Copyright(c)JCPDS-International Centre for Diffraction Data 2000,Advances in X-ray Analysis,Vol ISSN

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1 Copyright(c)JCPDS-International Centre for Diffraction Data 2000,Advances in X-ray Analysis,Vol AMBIGUITIES OF MICRO AND NAN0 STRUCTURAL DETERMINATION Giovanni Berti Dipartimento di Scienze della Terra Via S. Maria 53,56126 Pisa, Italy Abstract Microstructure determinations suffer from systematic experimental effects which can alter measurements on the physical, mineralogical and crystallographic characteristics of the crystalline material under investigation. Apart from such systematic artefacts, other effects are present that are related to diffraction physics and the geometrical arrangement of the specimen s lattice. The present paper focuses firstly on these latter effects and considers the crystalline lattices that may generate ambiguous diffraction patterns (i.e., different cells producing patterns with the same peak positions). Secondly, the author calculates the crystallite sizes and shapes that can modulate the reciprocal lattice node. This serves to demonstrate that, not only geometric ambiguities are admissible, but other, physical and crystallographic ones as well. In the specific case in question, ambiguity is revealed between the simulated difl%actograms for cubic F and orthorhombic I crystallites. The basic crystallographic computation of structural factors is carried out using potassium chloride as an example. 1. Introduction Some rather uncommon lattices possess the peculiar property of yielding geometrical ambiguities in the peak positions of diffraction patterns. By considering the diffraction pattern intensity modulation, such lattices are useful for underscoring that in micro and nano - structural determination there is real a risk of introducing ambiguities. In this case, such ambiguities are related to diffraction physics, and not to the lattice geometry alone. The risk of ambiguous interpretation of the diffraction pattern stems from the combinations of lattice symmetry and crystallite size and shape. This risk is related to the modelling computation not to the presence of systematic experimental aberrations. The consistent, inevitable presence of these systematic aberrations increases this risk. This paper focuses on the very simple models for micro/nano structures (crystallites size and shape) whose application can lead to ambiguous interpretations of diffraction patterns. It is commonly accepted that the ambiguities arising from geometrical lattice features can be resolved by resorting to evaluation of the peak intensity (or the intensity distribution). In fact, the structural and multiplicity factors alone should be able to guarantee resolution of possible ambiguous determinations by suitably modulating the lattice nodes in the reciprocal space. But what about the modulation due to the micro- and nano-structure of a real crystal (crystallite size and shape)? In order to simplify our approach, this paper outlines a sort of reductio ad absurdurn, which serves to demonstrate that suitably computed crystallite sizes and shapes are able to render an orthorhombic I configuration admissible for a KC1 powder sample, although such a structure is clearly unrealistic for KCl, which is, on the contrary, a face-centred cubic lattice.

2 This document was presented at the Denver X-ray Conference (DXC) on Applications of X-ray Analysis. Sponsored by the International Centre for Diffraction Data (ICDD). This document is provided by ICDD in cooperation with the authors and presenters of the DXC for the express purpose of educating the scientific community. All copyrights for the document are retained by ICDD. Usage is restricted for the purposes of education and scientific research. DXC Website ICDD Website -

