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Ewald's Sphere/Problem Ewald's Sphere/Problem 3.7 Studentproject/Molecular and Solid-State Physics Lisa Marx 0831292 15.01.2011, Graz Ewald's Sphere/Problem 3.7 Lisa Marx 0831292 Inhaltsverzeichnis 1 General Information 3 1.1 Ewald Sphere of Diffraction . 3 1.2 Ewald Construction . 4 2 Problem 3.7 / Ewald's sphere 5 2.1 Problem declaration . 5 2.2 Solution of Problem 3.7 . 5 2.3 Ewald Construction . 6 3 C++-Anhang 8 3.1 Quellcode . 8 3.2 Output-window . 9 Seite 2 Ewald's Sphere/Problem 3.7 Lisa Marx 0831292 1 General Information 1.1 Ewald Sphere of Diffraction Diffraction, which mathematically corresponds to a Fourier transform, results in spots (reflections) at well-defined positions. Each set of parallel lattice planes is represented by spots which have a distance of 1/d (d: interplanar spacing) from the origin and which are perpendicular to the reflecting set of lattice plane. The two basic lattice planes (blue lines) of the two-dimensional rectangular lattice shown below are transformed into two sets of spots (blue). The diagonals of the basic lattice (green lines) have a smaller interplanar distance and therefore cause spots that are farther away from the origin than those of the basic lattice. The complete set of all possible reflections of a crystal constitutes its reciprocal lattice. The diffraction event can be described in reciprocal space by the Ewald sphere construction (figure 1 below). A sphere with radius λ is drawn through the origin of the reciprocal lattice. Now, for each reciprocal lattice point that is located on the Ewald sphere of reflection, the Bragg condition is satisfied and diffraction arises. The Ewald sphere is a geometric construct used in electron, neutron, and X-ray crystallography which demonstrates the relationship between: the wavelength of the incident and diffracted x-ray beams, the diffraction angle for a given reflection, the reciprocal lattice of the crystal It was conceived by Paul Peter Ewald, a German physicist and crystallographer. Ewald himself spoke of the sphere of reflection.[1] Ewald's sphere can be used to find the maximum resolution available for a given x-ray wavelength and the unit cell dimensions. It is often simplified to the two-dimensional Ewald's¨ circlemmodel or may be Seite 3 Ewald's Sphere/Problem 3.7 Lisa Marx 0831292 referred to as the Ewald sphere. 1.2 Ewald Construction A crystal can be described as a lattice of points of equal symmetry. The requirement for constructive interference in a diffraction experiment means that in momentum or reciprocal space the values of momentum transfer where constructive interference occurs also form a lattice (the reciprocal lattice). For example, the reciprocal lattice of a simple cubic real-space lattice is also a simple cubic structure. The aim of the Ewald sphere is to determine which lattice planes (represented by the grid points on the reciprocal lattice) will result in a diffracted signal for a given wavelength, , of incident radiation. 2p The incident plane wave falling on the crystal has a wave vector Ki whose length is λ . The diffracted plane wave has a wave vector Kf . If no energy is gained or lost in the diffraction process (it is elastic) then Kf has the same length as Ki. The difference between the wave-vectors of diffracted and incident wave is defined as scattering vector ∆K =Kf - Ki. Since Ki and Kf have the same length the scattering 2p vector must lie on the surface of a sphere of radius λ . This sphere is called the Ewald sphere. The reciprocal lattice points are the values of momentum transfer where the Bragg diffraction condition is satisfied and for diffraction to occur the scattering vector must be equal to a reciprocal lattice vector. Geometrically this means that if the origin of reciprocal space is placed at the tip of Ki then diffraction will occur only for reciprocal lattice points that lie on the surface of the Ewald sphere. Abbildung 1.1: Ewald construction Seite 4 Ewald's Sphere/Problem 3.7 Lisa Marx 0831292 2 Problem 3.7 / Ewald's sphere 2.1 Problem declaration X-rays with an energy of 8 keV are used to analyze a simple orthorhombic lattice (a=0.6nm, b=0.8nm, c=0.4nm). The x-rays propagate in the (001) plane and also the detector scans in the (001) plane. Determine the number and the scattering angle of the scattered beams. The problem can be solved in two dimensions, i.e. Ewald's sphere becomes a circle. 2.2 Solution of Problem 3.7 • X-Rays: E = 8keV k c −34 8 E = h ∗ ν = h ∗ c ∗ λ = h ∗ 2∗π h = 6; 62 ∗ 10 c = 3 ∗ 10 ~ elastische Streuung ~ E∗2π 8∗103∗2π 10 −1 ) jkij = jkf j = h∗c = 6;62∗10−34∗3∗108 = 4; 05 ∗ 10 m Abbildung 2.1: Orthorombic lattice N N N • 0 0 1 − P lane ) m; n; p = [ n ; k ; l ) = (1; 1; 1) ) parallel zu x-y-Ebene ) 2D Seite 5 Ewald's Sphere/Problem 3.7 Lisa Marx 0831292 0a1 001 001 • real lattice: ~a1 = @0A ~a2 = @bA ~a3=@0A (Braucht man hier nicht) 0 0 c 011 B C @0A ! 0 1 ~ ~a2× ~a3 1 2π 0 recipr.: b1 = 2π ∗ ~a ∗( ~a × ~a ) = 2π ∗ a ∗ 1 = a ∗ ) 1 2 3 0 0! 0! ~ 2π 1 ~ 2π 0 b2 = b ∗ ) b3 = c ∗ 0 1 ~ ~ ~ 2π 2π 10 −1 x-y-Ebene: b1; b2 ) jb1j = a = 0;6∗10−9 = 1; 05 ∗ 10 m ~ 2π 2π 10 −1 ) jb2j = b = 0;8∗10−9 = 0; 79 ∗ 10 m 2.3 Ewald Construction Abbildung 2.2: Reciprocal lattice X-Rays propagieren mit beliebiger Richtung in der x-y-Ebene ~ ) Alle reziproken Gitterpunkte in einem Abstand von 2 ∗ jkij von (0,0) liegen irgendwann auf dem Rand der 2D-Ewaldsph¨are ) Zu jedem dieser reziproken Gitterpunkte ein gebeugter Strahl! ) Es wird nur ein Quadrant betrachtet (die doppelt gez¨ahlten Punkte auf Seite 6 Ewald's Sphere/Problem 3.7 Lisa Marx 0831292 der x−1bzw:y−1 Achse mussen¨ noch abgezogen werden) ) Anzahl der Punkte * 4 ) Winkel entsprechend berechnen ~ 10 −1 2 ∗ jkij = 8; 1 ∗ 10 m ~ 10 −1 8;1 jb1j = 1; 05 ∗ 10 m ) int1;05 = 7 Punkte im Ewaldkreis ~ 10 −1 8;1 jb2j = 0; 79 ∗ 10 m ) int0;79 = 10 Punkte im Ewaldkreis Abbildung 2.3: Ewald's sphere for problem 3.7 at a certain initial k-Vektor ) Anzahl der gebeugten Strahlen N=246 (Siehe C++-Anhang) ) Die Winkel der 70 Punkte eines Quadranten stehen im C++-Anhang und sind in allen 4 Quadranten gleich! Seite 7 Ewald's Sphere/Problem 3.7 Lisa Marx 0831292 3 C++-Anhang 3.1 Quellcode Seite 8 Ewald's Sphere/Problem 3.7 Lisa Marx 0831292 3.2 Output-window Seite 9 Ewald's Sphere/Problem 3.7 Lisa Marx 0831292 Seite 10.
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