Diffraction

1
Diffraction

• T. Ishikawa

Part 1 Kinematical Theory
2
Introduction

• What I intend to give in this lecture
• 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 3rd generation synchrotron facility in
  Japan.

3
X-Rays as Shorter Wavelength Electromagnetic Wave
4
Scattering of x-rays by a point charge (Thomson
Scattering)
Point Charge, massm, chargee
r
Electromagnetic Plane Wave
Lorentz Force
5
Scattering of x-rays by distributed charge (1/2)
K0 Incident Wave Vector
KS Scattered Wave Vector
KS
r(x) Number Density of Electron
r(x)
K0
x
Contribution from a volume element d3x at x
6
Scattering of x-rays by distributed charge (2/2)
3D Fourier Transform of Charge Density
Scattered Intensity
7
Electronic Charge Distribution in Crystal
Crystal 3D Regular Stacking of Molecules
Translation Symmetry
N
c
M
r(x) r(xlambnc) l, m, n integers
b
a
L
8
Fourier Transform of the Electronic Charge
Distribution in Crystal
9
Charge Density in Unit Cell
Rn
c
b
a
10
Atomic Scattering Factor, Structure Factor
Atomic Scattering Factor
Structure Factor
11
Laue Function (1/3)
12
Laue Function (2/3)
13
Laue Function (3/3)

• Scattering from a crystal is appreciable only
  when
• Then,

h, k, l integer
14
Reciprocal Lattice (1/3)
Scattering from a crystal is appreciable only
when Ka 2ph Kb 2pk Kc 2pl h, k, l
integer
Base Vectors of Reciprocal Lattice
15
Reciprocal Lattice (2/3)
16
Reciprocal Lattice (3/3)

• Reciprocal Lattice Vector

h, k, l integer
When the scattering vector, KKs-Ko, corresponds
to a reciprocal lattice vector, strong
diffraction may be observed (necessary condition,
but not a sufficient condition).
17
Ewald Sphere
Reciprocal Lattice
g
Ks
O
2q
Ko
r 2p/l
Bragg Condition of Diffraction
18
Forbidden Reflection (1/2)

• Kg is a necessary condition for observing
  diffraction, but not a sufficient condition...

If F(K)0, then Icrystal 0 even when Kg.
Example 1, Body Center Cubic Lattice (bcc)
c
b
a
19
Forbidden Reflection (2/2)
Example 2, Face Center Cubic Lattice (fcc)
20
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.
21
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
Scattered Intensity
Phase Retrieval
22
Reconstruction of Complex Real Space Images
Real Space
Phase Retrieval
Scattered Intensity
23
Original Image
Reconstructed Image After 5000 iteration
24
3D Diffraction Microscopy

• Miao et al. PRL (2002)

Two Layer Ni Pattern