3 Copyright(c)JCPDS-International Centre for Diffraction Data 2000,Advances in X-ray Analysis,Vol The diffraction peak intensity and geometric ambiguities There are some cases in which ambiguities are due to causes that are, rather than accidental, geometrical in nature. These occur because two or more lattices can yield powder difiaction patterns whose lines are at the same angular positions. Geometric ambiguities occur in lattices with symmetries of order greater than orthorhombic, or when particular relations exist between the parameters of triclinic, monoclinic and orthorhombic lattices. The present study addresses the geometric ambiguity existing between a face-centred cubic lattice (F cubic) and a body-centred orthorhombic one (orthorhombic I). The transformation that links these two lattices is I = MI l F, where MI is the transformation matrix: L -l/6 0 -l/6 I MI= l/2 0 -l/2 (1) When faced with such a problem of geometric ambiguity, the value sets of the reciprocal lattice vectors are equal and, consequently, the reflections of the two distinct difiaction patterns are in the same positions [ 11. More details have been reported by De Palo [2]. Therefore, the diffraction patterns from I and F are distinguishable in principle because: + the number of planes contributing to a given reflection in one or the other case may not be the same; + the structural factors, linked to electron densities, may also be different. In x-ray powder difficactometry, the combined contributions of terms determining the intensity values are: the structure factor (F), multiplicity (m), polarisation (p), the Lorentz effect (L), temperature (T), transmission (A) and the finite dimensions and shape of the diffraction coherent domains (a). If we use the intensity expression: I=mLp 1~1~~ (2) in which the crystallite dimensions have not been accounted for, we obtain percentage intensity differences of about 30%, and the diffraction patterns coming from I and F are therefore distinguishable, although they present peaks at the same angular positions. 3. The simulated microstructure of a real crystal and physical ambiguities A real crystal can be represented schematically as a mosaic of crystalline blocks of approximate dimensions 10S5 cm, shifted one from the other by an angle of the order of a fraction of a minute [3]. The interfaces between adjacent blocks can be considered extensive defects in the surface of the sample and are termed grain boundary or crystallite limits, while the holes between crystals give rise to corner-edges - dislocations that connect highly deformed portions of the lattice. Interference between diffracted X-rays occurs only within the interior of each single block and, due to the loss of cohesion between the diffracted waves of the various blocks, the difiaction intensity of the whole crystal is equal precisely to the sum of the single diffraction intensities. The real crystals generally studied in laboratories always present, apart from a non-ideal mosaic structure, also various other defects, which may be punctiform (vacancies, interstitial atoms), linear (dislocation), planar (stacking faults) or volumetric (small precipitates). Such lattice distortions cause modification of the mean dimensions of the element cell by altering the form and volume of the reciprocal lattice nodes.

4 Copyright(c)JCPDS-International Centre for Diffraction Data 2000,Advances in X-ray Analysis,Vol On the basis of the foregoing, it is impossible to state that the crystallite orientation is found at the exact Bragg angle for a given family of planes. Owing to its mosaic structure and to lattice distortions and defects, each reciprocal lattice node will have finite volume and be in contact with the reflection sphere for a well-defined range of angles. This implies that the microstructure modulates the intensity distribution around the lattice node and the structure factor as well. 4. Application of the Warren-Averbach theory to simulation of ambiguous diffraction patterns According to the Warren-Averbach method the dimensions of a crystallite (and even any lattice strain contained in it) will influence the shape of the reflection coming from the plane family perpendicular to a given direction and therefore, crystallite size and shape influence also the Fourier coefficient representing that peak. In the following, we shall apply the Warren-Averbach theory, not to its usual aim of determining crystallite dimensions from the intensity of a diffractogram, but instead to the inverse procedure, which serves to demonstrate (as a reductio to absursum that in certain circumstances the orthorhombic I configuration is admissible for KCl. In order to reach this target, we assume the dimensions to be known and then analyse their effects on the peaks. Crystallites generally have irregular shapes. However, for the sake of simplicity, we shall assume them to be regular by the following considerations:. crystallites with cubic symmetry can be reasonably simulated at a first approximation by a spherical external shape, that is, if we account for the isotropic behaviour demonstrated by cubic substances with regard to their various physical properties;. crystallites with orthorhombic symmetry, instead, can be modelled by cylindrical shapes, as they represent a fair compromise between the symmetry and the significance level of the calculations in question. Concerning the crystallite dimensions, for spheres, the apparent sizes are the same in all directions, and the number of cells per column is therefore constant. For cylindrical crystallites, on the other hand, we adopt the equation set forth by J.I. Langford,(l992) [4], which expresses the relation between the apparent size p, the cylinder s diameter D and its height H: cosp + 4sinp - B=( - - (3) H?iJD 1 where <p is the angle formed by direction (l&l) with the cylinder axis. Now, expressing angle <p as a function of indices (hkl), and introducing opportune values for the parameters involved, we calculate the power radiated from a diffraction plane according to the Warren-Averbach equation [5]. The P(h) values displayed in Figures 1 and 2 represent the radiated power and are obtained from the numerical expression (4) used by the Mathcad package [6]. These figures show how the intensity varies with increasing the number N. According to the Warren-Averbach theory N is the average number of cells perpendicular to the reflecting planes; h is the discrete variable of the power distribution in expression (4) and n is an integer number, the sum index and represents the couple of cells separated by a distance proportional to n. [5]. The numerical values of N, h and n have been reported here in the same format as required by Mathcad. The expression (4 ) A(n) represents the Fourier coefficients which are sensitive to the number of cell in the column (the crystallite dimension). The Matchad electronic sheet used for the power calculation is reported here following.

