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Introduction Into the Dynamical Theory of X-Ray Diffraction for Perfect Crystals

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Introduction Into the Dynamical Theory of X-Ray Diffraction for Perfect Crystals See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/313659217 dynamical theory-seminars 2015 Presentation · April 2015 DOI: 10.13140/RG.2.2.16280.19207 CITATIONS READS 0 52 1 author: Jürgen Härtwig Retired from European Synchrotron Radiation Facility 226 PUBLICATIONS 3,509 CITATIONS SEE PROFILE All content following this page was uploaded by Jürgen Härtwig on 13 February 2017. The user has requested enhancement of the downloaded file. Introduction into the dynamical theory of X-ray diffraction for perfect crystals Jürgen Härtwig European Synchrotron Radiation Facility, BP 220, 38043 Grenoble Cedex, France [email protected] 1 Outline 1. Introduction 2. Some results of the kinematical (geometrical) theory of X-ray diffraction 3. Dynamical theory of X-ray diffraction Short theoretical background Basic results and helpful tools (dispersion surfaces) The one “beam” case – refraction and reflection The two “beam” case 4. Some effects of dynamical X-ray diffraction Pendellösung length (Laue case), Pendellösung effect Anomalous transmission – Borrmann effect Bragg case and a bit X-ray optics Plane waves, monochromatic waves 2 1. Introduction What is/means “dynamical” diffraction theory? Why do we need it? Which are the differences to other, simpler theories? May I quantify if a simple theory is sufficient or not? Where it is applied? 3 Dynamical Diffraction – Applications today X-ray topography strain and defects in (single) crystals, e.g., dislocations, precipitates Crystal growth electronic & micro- electronic developement Bragg-diffracting optical elements for synchrotron radiation monochromators, phase plates High resolution x-ray diffraction single crystals, epitaxial films, superlattices ... Grazing incidence methods (reflection, GID, GISAXS) thin films & interfaces X-ray standing wave method (XRSW) secondary effects (photoelectrons, x-ray fluorescence etc), excited by a standing wave in a diffracting crystal Three-beam diffraction determination of phases of the structure factors Dynamical diffraction of light (λ ~ 1.5 mm) in a two-dimensional or three dimensional array of holes Optical photonic crystals 4 Some history The discovery of X-ray diffraction and first theories 5 The story of X-ray diffraction and of the dynamical theory (-ies) of X-ray diffraction started with Paul Peter Ewald (1888-1985) and Max Laue (1879-1960) 6 It started in 1910 with … … Paul Peter Ewalds thesis project Ewald asks Arnold Sommerfeld for a topic for a dissertation Proposal: to find out whether a lattice-like anisotropic arrangement of isotropic resonators might be capable of exhibiting light-optical birefringence (double refraction) Result: theory that relates macroscopic properties of dispersion and refraction in a crystal to the interaction of propagating waves with a microscopic distribution of resonators This was already a dynamical theory, only that is was formulated for visible light 7 January 1912 Ewald wanted to discuss his results with Max Laue During the presentation of his subject he mentioned: “ … 3-dimensional lattice …” ??!!! Laue 8 Laue: What are the distances between the lattice points ? Ewald: Maybe about one thousandth of the wavelength of light! Max Laue got the crucial idea: If crystals were indeed constructed like 3-D-lattices (with distances mentioned by Ewald), and if X-rays had the properties of waves (with distances estimated by Arnold Sommerfeld, ~0.6nm) then X-rays should be diffracted when passing a crystal!!! 9 around Easter 1912: Experiment by W. Friedrich, P. Knipping & M. Laue Friedrich’s and Knipping’s set-up to check Laue’s idea M. Laue, W. Friedrich, P. Knipping Sitzungsberichte der Königlich Bayerischen Akademie der Wissenschaften, June 1912, M. Laue, W. Friedrich, P. Knipping, Annalen der Physik (Leipzig), 41 971 (1913) 10 One of the first exposures, The famous exposure “Fig. 5”, taken with a CuS crystal adjusted ZnS crystal. plate (Friedrich et al. 1912) (Deutsches Museum, München) 11 1. X-rays are waves! 