Scattering of electrons

1
What happened during the electron beam/specimen
interaction?

• Scattering of electrons
• Elastic scattering
• Coherent
• Incoherent
• Inelastic scattering

2
Electron as particle vs. wave
Consider relativity
3
Electron as wave
The solution of the Schödinger equation at
constant V or in the absence of V
Plane wave
If V 0, then the equation can have spherical
wave for the solution
4
The definition of scattering vector
Start from a small volume in the atom
k wave vector of the incident beam k wave
vector of the scattered beam K scattering
vector O origin
5
Scattering from atom
Based on Huygens-Frenel principle
Atomic scattering factor
Potential function
Phase difference
Introduce the screen factor of electrons
f(q) increase as l increases increase as Z
increases decrease rapidly as q increases
6
The resulting spherical wave
7
Scattering from Unit cell
Atomic scattering factor (j atoms in one unit
cell)
X(x,y,z) coordinate of atom
F(q) structure factor
8
Scattering from crystal lattice
n number of unit cells
In real space
In reciprocal space
When
m is a integer
We have constructive interference
Which is
9
is integer for any given u, v, w
In reciprocal space
The Bragg Law
In real space
10
The construction of Ewald sphere
Direct correspondence of electron diffraction and
the reciprocal space
O
11
Then, What is an electron diffraction?
O
1/l
B
A
R
L
k
k
D
K
C
E
R
The enlarged projection of 2D reciprocal space
12
O
Diffraction maximum occurs at
1/l
But reflection takes place over a range of angles
k
k
Deviation from the reciprocal lattice point
K
s
r
Mn number of unit cell along the a, b and c
directions
13
So the diffraction is no longer a point
14
Effect due to structural factor
The choice of non-primitive lattice
BCC (0,0,0), (1/2, 1/2, 1/2)
15
When hkl2n, F2f hkl?2n, F0
c
b
a
O(0,0,0)
16
Effect due to the systematic absence
The existence of symmetry element
Glide plane Screw axis
17
Example
b Glide plane (100), glide distance b/2
(-x,y1/2,z)
b
(x,y,z)
a
When k2n, F ? 0 k ? 2n, F0