282
Chapter 4. Particle scattering
classical transport equation. This is permissible for an amorphous
target, as typically the phase coherence of elastic scattering has
been lost due to the random distribution of scattering centers, and
the presence of inelastic processes. This section is based on earlier
published work by Snyder and Scott [81], Keil, Zeitler, and Zinn
[50], Crewe and Groves [21], and Groves [38].
This is distinctly different from elastic electron scattering in a
crystal, where phase coherence is maintained. Here constructive
interference occurs at the Bragg angles, giving rise to the familiar
diffraction patterns. The following discussion does not apply to
the diffraction case.
The mean free path is given by
1
µ =
,
(4.192)
N σ
where N is the number of atoms per unit volume, and σ is the
total scattering cross section. For fast electrons in a typical solid,
µ for elastic scattering is proportional to the incident energy in the
first Born approximation. Consequently, µ ranges from a few tens
of nanometers at an incident energy of 10 KeV to a few hundreds
of nanometers at 1 MeV. The sections observed in a transmission
electron microscope must be thin relative to the mean free path, in
order to avoid degradation of the image due to multiple scattering
of the beam electrons. As this is not always possible, multiple scattering must be considered in the image formation. In this section
we derive a method for understanding the scattering as a function
of the sample thickness, measured in units of the mean free path.
We define a dimensionless quantity n = z/µ, which we call the
reduced thickness, where z is the thickness, measured in units
of length. The probability P of an electron undergoing exactly j
scattering events in the reduced thickness n is governed by Poisson
statistics, namely
j
n
P j (n) =
e
−n .
(4.193)
j!
Chapter 4. Particle scattering
classical transport equation. This is permissible for an amorphous
target, as typically the phase coherence of elastic scattering has
been lost due to the random distribution of scattering centers, and
the presence of inelastic processes. This section is based on earlier
published work by Snyder and Scott [81], Keil, Zeitler, and Zinn
[50], Crewe and Groves [21], and Groves [38].
This is distinctly different from elastic electron scattering in a
crystal, where phase coherence is maintained. Here constructive
interference occurs at the Bragg angles, giving rise to the familiar
diffraction patterns. The following discussion does not apply to
the diffraction case.
The mean free path is given by
1
µ =
,
(4.192)
N σ
where N is the number of atoms per unit volume, and σ is the
total scattering cross section. For fast electrons in a typical solid,
µ for elastic scattering is proportional to the incident energy in the
first Born approximation. Consequently, µ ranges from a few tens
of nanometers at an incident energy of 10 KeV to a few hundreds
of nanometers at 1 MeV. The sections observed in a transmission
electron microscope must be thin relative to the mean free path, in
order to avoid degradation of the image due to multiple scattering
of the beam electrons. As this is not always possible, multiple scattering must be considered in the image formation. In this section
we derive a method for understanding the scattering as a function
of the sample thickness, measured in units of the mean free path.
We define a dimensionless quantity n = z/µ, which we call the
reduced thickness, where z is the thickness, measured in units
of length. The probability P of an electron undergoing exactly j
scattering events in the reduced thickness n is governed by Poisson
statistics, namely
j
n
P j (n) =
e
−n .
(4.193)
j!