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ameter (d): l = 20Å, d = 150Å (dotted) and l = 40Å, d = 180Å (solid). The comparison shows the different effect on the reflections width according to the arm orientations and dimensions. Intensity 7000 6000 5000 4000 3000 2000 1000 0 Fraction (%) 0 0.990 4 5 6 -1000 10 20 30 40 50 90 60 30 2θ (deg) diameters (nm) Fig. 4. Iron oxide XRPD experimental dataset and relative calculated pattern. Background and difference profiles are included. Size & Strain distribution are shown in the inset. GoF = 2.43, Rwp = 6.79. 20000 15000 10000 5000 0 20000 15000 10000 5000 0 20000 15000 10000 5000 0 111 111 111 220 10 20 2θ (deg) 30 220 10 20 2θ (deg) 30 220 311 222 311 222 311 222 400 400 400 422 511 422 511 422 511 440 440 440 10 20 2θ (deg) 30 1.010 1.008 1.006 1.004 1.002 1.000 0.998 0.996 0.994 0.992 Strain TETRA110 620 533 444 TETRA111 620 533 444 TETRA100 620 533 444 Fig. 5. Calculated patterns for TPs with arms oriented The increase of the I440/I311 ratio as well as the (111) and (220) peak shift to higher angles, experimentally found, can be explained by a coexistence of TPs oriented both along the [111] and the [110], ruling out the [100] one. Tetrapods had been found with arms preferentially oriented along the [111] directions by transmission electron microscopy (TEM). Therefore, XRPD analysis confirmed TEM findings but adding a precious information. PERSPECTIVES Nowadays, the size and shape evolution of colloidal nanocrystals, driven by both thermodynamic parameters and kinetically-limited growth processes, is attempting to assemble different materials together in heterostructures (hybrid materials) playing with crystal miscibility, interfacial strain, and face-selective reactivity. The structural and compositional control of the resulting architecture is fundamental. Therefore powerful structural investigation tools are highly required. It is worth noting that fairly large amounts of material are produced and have to be controlled. In this respect, powder diffraction analysis has been showing to grant a rapid and cheap answer in terms of size, strain, shape and structure characterization. Relative data treatment methodologies are in continuous development and improvement to guarantee the most complete and reliable information. Critical aspects are the treatment of exotic shapes as well as the occurrence of defects. Further work is in progress to extend and complete the present theoretical approach. The most important parts of software – as a program suite – are to be made available under GPL (Gnu Public License). REFERENCES * On leave from the Istituto di Cristallografia -CNR- Bari, Italy [1] R. Lamber, S. Wetjen, I. Jaeger, Phys. Rev. B 51 (1995) 10968. [2] http://cst-www.nrl.navy.mil/lattice/prototype.html [3] F.L. Ten Eyck, Acta Cryst. A33 (1977) 486. [4] A. Cervellino, C. Giannini, A. Guagliardi, J. Appl. Cryst. 36 (2003) 1148. [5] A. Cervellino, C. Giannini, A. Guagliardi, J. Comp. Chem. 27 (2006) 998. [6] A. Cervellino, C. Giannini, A. Guagliardi, CPD Newsletter 30 (2005). [7] L. Cademartiri, E. Montanari, G. Calestani, A. Migliori, A. Guagliardi, G.A. Ozin, J. Amer. Chem. Soc. (2006) in press. [8] A. Cervellino, C. Giannini, A. Guagliardi, M. Ladisa, Phys. Rev. B 72 (2005) 035412. along the [110], [111] and [100] orientations. [9] P.D. Cozzoli, E. Snoeck, M.A. Garcia, C. Giannini, A. Guagliardi, A. Cervellino, F. Gozzo, A. Hernando, K. Achterhold, F.G. Parak, R. Cingolani, L. Manna, Nanoletters 6 (2006) 1966. 6
PRACTICAL ASPECTS OF PARTIAL COHER- ENCE OF NANOCRYSTALLINE DOMAINS David Rafaja, Volker Klemm and Milan Dopita * Institute of Materials Science, TU Bergakademie Freiberg, Gustav-Zeuner-Str. 5, D-09599 Freiberg, Germany, E-mail: rafaja@ww.tu-freiberg.de ABSTRACT Partial coherence of crystallites is often observed in supra-hard nanocomposites having a strong texture as an effect causing the diffraction line “narrowing”. The degree of the partial coherence depends on the crystallite size, on the local preferred orientation of crystallites and on the size of the diffraction vector. Consequently, the microstructure parameters influencing the partial coherence of crystallites can be determined from the dependence of the diffraction line broadening on the size of the diffraction vector. In some thermodynamically unstable systems, like in Ti-Al-Si-N and Cr- Al-Si-N thin films, the strongly correlated orientation of adjacent crystallites that is a necessary condition for their partial coherence can be understood as an indicator of the spinodal