Use of an ellipsoid model for the determination of average crystallite shape and size in polycrystalline samples

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1 Use of an ellipsoid model for the determination of average crystallite shape and size in polycrystalline samples Tonci Balic Zunic Geological Institute, 0ster Voldgade 0, DK-350, Copenhagen K, Denmark Jesper Dohrup Haldor Topsie Research Laboratories, Nymillevej 55, DK-2800 Lyngby, Denmark (Received 26 May 998; accepted 5 December 998) A mathematical model for interpreting the anisotropical broadening of the powder diffraction lines by an average crystallite in the form of a triaxial ellipsoid is developed. The model covers satisfactorily a broad range of averaged crystallite shapes in polycrystalline samples of all crystal symmetries and provides simple formulas for use in powder pattern fitting routines. When r a, r b, r c are the principal ellipsoid radii, and c a, c b, c c direction cosines of diffraction vector related to the principal axes of ellipsoid, the average dimension of crystallites along the diffraction vector (D hkl ) is: D hkl = K/^c 2 /r 2 + c\lr l b + c 2 clr 2. The coefficient Khas the value 3/2 if D m is the volume average dimension of crystallites along the diffraction vector, or 4/3 in the case of the surface average dimension. The appropriate expression for use in whole pattern fitting routines is: b n h 2 + b 2 2k 2 + b 33 l 2 + 2b u hk + 2b l3 hl + 2b 2 3kl= K 2 L 2 hkld 2 m, where b tj are the elements of a second-rank symmetric tensor. Finding eigenvalues and vectors of tensor b gives dimensions and orientations of the principal ellipsoid radii in reciprocal lattice values. 999 International Centre for Diffraction Data. [S (99)0040-7] INTRODUCTION X-ray powder diffraction can be used to determine crystallite sizes ranging from several tens to a few thousands of Angstroms (in other words, for homogeneous diffraction domains between ~ nm and ~ /an). As derived by Scherrer (98) the broadening of a diffraction maximum is inversely proportional to crystallite size. The measured width of diffraction lines in a powder pattern allows calculation of the average crystallite sizes in polycrystalline samples, after a correction is made for other contributions (instrumental broadening and strain-broadening). In an indexed powder pattern, the calculated parameter is a vectorial property showing both a size in direct values [D(A)] and orientation (hid) related to the crystal lattice. This result provides not only a possibility to calculate an average diameter but also to determine the average shape of crystallites by combining measurements on a number of diffraction maxima. If shape parameters of crystallites are not accounted for in determining their size, one tacitly assumes an average spherical shape. Considering crystallite shape parameters, several specific cases have been treated in the powder diffraction literature. Langford and Wilson (978) have analyzed several simple polyhedral shapes with cubic symmetry, giving parameters for reflections with different hkl combinations and for various measures of line broadening. Langford and Louer (982) gave parameters for crystallites of cylindrical shape, and Vargas et al. (983) for crystallites in the form of a hexagonal prism. Grebille and Berar (985) described a method for calculating the diffraction profile function using the particle shape/size integration of Wilson (962) for crystallites of any presumed polyhedral shape. If ignored during Rietveld refinement, a strong deviation of crystallites from a spherical shape can severely influence the accuracy of fitting. On the contrary, proper modeling of the influence of crystallite shape on broadening not only improves the fitting, but also gives additional structural information. Apart from the work of Toraya (989) who used the results of Langford and Louer (982) for crystallites of cylindrical shape, the other approaches that incorporated anisotropic line broadening parameters in the full pattern fitting routines have concentrated on applying the parameters of a second-rank symmetric tensor similar to that used for the temperature factors, however, without a background in the analytically developed crystallite shape model. The latter mentioned approaches are works of Lutterotti and Scardi (990), Le Bail and Jouanneaux (997), and Solovyov (998). An ellipsoidal model would be a logical choice for a general and unified approach in determining average crystallite size and shape. Its continuous change from a sphere through an elongated or flattened rotation ellipsoid to a general triaxial shape approximates a full range of possible convex crystallite shapes. At the same time, they all can be handled by the same, relatively simple expressions derived for a general triaxial ellipsoid. While the determination of actual crystallite shapes (e.g., presence of crystal forms) falls primarily under the scope of electron microscopy, powder diffraction gives an average value over a range of shapes and dimensions present in a sample, and also in this respect the use of ellipsoid is fully justified. In this paper, the parameters of a broadening function due to ellipsoidally shaped crystallites are calculated, and an adequate expression for use in Rietveld refinement and other pattern-fitting methods is given. MATHEMATICAL MODEL A triaxial ellipsoid with principal axes directed along Cartesian axes x, y, z is described by Eq. () (r a, r b, r c being principal radii): 203 Powder Diffraction 4 (3), September /99/4(3)/203/5/$6.00 '999 JCPDS-ICDD 203

