The recently developed RITA concept combines algorithms of the Rietveld structure analysis with those from modern texture analysis in order to get texture information from d-spacing diffraction spectra. It is demonstrated for a low symmetry material (quartz) and a minimum pole sphere covering by the measured spectra. The quality of the results underlines the efficiency of this new approach especially for time-of-flight neutron diffraction studies where access to the beam is limited. New interesting aspects for crystal structure refinement are discussed

Hans-Rudolf Wenk

K. Ullemeyer

Luca Lutterotti

Siegfried Matthies

English

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Textures and Microstructures, Vol. 33, pp. 139-149Reprints available directly from the publisherPhotocopying permitted by license only(C) 1999 OPA (Overseas Publishers Association) N.V.Published by license underthe Gordon and Breach SciencePublishers imprint.Printed in Malaysia.TEXTURE ANALYSIS OF QUARTZITE BYWHOLE PATTERN DECONVOLUTIONS. MATTHIES a,b, L. LUTTEROTTI a,e, K. ULLEMEYERdand H.-R. WENK a,.Department of Geology and Geophysics, University of California, Berkeley,CA 94720, USA; b GKSS, D-21502 Geesthacht, Germany; Dipartimento diIngegneria dei Materiali, Universita’ di Trento, 38050 Mesiano (TN), Italy;d Frank Laboratory of Neutron Physics, JINR, Dubna, RussiaThe recently developed RITA concept combines algorithms of the Rietveld structureanalysis with those from modern texture analysis in order to get texture information fromd-spacing diffraction spectra. It is demonstrated for a low symmetry material (quartz) anda minimum pole sphere covering by the measured spectra. The quality of the resultsunderlines the efficiency of this new approach especially for time-of-flight neutron dif-fraction studies where access to the beam is limited. New interesting aspects for crystalstructure refinement are discussed.Keywords: Time-of-flight method; Rietveld analysis; Texture factor;Whole pattern fitting; QuartzINTRODUCTIONThe determination of the orientation distribution (OD) of texturedpolycrystals has largely relied on the analysis of pole figures that weremeasured with diffraction techniques. Various methods exist to deter-mine the OD from a small number of complete or incomplete polefigures. This is efficient if diffraction peaks are well separated, as is thecase, e.g. in cubic metals. Many materials, particularly those of lowcrystal symmetry and composites, have complex diffraction spectraCorresponding author.139140 S. MATTHIES et al.with many closely spaced and overlapping diffraction peaks. In thesecases a different approach is more adequate.Rietveld (1967; 1969) introduced a method to determine the crystalstructure from a continuous diffraction spectrum of a powder sample.His method relies on the premise that a diffraction spectrum is given bythe incident beam characteristics, line-broadening/instrumental param-eters (such as peak shape, background function, Lorentz-polarizationcorrections etc.), and structural parameters (cell parameters, atomicpositions, temperature factors etc.). From the measured spectrum theseparameters can be refined using a non-linear least squares method.Analytical functions, like the March-Dollase correction (Dollase,1986), were incorporated to take into account intensity deviations dueto preferred orientations. More recently this method has been expandedto include elements of modern texture analysis of polycrystalline sam-pies, using the orientation distribution function (ODF) (Popa, 1992;Ferrari and Lutterotti, 1994; Lutterotti et al., 1997; Matthies et al.,1997; Von Dreele, 1997). With examples analysed so far, it could beshown that it is possible to determine (or to refine) simultaneously (orstep by step) crystal structure, texture, microstructural parameters andresidual stresses from a series of diffraction spectra, measured in dif-ferent sample orientations.In the conventional texture analysis many sample directions aremeasured on a few pole figures. With the new approach, a large numberof diffraction peaks are considered, using possibly diffraction spectrafor a few sample directions only. The OD resolution of both methodswill be similar for an equivalent amount of experimental textureinformation (Matthies et al., 1997), but the "Rietveld Texture Analysis"(RITA) is particularly efficient for complex spectra and TOF neutrondiffraction where the whole spectrum is recorded anyway.EXPERIMENTALIn this report we apply RITA to a sample of metamorphic quartzite.The quartzite is a clast from a metaconglomerate in Wildrose Canyon(Death Valley