Course detail

The Fourier Transform of Lattices and the Kinematical Theory of Difraction

FSI-9KTDAcad. year: 2024/2025

The course deals with the Fourier transform of functions of several variables and its use in diffraction theory and in structure analysis. The introductory parts are focused on the definition of the Fourier transform, spatial frequencies, spectrum of spatial frequencies, and on the relevance of the Fourier transform to the diffraction theory. Then, the properties of the Fourier transform are presented via mathematical theorems and are illustrated by the Fraunhofer diffraction patterns. In this way a view of the general properties of the diffraction phenomena of this type is obtained. At the end the kinematical theory of diffraction by crystals is presented as an application of the Fourier transform of three-dimensional lattices.

Language of instruction

Czech

Mode of study

Not applicable.

Guarantor

prof. RNDr. Jiří Komrska, CSc.

Department

Institute of Physical Engineering (ÚFI)

Entry knowledge

Basic mathematical description of light propagation (diffraction), basic knowledge of Theory of solid state physics (structural analysis).

Rules for evaluation and completion of the course

Examination: Oral. Both practical and theoretical knowledge of the course is checked in detail. The examined student has 90 minutes to prepare the solution of the problems and he/she may use books and notes.
The presence of students at practice is obligatory and is monitored by a tutor. The way how to compensate missed practice lessons will be decided by a tutor depending on the range and content of the missed lessons.

Aims

Knowledge of the kinematical diffraction in structure analysis.
Practice in analytical calculations of the Fourier transform.

Ability to calculate Fourier transform.
Knowledge of kinematic theory of diffraction in structural analysis

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Bracewell R. N.: The Fourier Transform and its Applications. 3rd ed.. McGraw-Hill Book Company, New York 1999. (EN)
James J. F.: A students guide to Fourier transforms. Cambridge University Press, Cambridge 1996. (EN)
Komrska J.: Fourierovské metody v teorii difrakce a ve strukturní analýze. VUTIUM, Brno 2007. (CS)
Papoulis A.: Systems and Transforms with Applications in Optics. McGraw-Hill Book Company, New York 1968. (EN)

Recommended reading

Brigham E. O.: The Fast Fourier Transform. 2nd ed.. Prentice-Hall, Inc., Engelwood Clifs, New Jersey 1987. (EN)
Komrska J.: Matematické základy kinematické teorie difrakce. Fourierova transformace mřížky. Ve sborníku Metody analýzy povrchů. Elektronová mikroskopie a difrakce (L.Eckertová, L.Frank eds.). Academia, Praha 1996.

Classification of course in study plans

  • Programme D-FIN-P Doctoral 1 year of study, summer semester, recommended course
  • Programme D-FIN-K Doctoral 1 year of study, summer semester, recommended course

Type of course unit

 

Lecture

20 hod., compulsory

Teacher / Lecturer

prof. RNDr. Jiří Komrska, CSc.

Syllabus

1. Summary of the crystal lattice geometry.
2. The Dirac distribution.
3. The Fourier transform of functions of several variables and its relevance for structure analysis.
4. Linearity of the Fourier transform and the Babinet theorem.
5. The Fourier transform of the lattice function and the reciprocal lattice.
6. Symmetry of the Fourier transform and the Friedel law.
7. Convolution and the Fourier transform of convolution. Cross-correlation and autocorrelation.
8. Kinematical theory of diffraction.
9. The Laue equations and the Bragg equation.
10. Calculations of the shape amplitudes.
11. Addenda.