O6 O7 O8 BVS Al Al Si3 3.5 Al O5.90 a O6.0 a O O a Not counting Al4. contents, but I chose to use the charge sum as an internal consistency check. The ratio of the occupancies of Al and Si3 was constrained, but the sum of the occupancies of Al, Si3, and Al4 was not, and equals.008. As we expect, either site /3 or 4 is occupied, but not both simultaneously. The standard uncertainties on the fractional coordinates are reasonable. The bond valence sums calculated from the bond distances Table V are close to the expected values for Al, Al4, and the oxygens, but not for Al/Si3. The differences from the expected values of 3 and 4 help rationalize the multiple and partial occupancy at this site. The agreement of the bond valence sums with their expected values is another way of saying that the bond distances fall within the expected ranges. The average deviation of the bond angles at the highly occupied metal sites Table VI from their ideal values is only. 35 degrees and degrees for the less-occupied Al4. The displacement coefficients of the atoms in mullite Al/Si=0.0093, O= Å, quartz Å, and silicon Å are quite reasonable for such a rigid inorganic solid. With a large angular range of data and careful specimen preparation, it is possible to obtain reasonable displacement coefficients even using laboratory X-ray powder data. The refined orthorhombic lattice parameters for mullite are the following: a= , b= , and c= Å. A search of the Inorganic Crystal Structure TABLE VI. Bond angles in mullite. Angle O5-Al-O5 O5-Al-O5 O5-Al-O5 O5-Al-O6 O5-Al-O6 O6-Al-O6 O5-Al/Si3-O6 O5-Al/Si3-O7 O6-Al/Si3-O6 O6-Al/Si3-O7 O5-Al4-O6 O5-Al4-O8 O6-Al4-O6 O6-Al4-O8 Value Powder Diffr., Vol. 4, No. 4, December 009 A Rietveld tutorial Mullite 359

10 TABLE VII. Mullites in the ICSD. Al frac Si3 frac Al4 frac Formula Average Al Si.4 6 O This work Al 4.85 Si.8 O 9.77 Database 008/ Hellenbrandt, 004 for structures having reduced cell parameters within 0.03 Å of these values yielded 4 aluminosilicate mullites Table VII. In almost all of these refinements, the occupation of Al at the tetrahedral site was fixed at and the Si occupancy was refined. This sample seems to be a typical and chemically reasonable mullite, which is slightly more Al-rich than average. The a lattice parameter corresponds to 75 wt % Al O 3, and the cell volume to 5 wt % SiO, according to the correlations in Deer et al. The mullite profile coefficients X and ptec correspond to an average crystallite size of Å consistent with the needle morphology observed in the starting material. The texture index derived from the spherical harmonic coefficients is only.004, so there is no significant preferred orientation. The mullite profiles are dominated by strain broadening as we might expect from its solid solution nature and the fact that it was micronized twice. The most convenient way to view the strain is a constant microstrain plot Figure 9. I tend not to interpret microstrain on an absolute basis, but look for changes/differences among related samples. The refined specimen displacement coefficient shft =.73 can be used to calculate the actual displacement of the effect specimen surface from the correct position diffractometer radius shft/ This value corresponds to the effective specimen surface being 36 m too low. No matter how hard I try, I almost always manage to front pack a specimen too high and thus observe negative shft. The small positive value here indicates that I did a good job at specimen preparation and that the X-rays penetrate significantly into the specimen. Even the diffuse scattering function can be useful. From the refined coefficients, POWPLOT can calculate the observed and calculated reduced pair correlation functions Figure 0. Except for the artifact at 0.38 Å, the plots are not unreasonable for an aluminosilicate glass. It is certainly possible to track changes in the amorphous phase with processing by looking at such functions. Figure 9. Constant microstrain plot for mullite. mullite/8.35% Si