2. Crystals have a discrete, 3-D-periodic (lattice) structure! Nobel prize 1914 for Max von Laue "For his discovery of the diffraction of X-rays by crystals" “Laue technique” Crystal orientation (SR) Structure analysis X-ray topography (oscillation method) (white beam topography) 12 Nowadays: Laue image of an AlPdMn icosahedral quasicrystal showing five-fold symmetry “Laue spots” (position, intensity) information about crystal symmetry In general white beam used, sample position fixed Integrated intensity recorded 13 ID19 ~ 40 cm 18 cm Typical Laue pattern from a small silicon sample. Δω=0° and λ=“infinite” (white beam). Typical “Laue pattern” from a tetragonal HEW lysozyme. Higher harmonics “contamination” Δω=1° and λ=0.8Å. for white beam λ/n = d(hkl)/n · sin θ(hkl) 14 First X-ray diffraction theories promptly followed the discovery of X-ray diffraction: 1912/13 Laue’s geometrical (or kinematical) theory, Laue equations 1913 Bragg’s law 1914 1916/17… Diffraction by a 3D-lattice (crystal) a (sina - sina0) = h l b (sinb - sinb0) = k l c (sing - sing0) = l l Laue equations h, k, l – “Miller” (Laue) indices 15 William Lawrence Bragg (son, left) and Sir William Henry Bragg (father, right) (Courtesy Edgar Fahs Smith Memorial Collection, Department of Special Collections, University of Pennsylvania Library.) Nobel Prize in Physics 1915: "For their services in the analysis of crystal structure by means of X-rays" 16 W H Bragg, W L Bragg, Proc Roy Soc A88, 428 (1913) Bragg’s law - 1913 W L Bragg - 22 years old student! Equivalent to Laue equation real space reciprocal space incident wave, l diffracted wave q q Khkl d h d· sinq K 0 l = 2dhklsinqhkl Kh = K0 + h 0 scalar form vectorial form of Bragg’s law of Bragg’s law Ewald’s construction 17 Determination of the first crystal structures NaCl (rock-salt) fcc, KCl (Sylvine) pc, ZnS (Zincblende) fcc, CaF2, (Fluorspar) fcc, CaCO3 (Calcite) rhombohedral W H Bragg, W L Bragg, Proc Roy Soc A89, 248 (1913), submitted 21 June, accepted 26 June !!! Diamond W H Bragg, W L Bragg, Nature 91, 557 (1913), submitted 28 July, publishing date 31 July Proc Roy Soc A89, 277 (1913), submitted 30 July!!! 18 First X-ray diffraction theories promptly followed the discovery of X-ray diffraction: 1912/13 Laue’s geometrical (or kinematical) theory 1913 Bragg’s law 1914 Darwin’s geometrical and dynamical theories C. G. Darwin, The theory of X-ray reflection, Phil. Mag. 27, 315, 675 (1914) 1916/17 Ewald’s extension of his theory to X-rays (fully dynamical theory) P. P. Ewald, Ann. Phys. (Leipzig) 49, 1, 117 (1916); 54, 519 (1917) (English translations exist!) all for perfect crystals! 19 2. Kinematical (geometrical) theory of X-ray diffraction Amplitude of the diffracted wave derived by: - adding the amplitude of the waves diffracted by each scatterer - by simply taking into account the optical path differences ( “geometrical”) - neglecting the interaction of the propagating waves with matter (only one scattering process, no absorption, no refraction, energy conservation law violated!!!) 20 Repetition of some results to better see the differences to the dynamical theory Scalar waves, single scattering P h- scattering vector, k h k 0 h (not necessary a crystal) h = 2 sinB/l, k0 = kh Amplitude in point P r r r r h k h r source k 0 0 21 Optical path difference: kh r k 0 r 1 (kh k 0 )r l hr k 0 k 0 k 0 k 0 r k 0 r r r r k h r h k k h h r source k 0 0 22 Used approximations: Fraunhofer approximation: rsource r and r r elastic (coherent) scattering: k 0 k h Amplitude in point P: 3 (r) = 0(r) + 1(r) = 0(r) + G(r|r’) V(r’) 0(r’) d r’ exp(2ik 0 r r ) Exact solution with: with: Gr | r 4r r (r’) instead of 0(r’) Vr 4reNr - for photon scattering on electrons (re– classical electron radius, N(r) – electron density) 2 Vr VFermi r b(2 /m)r - for thermal neutron scattering on the nuclear potential (b – scattering length) 2m Vr U r - for electron scattering on the Coulomb potential 2 C 23 Example – X-rays 0(r) – plane wave electrical field exp(2ik r) Scattered wave: (r) E r 0 F(h) 1 h r 3 Scattering amplitude: F(h) reE0P N(r)exp(2ihr)d r 1 for s-polarization P = cos(2B) for -polarization polarization factor for amplitudes Rayleigh scattering Special cases: 1. Scattering on 1 electron: N(r) (r) 2 2 r 1 IT (r) ET e C I C [1 cos2 (2 )] h h r 2 0 2 B polarization factor for intensities Thomson scattering 2. Scattering on 1 atom: N(r) Natom (r) Eatom (h) f – atom form factor h N(r)exp(2ihr)d3r f(h) T (amplitude) Eh Iatom IT f 2 Remark: Fourier h h transform of N(r)!!! 3. Scattering on a crystal: N(r) Ncrystal (r) Icrystal IT G 2 F 2 Like in classical optics – Fraunhofer h h h diffraction by a grating G - lattice amplitude Fh - structure amplitude (factor) (factor) Fourier transform of N(r)! 3 Fh (h) N(r)exp(2ihr)d r unitcell Structure analysis Kinematical scattering on a crystal: Like in classical optics – Fraunhofer diffraction by a grating G - lattice amplitude Fh - structure amplitude (factor) (factor) Icrystal ~ F 2 Such dependence only in h h kinematical approximation crystal T 2 2 Ih Ih G Fh Fourier transform of N(r)! Structure analysis 3 Fh (h) N(r)exp(2ihr)d r unitcell 27 Valid only under the following conditions: 1. Only one scattering process (no multiple scattering) rsource r r r 1(rk) 0 k h 1 << 0 k0 2 negligible 0(r) 1st Born approximation 2(r) 2. Plane wave approximation (Fraunhofer approximation): and 3.
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