2 =. () An ellipsoid in general orientation is expressed by Eq. (2) employing also mixed terms and is characterized by six a^ coefficients: (2) Coefficients can be calculated from the cosines of angles between the principal axes of the ellipsoid, and coordinate axes (c ax represents cosine of angle between r a and x, etc.): a n~ r 2 - r a + c_l l + r l r l c bx c by rl r 2 - r c r c cy 2 23y (3a) (3b) (3c) (3d) Depending on the method of measurement, the size obtained from a powder diffraction diagram is expressed as a relation of the average squared diameter to the average diameter along the diffraction vector (if the diffraction maximum width is used) or simply as an average diameter along the diffraction vector (if a Fourier analysis of the diffraction profile is used) (Bertaut, 950). The two measures are usually called the volume average dimension and the surface average dimension, respectively. The volume average dimension of an ellipsoid, say in direction z, can be calculated as an integral over the squared dimension (T z ) along z, divided by the ellipsoid volume (Bertaut, 950): the volume of ellipsoid being: j TUxdy, V= 4 j7rr a r b r c. (4) Expressing Eq. (2) as a quadratic equation in z: C bx C bz (3e) a i3 z 2 + 2(a 2 i,y + 2a n xy - ] = 0, (5) bz CcyCcz r 2 ' (3f) and z\,z 2 being the roots, the double integral can be developed as /$(z\-z 2 ) 2 dxdy which gives: ^ T dx '33 dy. Integrating over x and y and introducing the value for Eq. (4) leads to a simple expression: or, grouping variables in the usual form for an elliptic equation: (6) 2 «23~ a 22«33, ^ «3 a 23 TH y +2 xy=. Substituting a 33 using Eq. (3c) gives finally: 3 (7) The ellipse's area can be calculated from the coefficients of this equation as: The surface average dimension (M) of the ellipsoid is obtained as: V M=-, (8) where A is the projection area of the ellipsoid along the diffraction vector. Let M z be the dimension along z. We are interested in the area of an ellipse in the x,y plane defined by the condition z\=z 2 for Eq. (5). In this case: Introducing now the values for a tj from Eq. (3) and using specific relations between direction cosines for a transformation of Cartesian axes leads to the final expression: «33 fp c 2 ~P- Introducing this equation and Eq. (4) into Eq. (8), the surface average dimension along z is obtained: 204 Powder Diffr, Vol. 4, No. 3, September 999 T. B. Zunic and J. Dohrup 204

3 (9) or, equivalent to Eq. (6): M =- (0) The ellipsoid radius in direction z is: be the ellipsoid equation in terms of a reciprocal lattice (x c, y c, z c are coordinates relative to reciprocal lattice periods a*, b*, c*). The results obtained previously in Eqs. (7) and (9) were based on integration in the Cartesian axial system relative to the orientation of the diffraction vector. While Eq. (2) gives ellipsoid equation relative to the latter system, we would like to express coefficients a t j in terms of reciprocal axes and coefficients b t j. For this derivation we introduce in Eq. (): x c =p t x+p 2 y + p 3 z, cl t and our results show that the average dimension of the ellipsoid corresponds to \ diameter in the same direction or \r in the case of volume average, while for the case of surface average it corresponds to \ diameter or \r. It is, of course, not surprising that the corresponding values were obtained also for the sphere (Smith, 972), which is a special case of ellipsoid. The average sizes of an ellipsoid in directions of its principal axes are therefore: L b =\r b, M a =\r a, M b =\r h, M c =lr c. Introducing these values in Eqs. (7) and (9) we obtain: L = c n, M=- V c az c bz c cz This relation proves what intuitively Ml + could Ml + be expected, that the average size of ellipsoid follows an equivalent Ml ellipsoidal function, such as Eq. (2). Only, in calculating a^ coefficients, L a, L h, L c, or M a, M b, M c should be used instead of r a, r b, r c in Eqs. (3). (x,y,z being coordinates relative to the Cartesian system). According to Eqs. (6) and (0), we are interested only in a 33, and grouping coefficients with z 2 we obtain: + 2b 23 q 3 r 3. (2) Because z is directed along the diffraction vector, p 3 a*, q 3 b*, r 3 c* are components of vector of unit length parallel to this direction (according to the basis transformation rules). Because the diffraction vector is parallel to reciprocal vector: with the length \ld hkl, we find that: P3 = d hkt h, q 3 = d hkl k, r 3 = d hk,l. (3) Finally, combining Eq. (6) or (0) with Eqs. (2) and (3) we obtain (writing L hkl instead of L z, and M hkl instead of M z ): or ' 6 (4) 9M 2 hkld 2 hki (5) Equations (4) and (5) represent properly normalized expressions for anisotropic line broadening due to the volume or surface average crystallite sizes, by coefficients being related to reciprocal axes, and referred to the principal radii r a, r b, r c of the shape ellipsoid. As was shown earlier, if the normalizing factors f or ^ are excluded from the right side of the equation, the coefficients refer to the average dimensions (L a, L b, L c or M a, M h, M c ) along the principal directions. APPLICATION IN THE WHOLE-PATTERN FITTING An expression for a general orientation of an ellipsoid with respect to crystallographic axes (triclinic case) can be formulated in the following way. Let (ID CONCLUSION It is possible to obtain the parameters (principal radii and their orientation) for an average ellipsoidal crystallite from an analysis of the powder diffraction pattern. Equations (4) and (5) give the relation between the six by elements of a second-rank symmetric tensor describing the ellipsoid in relation to reciprocal lattice, and its average dimensions along 205 Powder Diffr, Vol. 4, No. 3, September 999 Ellipsoid model for polycrystalline samples 205

