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Mueller, Peter, 5.069 Crystal Structure Analysis, Spring 2010. (Massachusetts Institute of Technology: MIT OpenCourseWare), (Accessed 09 Jul, 2010). License: Creative Commons BY-NC-SA

Crystal Structure Analysis

Spring 2008

Molecular model of the amino acid tyrosine with experimental electron density in front of an X-ray diffractometer at MIT. The tyrosine is part of the crystal structure of phosphoglycerate mutase from M. tuberculosis. See Mueller, P., et al. Acta Cryst D61 (2005): 309-315. (Figure and photograph by Dr. Peter Mueller.)

Course Description

This course covers the following topics: X-ray diffraction: symmetry, space groups, geometry of diffraction, structure factors, phase problem, direct methods, Patterson methods, electron density maps, structure refinement, how to grow good crystals, powder methods, limits of X-ray diffraction methods, and structure data bases.

*Some translations represent previous versions of courses.



Recommended Textbooks

While it is not required to own or read a textbook on crystallography in order to attend 5.069, some of you may still be interested in a list of recommendations. There are many books on the market and everybody has different priorities and preferences, therefore this short list is highly biased and by no means complete.

For starters, the book by Werner Massa (Crystal Structure Determination, English by Bob Gould, Springer) is an excellent choice. Everything important is explained and the book starts from scratch. This book is sufficient as a companion for 5.069.

Massa, Werner. Crystal Structure Determination. 2nd ed. Translated into English by R. O. Gould. New York, NY: Springer, 2004. ISBN: 9783540206446.

The somewhat more advanced student may like the Fundamentals of Crystallography by Carmelo Giacovazzo (Oxford University Press). Even though the word "fundamentals" appears in the title of the book, it is very helpful to have prior knowledge, when attempting to read the Giacovazzo. This book covers all the basics and should be sufficient for most PhD students.

Giacovazzo, C., ed. Fundamentals of Crystallography. Oxford: Oxford University Press, 1992. ISBN: 9780198555780.

If you want to knock yourself out, you may consider reading the four books edited by the master himself: Sir Lawrence Bragg. The books are called The Crystalline State and consist of the following volumes:

  1. The Crystalline State - A General Survey by L. Bragg
  2. The Optical Principles of Diffraction of X-Rays by R. W. James
  3. The Determination of Crystal Structures by H. Lipson and W. Cochran
  4. Crystal Structures of Minerals by L. Bragg, G. F. Claringbull and W. H. Taylor

Bragg, Sir Lawrence, et al. The Crystalline State. Vols. I-IV. Ithaca, NY: Cornell University Press, 1965.

You may also want to try:

Hahn, Theo, ed. International Tables for Crystallography. Vol. A: Space-Group Symmetry. 5th revised ed. Dordrecht, Holland: Springer, 2002. ISBN: 9780792365907.

Lisensky, George C., et al. Optical Transform Kit. Madison, WI: University of Wisconsin Board of Regents, Institute for Chemical Education, 1994.

MacGillavry, Caroline H. Fantasy and Symmetry: The Periodic Drawings of M. C. Escher. New York, NY: Harry N. Abrams, In., 1976. ISBN: 9780810908505.

Schattschneider, Doris. Visions of Symmetry: Notebooks, Periodic Drawings, and Related Work of M. C. Escher. New York, NY: W.H. Freeman & Co., 1990. ISBN: 9780716721260.

Kleber, Will. Einführungin die Kristallographie. Berlin, Germany: Veb Verlag Technik, 1965.

Other Recommendations

For the more biologically oriented student, an excellent "classic" is:

Glusker, Jenny Pickworth, and Kenneth N. Trueblood. Crystal Structure Analysis: A Primer. 2nd ed. New York, NY: Oxford University Press, 1985. ISBN: 9780195035438.

Chemists that want it in a nutshell may like:

Clegg, William. Crystal Structure Determination New York, NY: Oxford University Press, 1998. ISBN: 9780198559016. With only 82 pages, this is probably the shortest Crystallography textbook ever published.

The list goes on and on and on…


This class consists of 12 lectures, five problem sessions, one hour devoted to a brief recap and one exam. The problem sessions are structured as follows:

During the Wednesday lecture, a homework sheet is handed out, which consists of several problems (ca. five) that the students are supposed to work out (in writing) during the following nine days. On the next Friday, the answer to each homework question is to be presented by a student or a small group of students in front of the class. Students will know in advance who will be presenting each problem so that they can prepare their presentations. After the presentations and a discussion about the problems, the students are supposed to hand in their problem sets, to be graded (one can get up to five points per set of problems).

Obviously the students can correct their answers during the presentations, which makes it relatively easy to get full points for the homework sets. This is not a problem, as the goal of having homework in this class is to make the students think about the subjects taught, and to do so outside of the classroom and independently. Even if a student comes to the wrong answer for a problem, the fact that he or she thought about it at home for some time will make correcting it during the homework session an educational experience that has nothing to do with cheating to get full points.


The problems and the exam are graded as follows:

A student can get a maximum of 25 points from the problem sets (up to 5 points each) and a maximum of 65 points from the exam. Handing in the homework late results in subtraction of points (more than one week late leads to one point being subtracted, more than two weeks late takes two points away).

Altogether, a student can get up to 90 points. The breakdown into grades is determined after the exam, taking into account the number of points actually accumulated by the students (i.e. if everybody does really badly in one particular exam question, the results for this question can be ignored for the grading altogether, etc.). In any case, it will be possible for a student to pass with the exam alone (at least arithmetically), but not with the homework alone.



Overview, textbooks, history of crystallography


Symmetry in 2D
Definition of symmetry, introduction of symmetry operators
Compatibility of symmetry operators with translation
Combining symmetry operations and determination of plane groups


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


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

Geometry of diffraction
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

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

Structure factors II
Complex numbers, Euler's equation and the argand plane
Introduction of the phase problem


Symmetry in reciprocal space
Introduction of Friedel's law and laue groups
Space group determination: |E2-1| 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

Structure refinement
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"

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

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


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

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.   Tell A Friend