Electrons as Waves
λ = h/ρ = h/mv
De Broglie, 1924
v =c
m=
mo
1-v2/c2









1
 2



2
 
 
2  
o
 
1
1− 1+ meVc




Lorentz, 1905
λ = hc(2eVmoc2 + e2V 2)-1/2
E = mc2 = moc2 + eV
Einstein, 1905
MSE 421/521 Structural Characterization
Bragg Diffraction
incident waves
1
θ
d
2
Path difference = 2dsinθ
specimen
θ
θ
θθ
θθ
dsinθ
dsinθ
If 2dsinθ is a multiple of λ
then waves are in-phase
d
Condition for diffraction:
2dsinθ = nλ
crystal planes
2θ
transmitted
waves
diffracted
waves
MSE 421/521 Structural Characterization
Diffraction in the TEM
Sample
(d = planar spacing)
2dsinθ = λ
sinθ ≈ θ
∴2θ = λ / d
2θ
tan(2θ) = R/L
tan(2θ) ≈ 2θ
∴2θ ≈ R /L
L
∴Rd = Lλ
T
D
Spacing “R” in diffraction pattern
is inversely proportional
to real crystallographic
interplanar spacing “d ”.
R
MSE 421/521 Structural Characterization
Spot Patterns
Single, perfect crystal.
SADP is regular array of spots.
Large number of randomly-oriented grains
SADP contains continuous Debye rings
Small number of grains (crystals) with
different orientations.
SADP contains discrete spots, each lying
on Debye ring.
MSE 421/521 Structural Characterization
Amorphous material
No distinct SADP
Reciprocal Lattice
In reciprocal space,
no one can hear you scream.
[001]
[111]
(002)
(022)
(202)
(222)
(111)
000
[010]
(020)
(200)
[100]
[110]
a
fcc
(220)
1/a
bcc
MSE 421/521 Structural Characterization
Ewald Sphere
Ewald Sphere
1/λ
Incident
Beam
1/λ
SOLZ
FOLZ
Reciprocal
Lattice Planes
MSE 421/521 Structural Characterization
ZOLZ
Symmetry of the HOLZ
ZOLZ
6-fold symmetry
FOLZ
true 3-fold symmetry
Nd2Hf2O7 [111] zone axis
MSE 421/521 Structural Characterization
Diffraction Theory
Kinematic Theory
(Zero-Order Approximation)
Intensity of diffracted waves
is small
Electrons scattered only once
No interaction between
incident and scattered beams
Validity:
– very thin crystals
– far from exact Bragg angle
(weak diffracted beam)
MSE 421/521 Structural Characterization
Dynamical Theory
Intensity of diffracted
waves is large
Accounts for absorption
No limit to number of
electron scattering events
Incident and scattered
waves interact
Always valid
Double Diffraction
006
442
002 111 220
111
111
220 111 002
006
442
Fd3m
MSE 421/521 Structural Characterization
Kikuchi Lines
S. Kikuchi, 1928
β
Deviation Parameter
s = λx/Rd2 = x/Ld
2θ
2θ
E
g
D
s>0
α
T
R = λL/d
x = sLd
s>0
s = 0 (Bragg)
s<0
MSE 421/521 Structural Characterization
Uses of Kikuchi Patterns
440
19 5 13
220
Structural Analysis
044
13 19 5
Symmetry is precisely
that of the crystal
Contrast Work
404
A road map
through reciprocal space
5 13 19
404
Stereo Microscopy
Measuring parallax and
obtaining 3D information
202
044
440
Nd2Hf2O7 [111] zone axis
MSE 421/521 Structural Characterization
Crystal Shape Factor
z
y
real space
x
reciprocal space
rel rod
MSE 421/521 Structural Characterization