decomposition. INTRODUCTION Nanocrystalline materials are employed in many industrial applications, e.g. in materials for catalytic converters, in production of self cleaning surfaces, in magnetic nanostructures or in supra-hard nanocomposites. Nano-sized structures combine advantages of excellent chemical and physical properties that are tailored by modification of the material’s microstructure. For these reasons, the microstructure analysis on nanocrystalline materials and nanocomposites is a very important issue for materials science. Obligatory methods for the microstructure analysis on nano-sized structures are the transmission electron microscopy (TEM) and the highresolution transmission electron microscopy (HRTEM). The X-ray diffraction (XRD) offers still some benefits, the non-destructivity, a fast and easy sample preparation, and excellent statistics being few of them. Thus, XRD is still regarded as a very important experimental method for the microstructure analysis on nanocrystalline materials and nanocomposites [1 – 9]. X-RAY SCATTERING ON NANOCRYSTALS Within the kinematical diffraction theory, the diffracting volume of a material under study is assumed to consist of individual coherently diffracting domains called crystallites. The X-ray scattering on atomic structures is usually described by the amplitude of the scattered wave, E, which is equal to the Fourier transformation (FT) of the electron density ( r ) r ρ : r r r r r r E( q) = ρ( r ) exp( iq ⋅ r ) dr = FT[ ρ( r ) ] , (1) ∫ V where q r is the diffraction vector and r r the positions of the scattering centres. Electron density of an infinite crystal can be described as a convolution of the electron density of the unit cell cell( ) function L( r ) r , see, e.g. [10]: r r r ( r ) = ρ ( r ) ∗ L( r ) where ∞ cell r ρ with the lattice ρ , (2) ( ) ∑ ∞ r r r = δ r − rn 2, n ). (3) 1,n 3 n1, n2, n3= −∞ In Eq. (3), δ means the Dirac delta function, n1, n2 and n3 are integers. Analogously, the electron density of an L r ( individual crystallite ρ Xtal( ) can be described by convolution the electron density of the unit cell ( r ) r ρ r r with the lattice function L (r) r multiplied by a shape function ( r ) r Ω : r r r r ρ Xtal( r ) = ρcell( r ) ∗[ Ω( r ) ⋅ L(r)] (4) ( r ) r Ω is equal to unity within the crystallite and zero outside. The amplitude of the wave scattered by the crystallite is then given by the Fourier transformation of the electron density (r ) r ρ Xtal : r r r r r E( q) = FT[ ρXtal ( r ) ] = FT{ ρcell( r) ∗ [ Ω( r ) ⋅ L( r ) ] } = r r r (5) = FT[ ρcell( r ) ] ⋅ FT[ Ω( r ) ⋅ L( r ) ] The last term in Eq. (5), FT( Ω ⋅ L) , describes the shape of three-dimensionally periodic reciprocal lattice points that are broadened due to the finite crystallite size. The Fourier transformation of the electron density of a unit cell is equal to the structure factor, which in the first approximation can be treated as a constant within the size of individual reciprocal lattice points: [ ( ) ] ( ) ( l) r r FT ρ cell r = F q ≈ F hk (6) Within one reciprocal lattice point, the amplitude of the wave scattered by a finite crystallite is consequently given by a product of the structure factor of the unit cell and the Fourier transformation of the shape factor: r r E ( q) = F( hkl) ⋅ FT[ Ω(r)] (7) The intensity scattered by the diffracting volume of material is proportional to the modulus of the sum of the amplitudes of waves scattered by individual crystallites (domains): N N r I( q) ∝ * ∑ [ Fhkl ⋅ FT( Ω) ] n ⋅∑ [ Fhkl ⋅ FT( Ω) ] n (8) n= 1 n= 1 Asterisk in Eq. (8) denotes the complex conjugate. The summation is performed over all crystallites within the irradiated volume of the sample. The product of the sums in Eq. (8) can easily be rewritten into the following form, which is more suitable for explaining the coherence phenomena in nanocrystalline materials: N r I( q) ∝ ∑[ F ⋅ ( Ω) ] * hkl FT n ⋅[ Fhkl ⋅ FT( Ω) ] n + n= 1 (9) N −1 N −m + 2∑∑[ F ⋅ ( Ω) ] * hkl FT n ⋅[ Fhkl ⋅ FT( Ω) ] n+ m m= 1 n= 1 The first term in Eq. (9) describes the scattering of Xrays on identical domains, i.e. the interference of waves scattered by the same crystallites. The second term describes the interference of waves scattered by different cell 7
Page 1 and 2: IN THIS ISSUE Microstructur
Page 3 and 4: THE IUCR COMMISSION ON POWDER DIFFR
Page 5: Once the sampled interatomic distan
Page 9 and 10: E1 = FT[ Ω( q + Δq) ] = r r 4π
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Page 25 and 26: Čížek, I. Procházka, R. K. Isla
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