5 Copyright(c)JCPDS-International Centre for Diffraction Data 2000,Advances in X-ray Analysis,Vol h : = 3.5, n := N :=5 P(h) := ( 8*101g ~~l+~~s~8.3~~)2~.~(*(~~.cos(2.~.~h~i sin L n 180 (4) A(n) := 2d+ 4-N InI + 3a2 *e 2-g (4 ) 4*1025 I P(h)2=lo25 - h : = 3.5, n := N :=loo Figure 1: peak pattern vs. number of cells 4 h P(h) - S(h) -- Figure 2: peak h intensity vs. number of cells In essence, the equation (4) describes a difli-action maximum developed through the Fourier series. For the cubic case, the crystallites are assumed to have a spherical shape, and the apparent sizes are therefore independent of the direction considered. Assuming an apparent size equal to 10 unit cells, by applying equation (12), we obtain the percentage intensities presented in Table 1. In the orthorhombic case, instead, the crystallites are assumed to be cylindrical in shape, and the choice of ratio D/J3 can be made on the basis of the following considerations: + the first reflection is (002), which yields information on the apparent dimensions of the direction perpendicular to the xy plane, i.e., along z; + the second reflection comes from the (020) plane and is therefore linked to the dimensions along the plane xy.

6 Copyright(c)JCPDS-International Centre for Diffraction Data 2000,Advances in X-ray Analysis,Vol H=lO 1: /2-+ Y Figure 3: shape of cylindrical crystallite in the orthorhombic I case Peak n hm cubic F 1% cubic F hkl orth I 1% orth I <l Table 1: percent intensities obtained for a face-centred cube and calculated for an ort. I with D/H= m; \ \ 75 '\ \ 0; I% 50 \ t \ j I ~;..-~.--.~--~-..~.-;----i.._. ~ y--*+-..m ;ealc : Figure 4: Percent intensities relative to the two cases examined: F-Cubic (dotted line with black diamonds), I-Orth (solid line with grey squares).

7 Copyright(c)JCPDS-International Centre for Diffraction Data 2000,Advances in X-ray Analysis,Vol By attributing a fictitious dimension of 10 along the z axis in order to determine a percentage intensity of the second peak comparable to that in the cubic case F, we must set p equal to 19 unit cells in the xy plane. From equation (1 l), it therefore holds that D=23 unit cells, and that the crystalline dimensions are D/H=24/10= 2.4, as illustrated in figure 3. Table 1 reports the percentage intensities obtained for the 14 peaks of KC1 under the assumption that it is orthorhombic I with cylindrical crystallites having D/H=2.4. As Figure 4 clearly shows, there are only small discrepancies between the percentage intensities relative to the two cases examined. Conclusions KCl, amongst various other samples, can be used to demonstrate that, when other unfavourable circumstances exist, some ambiguities in pattern interpretation depend upon Dieaction Physics. Potassium chloride, F-cubic and I-orthorhombic lattices have been considered here, as they can render extreme situations admissible. To this end, the similarity between the F-cubic and the I-orthorhombic pattern of KC1 appears emblematic, because the orthorhombic structure is clearly unrealistic for KCl. Figure 4 shows that the difiaction pattern of the F-Cubic lattice is compatible with the one from an I-Orthorhombic arrangement (at far below 10% discrepancy), which is the random fluctuation limit). Due to the geometric ambiguities [S], such compatibility is allowed only for the position of the maximum. The structure factors and multiplicity which characterise the two distinct structures modulate the intensity in distinct ways and should, in principle, be capable of removing the ambiguity. The compatibility shown in Figure 4 is can be re-introduced by simply considering the size and shape of the crystallite as a further intensity-modulation item. Acknowledgements The author wishes to thank Dr. Fabrizia Toncelli and Dr. Sabina Citi, graduate students in the Course Tecniche Diffrattometriche (PI8F224, 1998 Dept. of Earth Science), for their assistance in preparing this work References [l] M&hell A.D, Santoro A., (1975), Geometrical Ambiguities in the Indexing of Powder Patterns, J. Appl. Cryst. 8, [2] De Palo S., Degree Thesis in Geology - University of Pisa, February, [3] Giacovazzo C., Fundamentals of Crystallography, Oxford University Press,1992. [4] Langford J.I., The Use of the Voigt Function in Dete rmining Microstructural Properties from Diffraction Data by means of Pattern Decomposition. NIST Spec. Pub. 846,1 1 O-126. [5] Warren B.E., X-ray Diffraction, Addison Wesley, London, [6] Mathcad 7. User s Guide, Math Soft, 1997.