Lorentz Force of Moving Electron in Magnetic Field
- Practical Electron Microscopy and Database -
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This book (Practical Electron Microscopy and Database) is a reference for TEM and SEM students, operators, engineers, technicians, managers, and researchers.



The Lorentz force, which a moving electron experiences in the electric and magnetic fields in the electron microscopes, induces deflection of the negatively charged electron q (-e), providing the physical basis for electron lenses. With an electric field strength E and a magnetic flux B (Tesla), the the well-known Lorentz force F is given by

                 Lorentz force in electron microscope ---------------------------------------- [4919a]

where −e is the charge of the electron and v the velocity of the electron.

Based on Equation [4919a], one can in principle use either electrostatic or magnetic lenses to focus an electron beam; however, magnetic lenses are preferred rather than electrostatic lenses because they are more convenient to use and have lower aberrations. That is why only magnetic lenses are used in electron microscopes. Without applying an electric field within the lens, the resulting F is a vector normal to v and B, which are inclined to one another at an angle Θ. As shown by right-hand rule, your thumb represents the direction of the force acting on a positive charge moving in the direction of the middle finger through a field in the direction of the index finger as shown in Figure 4919. So the force on the electron acts in the opposite direction to your thumb.

Figure 4919. Schematic of right-hand rule of Lorentz force (F) vector in a magnetic field.

The magnetic field action expressed by the vector cross-product of v and B results in a force vector that is normal to vectors v and B (see right-hand rule for cross products). Inserting Equation [4919a] into Newton’s law of motion

                 Newton’s law of motion--------------------------------------------------------------[4919b]

obtains the law of optics of electron “particle”. Here, we consider the electrons are particles rather than wave effect in quantum mechanics. Thus, the force is given by

                  Lorentz force of moving electron in magnetic field ----------------- [4919c]

In the objective lens of TEM, the magnetic flux generated by the lens coil is condensed at the tip (at the gap) of the rotationally symmetric polepiece, with a bore diameter b and a gap distance S between the poles. The electrons, which travel along the optical axis exactly, do not receive the Lorentz force due to the magnetic field. The incident electrons at a distance r from the axis receive the Lorentz force based on Fleming's left-hand rule, and thus the electron makes a rotational motion.




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