Fundamentals of crystallography and diffraction
Summary
Crystallography and diffraction are fundamental to much of condensed matter and materials physics research. By developing the relevant formalisms, this course builds a solid basis for utilizing crystallography and diffraction, extending to crystalline phenomena that impact physical properties.
Content
1. Introduction to the crystalline lattice.
Stacking of atoms: atomic planes, close packed structures. Tessellation: translational symmetry and the crystal lattice. Description by the unit cell.
2. Symmetry and the Bravais lattices.
Introduction to the basic symmetry elements. Definition of the 7 unit cell types. Lattice centering possibilities and the 14 Bravais lattices.
3. Lattice vectors, crystal planes and the reciprocal lattice.
Formalisms of lattice vectors, Miller indices. The spatial Fourier transform and reciprocal lattice.
4. Symmetry operators and formalisms.
Description and applications of point groups and space groups. Application to common cubic structure variants and polymorphs.
5. Diffraction basics.
X-ray diffraction and Bragg's law. Reciprocal lattice and the Ewald sphere. Kinematical (Laue) theory of X-ray scattering. Atomic scattering factors and the structure factor.
6. X-ray diffraction analyses.
Single crystal theta-2theta diffraction, powder diffraction, Reitveld refinement. Thin film analyses.
7. The hexagonal lattice.
Common hexagonal structures (hexagonal close packed wurtzite, graphene, dichalcogenide), and relevance to semiconductors and other condensed matter research. The 4-index notation for hexagonal lattice planes and vectors.
8. Other structure variants.
Insights into structures such as the distorted perovskite oxide lattice, oxide polymorphs, and superlattice reflections.
9. Crystalline defects.
1-D defects (crystal dislocations), 2-D defects (stacking faults and twins), crytal interfaces such as sigma boundaries.
10. Other diffraction types.
Electron diffraction, neutron diffraction, and their application to crystallography.
Keywords
Crystallography, X-ray diffraction, unit cell, Bravais lattice, lattice vector, Miller indices, reciprocal lattice, kinematical scattering, structure factor, Reitveld, 4-index notation, dislocations, interfaces, polymorphs
Learning Prerequisites
Required courses
MATH-206/MATH-207, PHYS-207
Important concepts to start the course
Fourier transform (e.g., MATH-206, MATH-207); Schrödinger wave equation (e.g., PHYS-207).
Learning Outcomes
By the end of the course, the student must be able to:
- Develop the nature of the reciprocal lattice
- Infer the point group symmetry of a 2D lattice
- Elaborate the connection between a crystal lattice and its space group
- Prove the kinematical theory of diffraction
- Formulate the unit cell structure factor
- Interpret single crystal and powder diffraction patterns
- Sketch 3 index and 4 index lattice vectors and planes
- Assess / Evaluate unit cell types and symmetries
Transversal skills
- Use both general and domain specific IT resources and tools
- Demonstrate the capacity for critical thinking
- Communicate effectively, being understood, including across different languages and cultures.
- Use a work methodology appropriate to the task.
- Assess one's own level of skill acquisition, and plan their on-going learning goals.
Teaching methods
Oral lectures. Weekly exercises/integrated exercises in the classroom. Use of open-source sofware and database resources.
Assessment methods
Written exam.
Resources
Bibliography
Structure of materials: an introduction to crystallography, diffraction and symmetry. Marc de Graef. (Oxford University Press)
Fundamentals of Crystallography. Edited by C. Giacovazzo. (Oxford Science Publications)
Moodle Link
In the programs
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Fundamentals of crystallography and diffraction
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 1 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Fundamentals of crystallography and diffraction
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 1 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Fundamentals of crystallography and diffraction
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 1 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Fundamentals of crystallography and diffraction
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 1 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Fundamentals of crystallography and diffraction
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 1 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Fundamentals of crystallography and diffraction
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 1 Hour(s) per week x 14 weeks
- Type: optional