Diffraction T Ishikawa Part 1 Kinematical Theory 9252020

Diffraction T. Ishikawa Part 1 Kinematical Theory 9/25/2020 JASS 02 1

Introduction n What I intend to give in this lecture: u n 9/25/2020 Basic Concepts of "Diffraction" « Simpler Case: Kinematical Theory (Part 1) « Complicated Case: Dynamical Theory (Part 2) The speaker has been worked on experimental dynamical diffraction for 25 years. Now, he is in charge of X-ray optics for SPring-8, world's largest 3 rd generation synchrotron facility in Japan. JASS 02 2

X-Rays as Shorter Wavelength Electromagnetic Wave 9/25/2020 JASS 02 3

Scattering of x-rays by a point charge (Thomson Scattering) Point Charge, mass=m, charge=e r Electromagnetic Plane Wave Lorentz Force 9/25/2020 JASS 02 4

Scattering of x-rays by distributed charge (1/2) K 0 : Incident Wave Vector KS : Scattered Wave Vector KS r(x) : Number Density of Electron r(x) K 0 x Contribution from a volume element d 3 x at x 9/25/2020 JASS 02 5

Scattering of x-rays by distributed charge (2/2) 3 D Fourier Transform of Charge Density Scattered Intensity 9/25/2020 JASS 02 6

Electronic Charge Distribution in Crystal = 3 D Regular Stacking of Molecules Translation Symmetry N c M b a 9/25/2020 r(x) = r(x+la+mb+nc) l, m, n: integers L JASS 02 7

Fourier Transform of the Electronic Charge Distribution in Crystal 9/25/2020 JASS 02 8

Charge Density in Unit Cell Rn c b a 9/25/2020 JASS 02 9

Atomic Scattering Factor, Structure Factor Atomic Scattering Factor Structure Factor 9/25/2020 JASS 02 10

Laue Function (1/3) 9/25/2020 JASS 02 11

Laue Function (2/3) Laue Function, L=20 9/25/2020 JASS 02 12

Laue Function (3/3) n Scattering from a crystal is appreciable only when h, k, l; integer n 9/25/2020 Then, JASS 02 13

Reciprocal Lattice (1/3) Scattering from a crystal is appreciable only when K·a = 2πh K·b = 2πk K·c = 2πl h, k, l : integer Base Vectors of Reciprocal Lattice 9/25/2020 JASS 02 14

Reciprocal Lattice (2/3) c* c a* 9/25/2020 b b* a JASS 02 15

Reciprocal Lattice (3/3) n Reciprocal Lattice Vector h, k, l; integer When the scattering vector, K=Ks-Ko, corresponds to a reciprocal lattice vector, strong diffraction may be observed (necessary condition, but not a sufficient condition). 9/25/2020 JASS 02 16

Ewald Sphere Reciprocal Lattice g Ks r = 2 p/l 2 q Ko O Bragg Condition of Diffraction 9/25/2020 JASS 02 17

Forbidden Reflection (1/2) K=g is a necessary condition for observing diffraction, but not a sufficient condition. . . If F(K)=0, then Icrystal = 0 even when K=g. Example 1, Body Center Cubic Lattice (bcc) c b 9/25/2020 a JASS 02 18

Forbidden Reflection (2/2) Example 2, Face Center Cubic Lattice (fcc) c b a 9/25/2020 JASS 02 19

Special Topics What we can measure in diffraction/scattering experiment = Intensity All phase information is lost! Non-Crystalline Charge Distribution: If is obtained, we can calculate r(x) by Fourier inversion. 9/25/2020 JASS 02 20

Iterative Phase Retrieval (Jianwei Miao & David Sayre) X-ray intensity data: Phase Information is Lost! Scattered pattern in Far Field with Coherent Illumination, Phase can be retrieved. Phase Retrieval → Iterative Algorism developed by　Gerchberg & Saxon, followed by the improvement by Fienup (Opt. Lett. 3 (1978) 27. ) Real Space Image 9/25/2020 Scattered Intensity JASS 02 Phase Retrieval 21

Reconstruction of Complex Real Space Images Real Space Phase Retrieval Scattered Intensity 9/25/2020 JASS 02 22

Original Image Reconstructed Image After 5000 iteration 9/25/2020 JASS 02 23

3 D Diffraction Microscopy Miao et al. PRL (2002) Two Layer Ni Pattern SEM image of Ni pattern on Si. N Coherent Scattering Pattern 2 D Reconstructed Image (<10 nm resolution) 9/25/2020 3 D Reconstructed Image (~50 nm resolution) JASS 02 24