Transcript ch3x - MyCourses

Chem E5225 – Electron Microscopy
P
Elastic scattering
Particles and waves
• The electron beam can be observed to be both as
particles and as a beam. As particles the electrons
have a scattering cross section and a differential
scattering cross section. The electrons can interact with
the nucleus and the electron cloud surrounding the
atoms from the TEM specimens. The electrons also act
with a wave nature because waves are created by
atoms diffracting waves creating scattering centres.
The strength of the wave scattering by an atom is
determined by the atomic scattering amplitude. Solids
create a more complicated diffraction but this is central
to TEM
Particles and waves
Mechanisms of elastic scattering
Elastic scattering around a single particle can
occur in two ways
Particles and waves
• The other method
of elastic electron
scattering happens
when the electron
wave interacts with
the TEM specimen
as a whole.
Elastic Scattering from Isolated Atoms
• The path of the electron can either interact
with the electron field resulting in a small
deviation from the original path or a strong
deviation from an interaction with the
nucleus.
𝑟𝑒 = 𝑒 𝑉𝜃(1)
𝑟𝑛 = 𝑍𝑒 𝑉𝜃 (2)
The Rutherford cross section
• This only concerns the electron – nucleus
interactions.
𝑒4𝑍2
𝑑 𝑜𝑚𝑒𝑔𝑎
𝜎𝑟 (𝜃) =
16(4𝜋𝜀0 𝐸0 )2 𝑠𝑖𝑛2 𝜃
2
Modifications to the Rutherford cross
section
•
There are many similar differential cross
sections, the Rutherford cross section
doesn’t take into account the screening
effects the electron cloud.
So far the equations 4 and 3 are non –
relativistic, since relativistic are significant
for electrons above 100 KeV
Net result of adding the screening and the
relativistic corrections is that the Rutherford
cross section changes to equation (6).
1
𝜃0 =
𝑎0 =
𝜎𝑟 (𝜃) =
0.117𝑍 3
𝐸0
1
2
ℎ 2 𝜀0
𝜋𝑚0 𝑒 2
𝑍 2 𝜆4𝑅
64(𝜋4 𝑎02 )2
(4)
(5)
𝑑Ω
𝜃 𝜃0 2 2
2
𝑠𝑖𝑛 2 ( 4 )
.
(6)
Modifications to the Rutherford cross
section
The atomic scattering Factor
• An aspect of the wave approach
is the atomic scattering factor
which can be related to the
differential elastic cross section
𝑓 𝜃 is complimentary to the Rutherford
differential cross section analysis, most notably
low angle elastic scattering where the
Rutherford approach is lacking
𝑓 𝜃
2
𝑑𝜎 𝜃
=
(8)
𝑑Ω
The origin of F(θ)
We can describe the incident beam as a wave
of amplitude
𝜑 = 𝜑0 𝑒 2𝜋𝑖𝑘𝑟 (10)
When the incident plane wave is scattered by
the atom a spherical wave is created around
that point with a different amplitude, but it
keeps the same phase
Simple Diffraction Concepts
Diffraction Equations
The Path difference between scattered waves
is AB-CD. B and C are the atoms In 2
dimensions the path difference is:
𝑎 𝑐𝑜𝑠𝜃1 − 𝑐𝑜𝑠𝜃2 = ℎ𝜆
And for 3 dimensions two more Laue
equations :
𝑏 𝑐𝑜𝑠𝜃3 − 𝑐𝑜𝑠𝜃4 = 𝑘𝜆
𝑐 𝑐𝑜𝑠𝜃5 − 𝑐𝑜𝑠𝜃6 = 𝑟𝜆
Diffraction Equations
• usually simpler
approach is
Used in TEM,
that waves are
reflected off
atomic planes
(bragg and
bragg 1913)
Bragg’s Law
𝑛𝜆 = 2𝑑𝑠𝑖𝑛𝜃𝐵