Chapter 1. Introduction

Chapter 1. Introduction X-rayDiffraction & Crystal StructureAnalysis Intensity Two-theta Powder XRD pattern Single-crystal XRD pattern

Diffraction spots on detector Incident X-ray beam x’tal

Intensity Two-theta

Type λ 對空氣穿透率 (10 dm path length) Hard X-rays < 5 Å> 80% Soft X-rays > 5 Å < 80% Applications: (i) Diffraction for powder, single crystals and partially crystalline materials (ii) Spectroscopy (iii) Spectrometry (medical and industrial)

incident wave scattered wave When the particle is large compared to the wavelength of the radiation, therev will be several cycles of the waves contained in the same volume of space as that occupied by the particle. The particle will experience the average electric field close to zero. If the particle is small compared to the wavelength of the radiation, it will “see” only a very small portion of a cycle of the wave, and willexperience a well-defined electric field E. It then, experience a force F given by F = qE. If the particle has a mass m, the force F will give rise to an accelerationa, as F = ma = qE with a = (q/m)E. Intensity of scattering I  |a|2 (q/m)2|E|2 The above equation states that the intensity of the radiation scattered by a particle of mass m and charge q depends on the ratio (q/m)2.

X-rays are scattered by electrons Neutrons have no charge and will not scatter X rays. Protons have the elemental charge e, and a mass mp, and electrons also have the elemental charge e, but a mass me. A proton, however, is 1837 times heavier than an electron, i. e. mp = 1837 me . The intensities of radiation scattered by an electron Ie and that by a proton Ip are Ie / Ip = (e/me)2(mp/e)2 = 18372 X-ray diffraction therefore looks at the electron distribution in a crystal, and not directly at the positions of the nuclei of atoms.

Chapter 2

Lattice:a set of identical points in identical surroundings. b a b a Lattice point –in identical enviroment – on the center of symmetry Translation vectors intersecting at lattice points 2D lattice: a and b are non-parallel 3D lattice : a,b, and c are non-co-planar

V = a x b.c V=abc(1- cos2 - cos2  -cos2 + 2cos - cos -cos)½ right-handed system b. 2.1.1 The Unit cell 6 cell constants (parameters): a, b, c, , , 

556 455 555 565

Select a unit cell for each of the following lattices. How many lattice points are in each unit? (a) (c) (b)

1  x  0 1  y  0 1  z  0

180o Horizontally clockwise rotation All two-dimensional lattices have two-fold rotation axis. Crystal Symmetry-- Three simple symmetry elementsi, m, n (i) inversion centeri (x,y,z)  (-x,-y,-z) (ii) mirror planem(x,y,z)  (x,y,-z) … (iii) rotation axesn, where n = 360o/α (α= angle of rotation) (x,y,z)  (-x,-y,z) …

The periodicity (i.e. operation of translation) will impose rotational symmetry to a lattice. If P and Q are lattice points. If the lattice possesses a rotation of  angle, the resulting P’ and Q’ are also lattice points. P Q’ a   P’ Q DPQ’ = ma = a + 2a sin ( - 90o) (m must be an integer).

DPQ’ = ma = a + 2a sin ( - 90o) (m must be an integer). After rearrangement, cos  = (m-1)/2 m must fall in the range of 3 to -1: m  n-fold rotation 3 360 o1 (one-fold) 2 60 o6 (six-fold) 1 90 o4 (four-fold) 0 120 o3 (three-fold) -1 180 o2 (two-fold) In all 2D and 3D lattices, only five kinds of rotation angles are allowed.

Five 2D Lattices cell parameters: a, b, and  Rotational Symmetry 2-fold  to plane one 2-fold  to plane two 2-fold // plane

same as above 4-fold  to plane 6-fold or 3-fold  to plane

The Seven Crystal Systems: cell parameters: a, b, c, , ,  no. independent parameters. 6 4 3 2 1 * 2 2 The symbol  implies that equality is not required by symmetry. *In hexagonal lattice. If in rhombohedral lattice: a = b = c;  =  =   90o

Basic Lattice Types: P, I, C, F, and R

The 14 Bravais Lattices :

Relationship between monoclinic B and P lattices Choice of C or I unit cells in monoclinic lattices

Relationship between tetragonal C and P lattices Possible choice of an orthorhombic unit cell in the hexagoanl system

1 2 3 4 6 1 2 3 4 6 i m Ten basic symmetry elements in crystals (i) Combination of simple symmetry elements of inversion and rotation They also represent ten point groups. (ii) Combination of symmetry axes – when two rotation axes are combined a third rotation axis is created. There are only six possible combinations of rotation axes of symmetry that can operate on crystal lattice. 222 223 224 226 233 234

The the External Symmetry of Crystals -- the 32 crystallographic point groups -- the 32 crystal class

Three Translational vectors Five rotation axes Four lattice types Three simple symmetry elements Seven crystal systems 10 basic symmetry elements 14 Bravais lattices Symmetry combinations 32 symmetry point groups 73 symmorphic space groups External symmetry of crystals

Chapter 1. Introduction

Chapter 1. Introduction

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Introduction – Chapter 1

Chapter 1: Introduction

Chapter 1: Introduction

Chapter 1: Introduction

Chapter 1: Introduction

Chapter 1: Introduction

Chapter 1 - Introduction

Chapter 1 - Introduction

Chapter 1: Introduction

Chapter 1: Introduction

Chapter 1: Introduction

Chapter 1 Introduction

Chapter 1. Introduction

Chapter 1 Introduction

Chapter 1: Introduction

Chapter 1 Introduction

Chapter 1: Introduction

Chapter 1—Introduction

Chapter 1: Introduction

Chapter 1: Introduction

CHAPTER 1: Introduction

Chapter 1: Introduction