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Partial fraction expansion of electron optical cardinal elements
Ultramicroscopy
|April 1, 1977
Summary
This study analyzes electron optical systems, revealing characteristic discontinuities when acceleration voltage changes. Partial fraction expansion and complex analysis methods are used to understand focal properties and their behavior under varying voltage.
Area of Science:
- Electron optics
- Applied physics
- Complex analysis
Background:
- Electron optical systems exhibit complex behavior with varying acceleration voltage.
- Understanding cardinal elements (focal points, widths) is crucial for system design.
- Characteristic discontinuities arise during voltage variations.
Purpose of the Study:
- Investigate the behavior of cardinal elements in electron optical systems under changing acceleration voltage.
- Analyze partial fraction expansions for magnetic systems (round lenses, quadrupoles).
- Establish convergence of series using complex analysis and R-functions.
Main Methods:
- Partial fraction expansion with respect to U-1/2.
- Complex analysis techniques.
- Application of E. Wigner's R-functions.
Main Results:
- Identified characteristic discontinuities in cardinal elements during acceleration voltage variation.
- Established convergence of series for magnetic systems.
- Demonstrated applicability to electric quadrupole systems.
Conclusions:
- The study provides a rigorous mathematical framework for analyzing electron optical system behavior.
- Results are applicable to both magnetic and electric quadrupole systems.
- Connections to analytical field models and inverse problems are highlighted.