Lecture

Kinematical Scattering: Bloch Wave Theory and Applications

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In course

PHYS-637: Electron Matter Interactions in Transmission Electron Microscopy

This course will present the fundamentals of electronâmatter interactions, as occuring in the energy range available in modern transmission electron microscopes, namely 60-300 keV electrons. Diffraction and high-resolution image formation as well as elec ...

Description

This lecture covers the principles of kinematical scattering from a crystal lattice, focusing on the Bragg law, Laue condition, and the Ewald sphere. It introduces the concept of scattering geometry and the first Born approximation, explaining how a unit cell scatters a spherical wave modulated by the scattering angle. The lecture also discusses the deficiencies of the kinematical model and transitions into the dynamical theory using the Bloch wave approach. The instructor presents the mathematical formulations for calculating scattering intensity and the integration over the sphere at a distance from the sample. The lecture concludes with an overview of the Bloch wave theory, particularly the two-beam approximation, and how it relates to diffracted beam intensities, emphasizing the decoupling of Bloch waves into plane wave components at the bottom of the crystal thickness. This comprehensive overview provides a solid foundation for understanding the kinematical and dynamical aspects of electron diffraction in crystal lattices.

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Instructors (3)

Thomas La Grange

Expertise Electron Microscopy, Aberration Correct Scanning Transmission Electron Microscopy, Electron Diffraction, Ultrafast Transmission Electron Microscopy, Electron Energy Loss Spectroscopy, Energy Dispersive Spectroscopy, Materials Science, High-speed deformation, Dislocation Dynamics, Skyrmionics, Martensitic Phase Transformation Theory, Solidification Expertise Electron Microscopy, Aberration Correct Scanning Transmission Electron Microscopy, Electron Diffraction, Ultrafast Transmission Electron Micro

Cécile Hébert

Cécile Hébert was born in France in 1970. She obtained her Ingeneer degree (physics) and her PhD degree (design of a new energy filter for transmission electron microscopy, under the direction of Prof. B. Jouffrey) at the Ecole Centrale in Paris. As a post doc with Prof. P. Schattschneider, she was working on the calculation of the fine structure of ionisation edges in EELS. As a research assistant at the Vienna University of Technology, she was one of the main participant to the CHIRALTEM project dealing with the measurement of magnetic circular dichroism in the electron microscope.

Duncan Thomas Lindsay Alexander

Duncan Alexander graduated with a PhD in Materials Science and Metallurgy at the University of Cambridge in 2003 for a thesis on the kinetics of metallurgical phase transformations under the direction of Lindsay Greer. From 2003 to 2007 he held a Royal Society Overseas Research Fellowship at the University of Sydney, and then post-doctoral positions at the University of Cambridge and Arizona State University, working first on materials electrochemistry with Carsten Schwandt and Derek Fray, and then the characterization of atmospheric aerosol particles with Peter Crozier and Jim Anderson. In 2007 he came to EPFL as a scientific collaborator in the group of László Forró, and from 2008 to 2017 he was a staff scientist at EPFL’s Interdisciplinary Centre for Electron Microscopy (CIME), specializing in the application of advanced transmission electron microscopy techniques and enabling electron microscopy research activities for groups across EPFL through teaching, training and support. In 2018 he joined the Electron Spectrometry and Microscopy Laboratory (LSME) as a full-time research scientist dedicated to advancing transmission electron microscopy techniques, and was promoted to the status of Research and Teaching Associate. In 2019, together with LSME director Cécile Hébert, he launched the first MOOC on transmission electron microscopy for materials science: https://www.coursera.org/learn/microscopy/For a complete list of peer-reviewed publications, see:https://orcid.org/0000-0003-4350-8587

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Related concepts (32)

Precession electron diffraction

Precession electron diffraction (PED) is a specialized method to collect electron diffraction patterns in a transmission electron microscope (TEM). By rotating (precessing) a tilted incident electron beam around the central axis of the microscope, a PED pattern is formed by integration over a collection of diffraction conditions. This produces a quasi-kinematical diffraction pattern that is more suitable as input into direct methods algorithms to determine the crystal structure of the sample.

Bragg's law

In physics and chemistry, Bragg's law, Wulff–Bragg's condition or Laue–Bragg interference, a special case of Laue diffraction, gives the angles for coherent scattering of waves from a large crystal lattice. It encompasses the superposition of wave fronts scattered by lattice planes, leading to a strict relation between wavelength and scattering angle, or else to the wavevector transfer with respect to the crystal lattice. Such law had initially been formulated for X-rays upon crystals.

Low-energy electron diffraction

Low-energy electron diffraction (LEED) is a technique for the determination of the surface structure of single-crystalline materials by bombardment with a collimated beam of low-energy electrons (30–200 eV) and observation of diffracted electrons as spots on a fluorescent screen. LEED may be used in one of two ways: Qualitatively, where the diffraction pattern is recorded and analysis of the spot positions gives information on the symmetry of the surface structure.

Scattering

Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including particles and radiation) in the medium through which they pass. In conventional use, this also includes deviation of reflected radiation from the angle predicted by the law of reflection.

Crystal structure

In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter. The smallest group of particles in the material that constitutes this repeating pattern is the unit cell of the structure.

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