Wave Optics of Electrons
A de Broglie wavelength can be attributed to each accelerated particle and the propagation of electrons can be described by means of the concept of a wave packet. The interaction with magnetic and electrostatic fields can be described in terms of a phase
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A de Broglie wavelength can be attributed to each accelerated particle and the propagation of electrons can be described by means of the concept of a wave packet. The interaction with magnetic and electrostatic fields can be described in terms of a phase shift or the notion of refractive index can be employed, leading to the Schrodinger equation. The interaction with matter can similarly be reduced to an interaction with the Coulomb potentials of the atoms. Many of the interference experiments of light optics can be transferred to electron optics. The most important are the Fresnel biprism experiment and Fresnel diffraction at edges. The diffraction pattern far from the specimen or in the focal plane of an objective lens can be described by means of Fraunhofer diffraction. As in light optics, the Fraunhofer diffraction amplitude is the Fourier transform of the amplitude distribution of the wave leaving the specimen where the lens aberrations are incorporated in the wave aberration function. The image amplitude can be described in terms of an inverse Fourier transform, which does not however result in an aberration-free image owing to the phase shifts introduced by the electron lens and to the use of a diaphragm in the focal plane.
3.1 Electron Waves and Phase Shifts 3.1.1 de Broglie Waves
In 1924, de Broglie showed that an electron can be treated as a quantum of an electron wave, and that the relation E = hv for light quanta should also be valid for electrons. As a consequence he postulated that the momentum p = mv is also related by p = hk to the wave vector k, the magnitude of which (the wave number) may be written lkl = 1/>.. (>..:wavelength); this is analogous top = hvjc = hk for light quanta. This implies that >.. = hjp (2.12) with the relativistic momentum p (2.11). Substitution of the constants in (2.12) results in the formula h
>..
1.226
= mv = [U(1 + 0.9788
L. Reimer, Transmission Electron Microscopy © Springer-Verlag Berlin Heidelberg 1997
x 1Q-6Ujl/2
(3 .1)
46
3. Wave Optics of Electrons Infinite plane wave
Wave packet
ljl=ljl0 cos(lnz)=ljl 0 cos(2ttkz)
o-).
--t
r-----!J.z
).
-----1
~¥M?v~ z-
Alkl
' ·-.l
Aik......__),- - - + -
a)
Fig. 3.1. (a) Infinite sine wave with a discrete k-spectrum and (b) a wave packet
of lateral width Lh and a broadened k-spectrum of width Llk =
1/Llz
with .X [nm] and U [V] (.X= 3.7 pm for U = 100 kV and 0.8715 pm for U = 1 MV). A stationary plane wave that propagates in the z direction can be described by a wave function '1/J, which depends on space and timer: 'ljJ = '!j;0 cos[21r(kz- vr)] = '1/Jocos ( 2: z- 21rvr) = 'l/Jocoscp.
(3.2)
where '1/Jo is called the amplitude and cp the phase of the wave. The phase changes by 21r for r = const if the difference between two positions (z2 - zl) is equal A (Fig. 3.1a). However, there are important differences between light quanta and electrons when the relations E = hv and p = hk are used. The energy E of a light quantum means the energy necessary to generate a quantum or which is transferred when a quantum
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