Download Transmission Electron Microscopy 3. Elastic Scattering Outline

Transmission Electron Microscopy
3. Elastic Scattering
EMA 6518
Spring 2009
01/21/09
EMA 6518: Transmission Electron Microscopy
C. Wang
Outline
•Particles and Waves
•Mechanisms of Elastic Scattering
•Scatter from Isolated Atoms
•The Rutherford Cross Section
•Modifications to the Rutherford Cross Section
•Coherency of the Rutherford-Scattered Electrons
•The Atomic Scattering Factor
•Diffraction equation
•Inelastic scattering
•Beam damage
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Particles and Waves
• Electrons are particles---X-ray and electron spectrometry
– They have a scattering cross section and differential
scattering cross section
– They can be scattered through particular angles
– The electrons interact with the nucleus through
Coulomb forces
– We can relate this process to scattering of other
particles, such as α particles, so lots of analysis can
carry over form other systems
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Particles and Waves
•
Electrons have a wave nature and the electron beam is
almost a plane wave---imaging, HRTEM, and diffraction
patterns
– Waves are diffracted by atoms or “scattering centers”
– How strongly a wave is scattered by an atom is
determined by the atomic scattering amplitude
– We can relate the process to the scattering of X-rays,
so lots of analysis already exists
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Mechanisms of Elastic Scattering
EMA 6518: Transmission Electron Microscopy
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Scattered by an Isolated Atom
• The electron may interact
with the electron cloud,
resulting in a small angular
deviation
• If an electron penetrates the
electron cloud and
approaches the nucleus, it will
be strongly attracted and may
be scattered through a larger
angle that in rare cases in the
TEM can approach 180º
(backscattering)
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Collective Scattering from Many Atoms
A plane coherent electron
wave generates secondary
wavelets from a row of
scattering centers (e.g.,
atoms in the specimen).
The secondary wavelets
interfere, resulting in a
strong direct (zero-order)
beam and several orders of
coherent beams scattered
(diffracted) at specific
angles.
EMA 6518: Transmission Electron Microscopy
C. Wang
Mechanisms of Elastic Scattering
(1) Electron scattering from isolated single atoms:
•Both two cases involve Coulomb forces
• In fact, many electron-electron interactions are inelastic. We
will ignore any inelastic effects now
(2) Elastic scattering when the electron wave interacts with the
specimen as a whole:
• Diffraction is particularly important at low angles
•The low angle elastic scattering distribution is modified by the
crystal structure of the specimen, and intense diffracted beams
emerge at certain specific angles
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Scatter from Isolated Atoms
• Elastic electron-electron interactions usually result in
a relatively low scattering angle, while electronnucleus interactions cause higher-angle scattering
e 2
)
Vθ
Ze
2
= πr n = π ( ) 2
Vθ
σ electron = πre 2 = π (
σ nucleus
EMA 6518: Transmission Electron Microscopy
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Rutherford Cross Section
• High angle electron-nucleus interaction is analogous to
the backscattering of α particles (composed of two
protons and two neutrons) from a thin metal foil.
1911, undergraduate research result!
• The positive matter in atoms was concentrated in an
incredibly small volume and gave birth to the idea of the
nuclear atom.
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Rutherford Cross Section
dσ (θ )
e4Z 2
=
(Rutherford differential cross section)
θ
dΩ
16( E0 ) 2 sin 4
2
Z
θ
(single isolated atom)
σ nucleus = 1.62 ×10− 24 ( ) 2 cot 2
E0
2
Qnucleust = ( N 0
ρ
A
t )σ = 1.62 ×10−24 ( N 0
ρ
A
t )(
Z 2
θ
) cot 2
E0
2
(scattering from atoms in a TEM specimen of thickness t)
The beam energy (E0), the angle of scattering (θ), and the atomic number (Z)
all affect the probability that an electron will be scattered by the nucleus.
EMA 6518: Transmission Electron Microscopy
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Modifications to the Rutherford Cross
Section
• When the electron does not pass close to the nucleus,
the scattering angle will be small (<5º) because
screening is important. sin 2 (θ / 2) → sin 2 (θ / 2) + (θ 0 / 2) 2
0.117 z1 / 3
θ0 =
1/ 2
E0
screening
parameter
h 2ε 0
a0 =
πm0e 2
When the scattering angle is greater than θ0,
we can neglect electron-electron interactions
and the nuclear interaction is dominant
Bohr radius of the scattering atom, 0.5Å
4
dσ (θ )
λR Z 2
Screened relativistic
=
θ
θ
dΩ
64π 4 (a0 ) 2 (sin 2 + ( 0 ) 2 ) 2 Rutherford cross section
2
2
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Screened Relativistic Rutherford Cross
Section
Limitation: voltages < 300-400V; Z<30
• Scattering is most likely to occur in the forward direction
• Higher-energy electrons are less likely to be scattered
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Screened Relativistic Rutherford Cross Section
If you keep your specimen below
100-nm thickness, you can then
approach the ideal of single
scattering
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Coherency of the Rutherford-Scattered Electrons
High-angle Rutherford-scattered electrons are incoherent:
The phases of the electron waves are not in step
• Z-contrast images:
The high-angle forward scattering can be used to form
exceptionally high-resolution images of a crystalline
specimen in which the image contrast is due to the value of
Z, not the orientation of the specimen
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C. Wang
Coherency of the Rutherford-Scattered Electrons
The high-angle backscattered electrons can be used to form
images of the beam entrance surface of the specimen in
which the contrast is not only due to differences in Z but also
to changes in surface topography of the specimen
Backscattered electron images are rarely used in the TEM!
For example: Cu, 103 incident electrons
Only 0.3% are backscattered from foil specimen
30% are backscattered from bulk sample
SEM
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Atomic Scattering Factor
Classical Rutherford differential
cross section ignores the wave
nature of the EB
• Atomic scattering factor f(θ)
2
f (θ ) =
dσ (θ )
dΩ
•f(θ) is a measure of the amplitude of an electron wave
scattered from an isolated atom
2
• f (θ ) is a proportional to the scattered intensity
2
E0 

)
m0c 2  λ 

 (Z − f x )
8π 2 a0  sin θ 



2
(1 +
f (θ ) =
EMA 6518: Transmission Electron Microscopy
C. Wang
Atomic Scattering Factor
•f(θ) depends on λ, θ, and Z:
•f(θ)
as θ
•f(θ)
as λ
•f(θ)
as Z
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Remember!
Both the differential cross section and the scattering
factor are simply a measure of how the electron
scattering intensity varies with θ
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Diffraction Equation
Bragg equation
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Inelastic Scattering
• Historically, the conventional TEM only used two elastic
signals, namely, the direct beam and the diffracted
beams.
• Inelastic scattering:
– Processes that generate X-rays
– Processes that generate other (secondary) electrons
• Slow secondary electrons, Auger electrons, electronhole pairs and cathodoluminescence
– Processes that result form collective interactions with many
atoms
• Plasmons and phonons
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Beam Damage
•
•
•
•
•
•
Electron dose
Specimen heating
Beam damage in polymers
Beam damage in covalent and ionic crystals
Beam damage in metals
Sputtering
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