4 a diffraction vector, and are directly applicable in the full pattern-fitting methods. The b t j elements would then appear as refinable parameters in an appropriate Rietveld-refinement program, and bear resemblance to parameters of anisotropic displacement factors. There have been applications for treatment of anisotropic line broadening in powder diffraction inspired by the latter analogy (Lutterotti and Scardi, 990; Le Bail and Jouanneaux, 997). The results presented here are derived directly from the assumption of an average ellipsoidal particle shape and confirm that the used tensor elements (bij) really describe an average ellipsoidal crystallite shape, showing at the same time how the results can be interpreted in terms of the ellipsoid size and orientation. In calculating the values of the principal ellipsoid radii and their orientation from b^ elements, one must remember that they are related to the reciprocal axes. General principles for determining eigenvalues and eigenvectors are described, e.g., by Giacovazzo (992). Because the shape of the ellipsoid should be expected to conform to the symmetry of the crystal class, the same restrictions to bjj should apply as, e.g., for the optical indicatrix, or the anisotropic displacement factors of atoms in positions with the maximal site symmetry for a given system. In the monoclinic system therefore > 2 = &23 = 0. For the orthorhombic system all three & 2, b n and > 2 3 should be zero, and the rest are directly related to crystallographic parameters: anisotropic crystallite size function was tied to diffraction profiles full-width at half maximum, so Eq. (4) would be applicable for this case. One can easily see that their formula () generally corresponds to the Eq. (4). While the absolute tensor elements (i/,y) are used instead of relative by what involved the inclusion of reciprocal periods in formula, and the factor f is missing, the main difference is in that the ellipsoid expression is made directly proportional to the diffraction maximum width parameter. As the latter is inversely proportional to the average size, this result is not in contradiction with Eq. (4). However, such parameters were included directly in Caglioti's formula (Caglioti et al, 958) which is not fully justifiable. From Scherrer's formula (Scherrer, 98) the width of diffraction maximum due to crystallite size is inversely proportional to cos 6, while in Caglioti's formula, the width is related to powers of tan 8. A fully justifiable treatment of ellipsoidal crystallite size effect on diffraction maxima widths is presented by Solovyov (998). Here, the width parameter due to crystallite size is made both proportional to ellipsoidal expression and inversely proportional to cos 6, and then added to the width obtained from Cagioti's formula that models the instrumental contribution, to obtain the FWHM. It should be noted that the factor f is missing in the size contribution, so the fc, ; parameters are again related to the average sizes, and not the ellipsoid radii along the principal axes (note also that the tensor elements of Solovyov are written in an unusual way in that the "mixed" b ti include the factor 2). ACKNOWLEDGMENTS In tetragonal and hexagonal case b n = b 2 2, for hexagonal additionally b n =\b xx. For a cubic case the anisotropy of the line broadening cannot be treated by this model, and only an average sphere radius can be obtained from b u ( = b 2 2 Commenting on the existing applications in the light of the present results, one can conclude that for the method described by Lutterotti and Scardi (990), Eq. (5) would be applicable, because they used the Fourier analysis approach in treating the diffraction maxima broadening. Although that conclusion was not stated in the paper, it seems that the authors used the expression for a probability ellipsoid in developing their formula (9) in which the anisotropic function of broadening due to crystallite size was modeled. The formula does not correspond to the present Eq. (5) because the ellipsoidal expression was made directly proportional to the squared average size, and not inversely proportional as required by Eq. (5). It should be noted that the main advantage of the Lutterotti and Scardi's method is the integration of both anisotropic size and strain distribution function based on Warren and Averbach method (Warren, 969) and the work of Nandi et al. (984) in a Rietveld-refinement. Therefore, their approach would be preferred in the case of a sizable strain contribution, but the use of the present Eq. (5) is suggested for modeling the crystallite size contributions. In the approach of Le Bail and Jouanneaux (997), the The first author is grateful to L. A. Solovyov and Dr. S. D. Kirik from Institute of Chemistry and Chemical Technology in Krasnoyarsk for an inspiring discussion during the 5th EPDIC conference in Parma. We thank C. Harm Sarantaris for typing the manuscript. Bertaut, E. F. (950). "Raies de Debye-Scherrer et Repartition des Dimensions des Domaines de Bragg dans les Poudres Polycristallines," Acta Crystallogr. 3, 4-8. Caglioti, G., Paoletti, A., and Ricci, F. P. (958). "Choice of collimators for a crystal spectrometer for neutron diffraction," Nucl. Instrum. 3, Giacovazzo, C. (992). Fundamentals of Crystallography (International Union of Crystallography, Oxford University Press, New York). Grebille, D., and Berar, J.-F. (985). "Calculation of diffraction line profiles in the case of a major size effect: Application to boehmite AOOH," J. Appl. 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