Transmission Electron Microscope_1

Scattering and Diffraction:
1. Elastic scattering usually occurs at relatively low angles (1-10^o), i.e., in the forward direction.
2. At higher angles (>10^o) elastic scattering becomes more incoherent.
3. Inelastic scattering is almost incoherent and relatively low angle (<1^o) forward scattering.
4. As the specimen gets thicker, less electrons are forward scattered and more are backscattered until primarily incoherent backscattering is detectable in bulk, nontransparent specimens.

A convenient definition of a small angle is 10 mrads:
1 mrad = 0.0573^o, 10 mrads is about 0.5^o.

The origin of Kikuchi lines:
Kikuchi patterns are formed by inelastically scattered electrons.
If the specimen is thick enough, we will generate a large number of scattered electrons which travel in all directions, i.e., they have been incoherently scattered but not necessarily in elastically scattered. They are sometimes refered to as diffusely scattered electrons.

• The diffusely scattered electrons have the same λ as the incident electrons since energy losses are small compared to E₀.


• When first formed, most of the diffusely scattered electrons travel close to the direction of the incident beam.


• The ideal specimen thickness will be such that we can see both the spot pattern and the Kikuchi lines. This is one of the few illustrations when thinner is not necessarily better.


• The Kukuchi lines consist of an excess line and a deficient line. In the DP, the excess line is further from the direct beam than the deficient line.


• The Kikuchi lines are fixed to the crystal so we can use them to identify orientation accurately.


• The trace of the diffracting planes is midway between the excess and deficient lines.
  

Resolution and Wavelength:
It is easier to think image resolution in TEM as the smallest distance that can be resolved, the concept is borrowed from the classical optical microscope Raley criterion:

δ=(0.61λ)/μsinβ

μ is the refractive index of the viewing materials;β is the angle of collection of the magnifying lens;
λ is the electron wavelength, which can be calculated in the following simple way ignoring relativistic effects:
λ=1.22/√E

Considering relativistic effects:

Electron Properties as a Function of Accelarating Voltage

Accelerating Voltage(kV)	relativistic wavelength (nm)	Mass(x m0)	Velocity(x108m/s)
100	0.0037	1.196	1.644
200	0.00251	1.391	2.086
300	0.00197	1.587	2.330

Microscopy Related websites:
http://cimewww.epfl.ch/
Scattering and Diffraction:
Diffraction: An interaction between a wave of any kind and an object of any kind.
Scattering: The process in which particles, atoms, etc., are deflected as a result of collision.
So scattering might best apply to particles and diffraction to waves.

Dislocation Density:
The dislocation density is a measure of how many dislocations are present in a quantity of a material. Since a dislocation is a line defect, this is defined as the total length of dislocation per unit volume. Consequently the units are m/m³ = m^-2. Equivalently, it is the number of dislocation lines intersecting a unit area. Dislocation density is usually of the order of 10¹⁰ m^-2 in a metal, increasing to ~10^-15 m^-2 after work hardening.

Relationship Between R and L:
L,Camera Length:distance of the film from the diffraction pattern on the specimen.

....| Incident Beam
..===== Specimen
....| ...L|  ....|   \Diffracted beam
....|    ....o__R__O <- Diffraction spot.

Rd=λL
R is the distance between the diffraction spots.
d is the atomic spacing.
Structure Factors:
F_(hkl)=∑f_ie^{2πi(hx_i+ky_i+lz_i)}
This is the key equation, it is completely general.
BCC:
F=2f if h+k+l is even
F=0 if h+k+l is odd

FCC:
F=4f if h,k,l are all even or all odd
F=0 if h,k,l are mixed even and odd.

HCP:
|F|²=0 if h+2k=3m and l is odd
|F|²=4f² if h+2k=3m and l is even,
|F|²=3f² if h+2k=3m+1 and l is odd,
|F|²=f² if h+2k=3m+1 and l is evn.

NaCl:
F={f_(Na)+f_(Cl)e^πi(h+k+l)}{1+e^πi(h+k)+e^πi(h+l)+e^πi(k+l)}
F=4(f_(Na)+f_(Cl)) if h,k,l are all even,
F=4(f_(Na)-f_(Cl)) if h,k,l are all odd,
F=0, if h,k,l are mixed.