micronized blend (KADU38) Hist Lambda.5406 A, L-S cycle 37 Radial dist. function D(r), arb. units X0u radius, A Figure 0. Color online Observed and calculated reduced pair correlation functions for the amorphous material. This refinement was a lot of work because the material is complicated. I have not said anything about the minor phases because we can expect to learn less about them from their small peaks. It is possible, however, to extract a great deal of information from an X-ray powder pattern. One of the beauties of the Rietveld method is that it forces us to model all of the features of the powder pattern in a physically meaningful way, so we can extract the maximum information from the pattern. This tour through the process of refinement is necessarily compact, but I hope that the descriptions of the process and what to look for may be helpful to new and experienced users of the Rietveld method. ACKNOWLEDGMENTS The author thanks his INEOS Technologies colleagues Gerry W. Zajac for the electron micrographs and Lina K. Bodiwala for the particle size distribution measurements. Ban, T. and Okada, K. 99. Structure refinement of mullite by the Rietveld method and a new method for estimation of chemical composition, J. Am. Ceram. Soc. 75, 7 30; ICSD collection code Brindley, G. W The effect of grain or particle size on X-ray reflections from mixed powders and alloys, considered in relation to the quantitative determination of crystalline substances by X-ray methods, Philos. Mag. 36, Brown, I. D. 00. The Chemical Bond in Inorganic Chemistry Oxford University, Oxford. Buhrke, V. E., Jenkins, R., and Smith, D. K A Practical Guide for the Preparation of Specimens for X-ray Fluorescence and X-ray Diffraction Analysis Wiley-VCH, New York. Chung, F. H Quantitative interpretation of X-ray diffraction patterns. II. Adiabatic principle of X-ray diffraction analysis of mixtures, J. Appl. Crystallogr. 7, Deer, W. A., Howie, R. A., and Zussman, Y. 98. Rock Forming Minerals, Volume A, Orthosilicates Geological Society of London, Oxford, p Dragoe, N POWDER4, VERSION.. Hellenbrandt, M The Inorganic Crystal Structure Database ICSD Present and future, Crystallogr. Rev. 0, 7. Kaduk, J. A Chemical accuracy and precision in structure refinement from powder data, Adv. X-Ray Anal. 40, Kaduk, J. A Chemical reasonableness in Rietveld analysis: Inorganics, Powder Diffr., Larson, A. C. and Von Dreele, R. B General Structure Analysis 360 Powder Diffr., Vol. 4, No. 4, December 009 James Kaduk 360

11 System (GSAS) (Report LAUR ) Los Alamos National Laboratory, Los Alamos, New Mexico. Li, J. F., Li, L., and Scott, F. H Crystallographic analysis of surface layers of refractory ceramics formed using combined flame spray and simultaneous laser treatment, J. Eur. Ceram. Soc. 4, ; ICSD collection code Madsen, I. C., Scarlett, N. V. Y., Cranswick, L. M. D., and Lwin, T. 00. Outcomes of the International Union of Crystallography Commission on Powder Diffraction round robin on quantitative phase analysis: Samples a to h, J. Appl. Crystallogr. 34, Materials Data, Inc Jade 8.0. National Bureau of Standards U.S Monograph 5, Part 3, p. 3. See EPAPS Document No. E-PODIE for supplemental data. This step-by-step description of a complex Rietveld refinement illustrates many of the common challenges faced in inorganic materials, and provides guidance on how to decide what to do next and how to interpret the refined parameters. For more information on EPAPS, see Siggel, A. and Jansen, M Roentgenographische untersuchungen zur bestimmung der einbauposition von seltenen erden, Z. Anorg. Allg. Chem. 583, 67 77; ICSD collection code Stephens, P. W Phenomenological model of anisotropic peak broadening in powder diffraction, J. Appl. Crystallogr. 3, Yang, H., Downs, R. T., Finer, L. W., Hazen, R. M., and Prewitt, C. T Compressibility and crystal structure of kyanite, Al SiO 5,at high pressure, Am. Mineral. 8, ; ICSD collection code Powder Diffr., Vol. 4, No. 4, December 009 A Rietveld tutorial Mullite 36