National Park, California). The texture has beendetermined previously by optical measurements of individual c-axes(L. Weiss, unpublished) and with conventional neutron diffractionTEXTURE ANALYSIS OF QUARTZITE 141methods (Helming et al., 1994). The c-axes form a crossed girdle as istypical of quartzites deformed at relatively low metamorphic grade(Sander, 1950).A cylindrical sample, 2 cm in diameter and 2 cm long was measured atthe neutron source of the JINR-Dubna (pulsed reactor IBR-2), usingthe technique described in detail by Walther et al. (1995). The 100mTOF-line has a flux of 106 neutrons/(s cm2) at the end of the neutronguide, and 7 detectors with an angular distance of 7.2. Two arrange-ments ofthe detectors and a rotation of the sample around one axis (by7.2 steps) lead to a complete coverage of the whole pole sphere. Eachspectrum contains about 2900 time channels. Spectra were recorded forthe regular 7.2 grid (650 sample directions) counting for 15 min at eachsample position.The arrangement ofthe detector banks at different 20 positions in thepresent experiment requires some non-trivial corrections. Different de-tectors see the same diffraction peak at different neutron wavelengths A.Hence, the spectra must be normalized with respect to the energy dis-tribution of the incoming neutrons, and the wavelength dependence ofthe scattering behavior must be considered (A3 correction). Addition-ally, the non-equivalent detector positions require the consideration ofthe Lorentz effect (1/sin O sin20). All these correction procedures,including the detector sensitivity, had to be included in the RITAalgorithm for the present project. The new Dubna TOF spectrometerwith all detectors at the same Bragg angle O will remarkably simplifythe application of RITA in the future.RITAThe RITA analysis follows the procedure characterized above and isdescribed in detail by Matthies et al. (1997). Experimental informationwas available in the d-range 0.5-3.5 containing 527 diffraction peaks.Though the technical run of the RITA program package was success-fully tested for the whole amount of the experimental data, a reducedvariant was chosen especially to demonstrate the characteristic power ofthe RITA concept that is able to work with a high degree of incom-pleteness ofpole figures. Ofthe 650 measured spectra less than half(301)were used. The corresponding sample directions cover the central142 S. MATTHIES et al.portion of the pole sphere up to a polar angle of about 40 only. Of the527 reflections only the (37) peaks having an intensity greater than 10%of the most intense reflection were used for the texture related calcu-lations. One trial with 62 reflections (5% threshold) did not showremarkable improvements of the resulting texture data. Typical TOFspectra for different sample directions are shown in Fig. 1. The changingpeak intensities are due to texture effects.The crystal structure was assumed to be that of c-quartz as reportedin Table I; this structure has been kept fixed initially and refined only inthe final stage. The a-quartz has a trigonal crystal symmetry (32). Thiscauses two complications: First, there are non-equivalent diffractionpeaks such as 101 and 011 with identical d-spacing but different inten-sities. However, these strictly overlapped peaks can properly be resolvedfor texture related questions, using intensity weights given by the crystal01.0 1.5 2.0d [AngstrSm]FIGURE Typical TOF spectra of the quartzite sample for different sample direc-tions (detectors). The solid lines give an impression on the fit quality using texturefactors.TEXTURE ANALYSIS OF QUARTZITE 143TABLE Values of the starting and refined crystal structurefor a-quartz (space group P3221) using RITA. Values in paren-theses are the estimated standard deviations as evaluated by theleast squares algorithmStarting structure Wyckoff 1982) Refineda () 4.91304 4.91393(5)c (A) 5.40463 5.40394(9)Si (x,x,O)x 0.465 0.46323(7)B (.A.2) 0 0.4(2)o (x,y,z)x 0.415 0.41047(5)y 0.272 0.26474(5)z 0.787 0.78471(3)B (A2) 0 0.7(1)structure (Baker et al., 1969). Secondly, quartz is enantiomorphic,existing in a fight- and left-handed form. Due to the centrosymmetry ofnormal diffraction effects, this cannot be resolved from the availabledata by any method. Therefore commonly it is assumed that both formshave the same frequency and orientation distribution. For texturerelated problems effectively a center of symmetry is added and theresulting crystal symmetry (Laue group) considered is 32/m, the samepoint group as for calcite.In a first step ofthe RITA scheme some instrumental parameters wererefined. In