Electron diffraction: SAED and CBED - CIME - EPFL

Electron diffraction: SAED and CBED 

P. Stadelmann 

CIME-EPFL 

Bât. MX-C, Station 12 

CH-1015 Lausanne 

Switzerland 

Pierre.Stadelmann@epfl.ch 

30 janvier 2009 

P. Stadelmann CIME-EPFL Bât. MX-C, Station 12 CH-1015 Lausanne Electron Switzerland diffraction: Pierre.Stadelmann@epfl.ch 

SAED and CBED 

Electron diffraction : important concepts 

1 Elastic scattering. 


SAED and CBED



2 Ewald sphere. 






3 Crystal lattice & reciprocal lattice. 






3 Crystal lattice & reciprocal lattice. 

4 Structure factor. 



Electron diffraction : elastic scattering 

Assumptions =⇒ elastic scattering 

�ki : incident wavevector 

�kg : scattered wavevector 

�g : reciprocal lattice vector 

�sg : deviation from exact Bragg condition 

1 Energy conservation : |�kg| = |�ki|. 



Electron diffraction : elastic scattering 

Assumptions =⇒ elastic scattering 

�ki : incident wavevector 

�kg : scattered wavevector 

�g : reciprocal lattice vector 

�sg : deviation from exact Bragg condition 

1 Energy conservation : |�kg| = |�ki|. 

2 Momentum transfer : �ki +�g + �sg = �kg. 



X-Ray diffraction : Ewald sphere picture 

�sg = 0 



X-Ray diffraction : Bragg law 

With energy conservation and momentum transfer (�sg = 0) : 

=⇒ Bragg law : 

|�ki +�g| = |�kg| 

k 2 

i + 2 × ki × g × cos(�ki,�g)+g 2 = k 2 g 

2ki × cos(�ki,�g) =−g 

2ki × cos(90 o − θB) =−g 

2 

λ × sin(θB) =g = 1 

dg 

2 × dhkl × sin(θB) =λ 



Electron difrraction : Ewald sphere picture 

Two beams at Bragg condition : �sg = 0 



Relrod shape 

Why 

Answer : 

� �2 sin(πζ) 

πζ 

= sinc 2 (ζ) ? 

Specimen is in a box b(x, y, z) given by : 

� � � � 

x y 

b(x, y, z) =rect rect 

naa nbb 

rect 

� � 

y 

ncc 

−→ relrod is the Fourier transform ˜B(ξ, ψ, ζ) of b(x, y, z) : 

relrod(ξ, ψ, ζ)) = ˜B(ξ, ψ, ζ) ˜B ∗ (ξ, ψ, ζ) 



Relrod shape 

Since na and nb are very large and nc pretty small (why ?) : 

˜B(ξ, ψ, ζ) =sinc(naaξ)sinc(nbbψ)sinc(nccζ) ≈ δ(ξ)δ(ψ)sinc(nccζ 

˜B(ξ, ψ, ζ) ≈ δ(ξ)δ(ψ)sinc(nccζ) 

(See Mathematica notebook relrodShape.nb). 



Relrod shape : a = b = c = 0.5 nm 

t = 1 nm 

1.0 

0.8 

0.6 

0.4 

0.2 

- 2 - 1 1 2 

2 / t 

(see Mathematica notebook relrodShape.nb). 



Crystal lattice : [u, v, w] indices 

z 

[0, 0, 1] 

[0, 0, 0] 



[0, 1, 1] 

[0, 1, 0] 

y 

[1, 0, 1] 

[1, 0, 0] 

[1, 1, 0] 

x 

- 1 

� nm

ED reciprocal space : lattice of relrods 



(h,k,l) lattice planes (view down [0, 0, 1]) 

(0, 1, 0) 

y 

(1, 0, 0) 

(h, k, l)? 



x

Stereographic projection : cubic 



(h,k,l) lattice planes : triclinic 



Stereographic projection : triclinic 



Structure factors 

The structure factor gives the scattering strength of (h,k,l) 

planes. 

where : 

Fhkl = ∑ 

i=atomes 

fi(shkl)e (2πı(hxi+kyi+lzi)) 

1 fi(shkl) is the atomic scattering amplitude. 

