5.069 | Spring 2010 | Graduate
Crystal Structure Analysis

Lecture Notes


Introduction (PDF)

Overview, textbooks, history of crystallography


Symmetry in 2D (PDF)

Definition of symmetry, introduction of symmetry operators

Compatibility of symmetry operators with translation

Combining symmetry operations and determination of plane groups


Symmetry in 3D (PDF)

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


X-rays and matter (PDF)

X-ray generation

Diffraction experiment with optical grids and laser pointers

Convolution theorem and Fourier transformation

Introduction of Bragg’s law and Miller indices


Geometry of diffraction (PDF)

Reciprocal space vs. real space

Ewald construction as a geometric interpretation of Bragg’s law


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


Structure factors II (PDF - 1.7MB)

Complex numbers, Euler’s equation and the argand plane

Introduction of the phase problem


Symmetry in reciprocal space (PDF)

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


Structure refinement (PDF)

Different types of electron density maps (Fo, Fc, Fo-Fc, etc.)

Introduction of anisotropic displacement parameters

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

Crystallographic parameters, constraints and restraints


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”


Anomalous scattering (PDF)

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


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)


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



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
As Taught In
Spring 2010