particular the incident intensity, some background param-eters, the instrumental resolution function and d-spacing relatederrors one set for each detector, apart from the intensity as a uniqueparameter. For these refinements the known quartz structure and anisotropic "powder" spectrum have been used. This spectrum with anassumed random OD can be appproximated by a weighted mean of allspectra for the different sample directions. After these preliminaryrefinements the texture is determined using an iterative procedure asfollows.Applying a modification of the Le Bail algorithm from the conven-tional Rietveld analysis originally used for structure factor extraction inab initio crystal structure determinations (Le Bail et al., 1988), theintensities can be determined for all reflections of the sample directiondependent spectra. Because the structure factors are known by the fixedcrystal structure, so-called texture factors can be determined for allreflections correspondingly. These texture factors, which are (initially as144 S. MATTHIES et al.a rule quite ambiguously) related to the corresponding pole figurevalues, serve then as INPUT data for the direct WlMV algorithm(Matthies and Vinel, 1982) in order to get a first ODF-approximation.From this ODF "recalculated" pole figures can be determined (com-patible in the sense ofprojections of one common OD) and will then beused as starting approximation of higher quality for the next Le Bailstep a.s.o.As in previous examples, this iterative process converges quiterapidly, i.e. the WIMV procedure leads always to better RP values(Matthies et al., 1988) characterizing the deviations between the"experimental" (in our case the Rietveld texture factors) and ODrecalculated pole figures. The inspection ofthe RP values can indicate alow texture sensitivity or poor compatibility between values for differ-ent reflections hkl, and these can then be omitted from the next iterationsteps. After few of these cycles (generally 2 or 3 are sufficient), also thecrystal structure and microstructure parameters were refined togetherwith the others.The results for the quartzite sample are illustrated by some selected"experimental" pole figures (Fig. 2), by pole figures recalculated fromthe ODF (Fig. 3), and by the ODF of the last WlMV step after con-vergence (Fig. 4). The ODF is displayed by a-sections that show mini-mal distortions ofcomponent forms due to the complicated orientationspace metrics (Matthies et al., 1990). Three iteration steps were suffi-cient in order to get RP values in the order of 5%. A comparisonof Figs. 2 and 3 gives a visual impression on the fit quality. Due tothe statistics of the experimental data and the transformation of the7.2-grid data into the 5-working grid of WlMV, the resulting OD ofFig. 4 is rather rugged. Therefore a Gaussian OD filtering procedurewith a width b 7.5 (Matthies and Vinel, 1994) was applied to this’noisy’ ODF preserving its texture index and order of RP values. Thecorresponding results shown in Figs. 5 and 6 are more comprehensible.For representations and post-processing some programs contained inthe BEARTEX package were used (Wenk et al., 1998).A specific property of the quartzite sample is its relatively high phonof 0.4. It means that about 40% of the crystallites of the sample arerandomly oriented. In such cases ari additional correction procedure of"remaining ghosts" in accordance with the WlMV concept is recom-mended and was already performed for the ODF shown in Fig. 4.TEXTURE ANALYSIS OF QUARTZITE 145FIGURE 2 Some of the used incomplete "experimental" pole figures constructed bytexture factors from the Le Bail step third iteration cycle. Equal area projection.Isolines: 0.6, 0.8., 1.0, 1.1, (0.1) 1.6. Dotted area below 1.103112 111110 011FIGURE 3 Recalculated pole figures from the WIMV-ODF (el. Fig. 4) thirditeration cycle. Representation like in Fig. 2.146 S. MATTHIES et al.3555FIGURE 4 The required ODF third iteration cycle. Sigma sections. Azimuth a,polar angle / of the g= {c,/3,7} variant of Euler angles; a=(c +7)/2. Equal areaprojection. Isolines: 0.5, (0.5) 3.5. Dotted area below 1.5 10 153545 50FIGURE 5 The 7.5-filtered ODF from Fig. 4. Same parameters of representation.TEXTURE ANALYSIS OF QUARTZITE 147113 121 210 103FIGURE 6 Recalculated pole figures using the ODF of Fig. 5 (cf. Fig. 3).From a given OD any pole figures can be calculated. Of particularinterest is the c-axis pole figure (Fig. 7(a)) that can be interpreted as the"sum" of all r-sections of the ODF shown in Fig. 5. It compares rea-sonably with that obtained by the conventional method of direct polefigure