Particle Optics of Electrons
The acceleration of electrons in the electrostatic field between cathode and anode, the action of magnetic fields with axial symmetry as electron lenses and the application of transverse magnetic and electrostatic fields for electron-beam deflection and e
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The acceleration of electrons in the electrostatic field between cathode and anode , the action of magnetic fields with axial symmetry as electron lenses and the application of transverse magnetic and electrostatic fields for electron-beam deflection and electron spectrometry can be analysed by applying the laws of relativistic mechanics and hence calculating electron trajectories. Lens aberrations can likewise be introduced and evaluated by this kind of particle optics. In the case of spherical aberration, however, it will also be necessary to express this error in terms of a phase shift, known as the wave aberration, by using the wave-optical model introduced in the next chapter.
2.1 Acceleration and Deflection of Electrons 2.1.1 Relativistic Mechanics of Electron Acceleration
The relevant properties of an electron in particle optics are the rest mass mo and the charge Q = - e (Table 2.1). In an electric field E and magnetic field B, electrons experience the Lorentz force F = Q (E
+ o x B)
= -
e (E + v x B) .
(2.1)
Inserting (2.1) in Newton's law mf=F
(2.2)
yields the laws of particle optics. We start with a discussion of the acceleration of an electron beam in an electron gun. Electrons leave the cathode of the latter as a result of thermionic or field emission (see Sect. 4.1 for details). The cathode is held at a negative potential C/>C = - U (U: acceleration voltage) relative to the anode which is grounded, c/>A = 0 (Fig. 2.1). The Wehnelt electrode, maintained at a potential c/>w = - (U + Uw), limits the emission to a small area around the cathode tip . Its action will be discussed in detail in Sect. 4.1.3. The electrode potentials create an electric field E in the vacuum between cathode and anode, which can also be characterized by equipotentials L. Reimer, Transmission Electron Microscopy © Springer-Verlag Berlin Heidelberg 1993
20
2. Particle Optics of Electrons
ifJ = const (Fig. 2.1). The electric field is the negative gradient of the potential E
=-
VA. 'f'
= - (~ ox ' ~ Gy , ~) oz
.
(2.3)
The existence of a potential implies that the force F tive and that the law of energy conservation E
+ V = const
=-
eE is conserva(2.4)
can be applied, as will be demonstrated by considering the electron acceleration in Fig. 2.1. The kinetic energy at the cathode is E = 0, whereas the potential energy V is zero at the anode . The potential energy at the cathode can be obtained from the work W that is needed to move an electron from the anode to the cathode against the force F:
V
=-
W
=-
c
f F . ds
A
c
= e fE
. ds
A
= -e(ifJc - ifJA) = eU .
=-
c
e f VifJ . ds A
(2.5)
In the reverse direction, the electrons acquire this amount eU of kinetic energy at the anode. This implies that the gain of kinetic energy E = eU of an accelerated electron depends only on the potential difference U, irrespective of the real trajectory between cathode and anode. Relation (2.5) can also be used to define the potential energy Ver) at each point r at which the potential is ifJ(r):
Ver) = QifJ(r) = - eif
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