2 (xi, yi, zi) are the fractional coordinates of atom i 

(0 ≤ xi < 1). 

3 shkl = sin(θ B) 

λ 

= 1 

2d hkl . 



Atomic scattering amplitude 

(0,0,1) 

(0,0,2) 

(0,0,4) 

Question : why is the atomic scattering amplitude decreasing with s ? 



Examples : extinction rules 

Simple cubic : (hkl) no condition. 

1 atom at (0, 0, 0). 

−→ Fhkl = fi(shkl) 

Body centered cubic : (hkl) : h + k + l = 2n 

2 atoms at (0,0,0) and ( 1 

2 

, 1 

2 

1 , 2 ). 

Fhkl = fi(shkl) 

� 

1 + e πı(h+k+l)� 

Face centered cubic : (hkl) all even or odd. 

4 atoms at (0, 0, 0), (0, 1 1 1 1 

2 , 2 ), (1 

2 , 0, 2 ), (1 

2 , 2 , 0) 

� 

Fhkl = fi(shkl) 1 + e πı(h+k) + e πı(h+l) + e πı(k+l)� 



FCC extinction rules : ZOLZ [001] 

ZOLZ (zeroth order Laue zone : �g ·�u = 0) 

h k l Fhkl 

0 0 2 4 

2 0 0 4 

-2 0 0 4 

0 2 0 4 

0 -2 0 4 

2 2 0 4 

2 -2 0 4 

... 



SAED : two beams and exact Bragg condition 



SAED : two beams and near Bragg condition 



�sg deviation from exact Bragg condition 



�sg deviation from exact Bragg condition 



Diffracted intensity 



Intensity : two beams and exact Bragg cond. 



Intensity : two beams and near Bragg cond. 



Diffraction with // beam : exact Bragg cond. 

How does the intensity changes with specimen thickness ? 



Intensity : two beams, Al, no absorption 

Note that the diffracted beam is out of phase by π 

2 . 



Intensity : two beams, Al, with absorption 

Note that the diffracted beam is out of phase by π 

2 . 



Intensity : rocking curves, two beams, Al, with 

absorption 

Specimen thickness 50, 100, 150, 200 nm. 



Intensity : electron channeling (why ?) 

Channeling highly dependent of specimen orientation → 

ALCHEMI. 



[uvw] zone axis diffraction 

�ki parallel à [uvw]. 



Al [001] zone axis diffraction 

P. Stadelmann �ki parallel CIME-EPFL à Bât. [001]. MX-C, Station Note 12 CH-1015 (hkl) Lausanne all even Electron Switzerland or diffraction: Pierre.Stadelmann@epfl.ch 

odd. SAED and CBED 

[uvw] zone axis pattern : 3-D diffraction 



Laue zone planes : side view 



SAED on zone axis 



Indexing Zone Axis Pattern (ZAP) 



SAED summary : good news 



SAED : bad news 1 



SAED : bad news 2 



SAED : conclusions 



How to overcome SAED limitations ? 

1 Conical illumination → full rocking curve at once. 

2 Focused beam → very small irradiated area. 



CBED : important concepts 



CBED : accessible information 

Advantages : 

1 Intensities. 

2 Symmetries. 

3 Very precise alignment of diffraction conditions. 

4 Very small volume (≤ 20 nm 3 ). 

Problems : 

1 Contamination. 

2 Dynamical behavior → thick specimen. 

3 Longer calculations. 

4 Energy filtering necessary. 



CBED : rays scheme 



CBED disks 



CBED disks : excess/deficiency lines 



CBED : rocking curve 



CBED : rocking curve 



CBED : lines and fringes 



CBED : HOLZ lines 



CBED : ZAP HOLZ lines 



CBED : convergence angle effect 



CBED : convergence angle effect 

Why increasing incident beam convergence is very useful ? 



LACBED : Large Angle CBED 



LACBED : overlap of disks 



LACBED : spherical aberration effect 



LACBED : superposition of information 



LACBED : SAED aperture 



LACBED : deficiency lines 



LACBED : dark field 



LACBED : independence of Δh 



LACBED : energy filtering 



LACBED : crystal direction mapping 



LACBED : conclusions

Electron diffraction: SAED and CBED - CIME - EPFL

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