measurements (Fig. 7(b), Helming et al., 1994) with a broad andasymmetric girdle. The concentration ofsome c-axes near the peripheryin the Dubna data is not present in the conventional analysis and in theoptical grain by grain measurements. The difference could have severalreasons like the greater amount ofexperimental data in the DubnaTOFmeasurements, problems with the background correction in the con-ventional analysis, relatively poor statistics in the optical experiment, orpossibly systematic aberrations in the Dubna spectra.Concerning the crystal structure refinement it should be mentionedthat it is possible to obtain very accurate results also in the case of atextured sample, as it is evident from Table I. The large amount ofdatais responsible for the very low estimated standard deviations obtained.Some indicators of the correctness of the method are given by the rea-sonable Debye-Waller factor values (the B’s in Table I), that normally148 S. MATTHIES et al.(a)FIGURE 7 (a) The 001 pole figure from the OD of Fig. 5 (RITA). (b) The 001 polefigure from a conventional analysis (Helming et al., 1994).are considered as not sufficiently reliable from diffraction experiments.The employment ofneutron radiation has also enhanced the accuracy ofthe determination of the oxygen position. That is the present metho-dology demonstrates the possibility not only of a better refinement ofknown structures but also for the determination of partially unknownor incompletely refined substances.A real textured sample can be considered as lying in between an idealpowder sample with a random OD and a single crystal. Using thedemonstrated procedure in combination with ab initio methodologies itwould be possible to solve complicated structures for which singlecrystals are not available. In such cases commonly the powder methodsfail due to the impossibility to separate heavily overlapping reflectionsor real coincidences. A textured sample with sample direction depen-dent spectra can in principle provide sufficient information in order todiscriminate such reflections for a successful solution.CONCLUSIONSIt was demonstrated that the RITA concept can also be used to deter-mine textures of low symmetry materials such as quartz from TOFdiffraction spectra. The analysis can be carried out with only a fewmeasured spectra, making it efficient for neutron diffraction studieswhere access to the beam is limited. The agreement between measuredTEXTURE ANALYSIS OF QUARTZITE 149and recalculated pole figures assesses the quality of the experimentaldata and of the OD determination. RITA opens some new perspectivesfor crystal structure refinement.AcknowledgementsWe are appreciative for support of the RITA project through NSFgrant EAR 94-17580, a grant from IGPP Los Alamos and the Centerof Materials Science at Los Alamos (through D. Parkin). Access tothe Dubna texture goniometer is gratefully acknowledged.ReferencesBaker, D.W., Wenk, H.R. and Christie, J.M. (1969). J. Geol. 77, 144-172.Dollase, W.A. (1986). J. Appl. Cryst. 19, 267-272.Ferrari, M. and Lutterotti, L. (1994). J. Appl. Phys. 76, 7246-7255.Helming, K., Wenk, H.R., Choi, C.S. and Schaefer, W. (1994). In Textures ofGeologicalMaterials, H.J. Bunge, S. Siegesmund, W. Skrotzki and K. Weber (Eds.), DMG.p. 303.Le Bail, A., Durroy, H. and Fourquet, J.L. (1988). Mater. Res. Bull. 23, 447-452.Lutterotti, L., Matthies, S. and Wenk, H.R. (1997). J. Appl. Phys. 81, 594-600.Matthies, S. and Vinci, G.W. (1982). Phys. Status Solidi B 112, 111-120.Matthies, S., Wenk, H.R. and Vinel, G.V. (1988). J. AppL Cryst. 21, 285-304.Matthies, S., Helming, K. and Kunze, K. (1990). Phys. Status Solidi B 157, 71-83;489-507.Matthies, S. and Vinel, G.W. (1994). Mater. Sci. Forum 157-162, 1641-1646.Matthies, S., Lutterotti, L. and Wenk, H.R. (1997). J. Appl. Cryst. 30, 31-42.Popa, N.C. (1992). J. Appl. Cryst. 25, 611-616.Rietveld, H.M. (1967). Acta Cryst. 22, 151.Rietveld, H.M. (1969). J. Appl. Cryst. 2, 65-72.Sander, B. (1950). Einffihrung in die Geffigekunde der Geologischen K(irper. SpringerVerlag, Vienna.Von Dreele, R. (1997). J. AppL Cryst. 30, 517-525.Walter, K., Heinitz, J., Ullemeyer, K., Betzl, M. and Wenk, H.R. (1995). J. Appl. Cryst.28, 503-507.Wenk, H.-R., Matthies, S., Donovan, J. and Chateigner, D. (1998). J. Appl. Cryst.(in press).Wyckoff, R.W.G. (1982). Crystal Structures. 2nd Ed., Vol. 1. R.E. Krieger Pub., Florida.

TEXTURE ANALYSIS OF QUARTZITE BY WHOLE PATTERN DECONVOLUTION

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TEXTURE ANALYSIS OF QUARTZITE BY WHOLE PATTERN DECONVOLUTION

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