Calendar

LEC # TOPICS KEY DATES
1

Introduction

Overview, textbooks, history of crystallography

 
2

Symmetry in 2D

Definition of symmetry, introduction of symmetry operators

Compatibility of symmetry operators with translation

Combining symmetry operations and determination of plane groups

 
3

Symmetry in 3D

Extension of the plane groups concept to the third dimension: space groups

Introduction of screw axes and glide planes

Point groups vs. space groups

The unit cell and crystallographic conventions

 
4

X-rays and matter

X-ray generation

Diffraction experiment with optical grids and laser pointers

Convolution theorem and Fourier transformation

Introduction of Bragg’s law and Miller indices

Student presentations of current homework assignment due two days after Lec #4
5

Geometry of diffraction

Reciprocal space vs. real space

Ewald construction as a geometric interpretation of Bragg’s law

 
6

Structure factors

Real atoms are no point atoms (atomic form factors) and show thermal motion (atomic displacement factors)

Having more than one atom per unit cell leads to structure factors

Fourier transformation gives rise to electron density; crystallographic resolution

Student presentations of current homework assignment due two days after Lec #6
7

Structure factors II

Complex numbers, Euler’s equation and the argand plane

Introduction of the phase problem

 
8

Symmetry in reciprocal space

Introduction of Friedel’s law and laue groups

Space group determination: |_E_2-1| statistics, systematic absences, crystallographic directions for triclinic, monoclinic, orthorhombic and tetragonal systems

Introduction of the Patterson function and Harker sections, as well as direct methods for structure solution

Student presentations of current homework assignment due two days after Lec #8
9

Structure refinement

Different types of electron density maps (_F_o, _F_c, _F_o-_Fc_, etc.)

Introduction of anisotropic displacement parameters

Minimization functions: the least-squares approach and different R-factors

Crystallographic parameters, constraints and restraints

 
10

Structure refinement II

Problems and pitfalls: wrong space group, atom type assignment (all electrons are blue), disorder, twinning

What are artifacts (libration, C-C triple bonds, Fourier truncation ripples, etc.)?

Finding the hydrogen atoms, “riding model”

Student presentations of current homework assignment due two days after Lec #10
11

Anomalous scattering

Absorption of X-ray photons leads to loss of symmetry in orbital geometry, which results in a violation of Fridel’s law

 
12

Practical aspects and related methods

Growing crystals and keeping them alive (never remove the mother liquor!)

Mounting crystals onto the diffractometer

Short introduction of powder diffraction, neutron diffraction and EXAFS

Crystallographic data bases (ICSD, CSD, PDB, reciprocal net)

Student presentations of current homework assignment due two days after Lec #12
13

Quick recap

Symmetry, Bragg’s law, Miller indices, real space vs. reciprocal space, Ewald construction, structure factors, electron density, symmetry in reciprocal space, laue groups vs. point groups vs. space groups, space group determination, Patterson function, structure refinement, parameters/constraints/restraints, anisotropic displacement parameters, libration, hydrogen atoms

 
14

Exam

You have 50 minutes to answer all questions. You can use pens, a calculator, ruler and compass, as well as a letter sized piece of paper with anything written on it. No books or other material is allowed.

 

Course Info

Learning Resource Types

notes Lecture Notes