Two-beam dynamical electron diffraction
Francisco Lovey
Centro Atómico Bariloche
Instituto Balseiro
PASI on Transmission Electron Microscopy
Santiago de Chile, July 2006
Bibliography
Electron Microscopy of Thin Crystals. P.B. Hirsh, A. Howie, R.B.
Nicholson, D.W. Pashley and M.J. Whelan. Butterworths (1965)
The Scattering of Fast Electron by Crystals. C.J. Humphreys,
Reports on Progress in Physics (1979).
Electron Microdiffraction, J.C.H. Spence and J.M. Zuo. Plenum
Press (1992).
Diffraction Physics. J. M. Cowley. Elsevier (1995).
Transmission Electron Microscopy (I-IV). D.B. Williams and C.B.
Carter. Plenum Press (1996).
Transmission Electron Microscope. L. Reimer. Springer (1997).
Introduction to Conventional Transmission Electron Microscopy.
M. De Graef. Cambridge (2003).
DYNAMICAL THEORY OF ELECTRON DIFFRACTION
Time independent Schrödinger equation.
2
8

m0e
2
E  V(r)(r)  0
  (r ) 
2
h
The solution in the vacuum [V(r) = 0]
(r )  e
2
h 2k 0
 eE
2m
2 ik 0 .r
Relativistic correction
eE  (m  m 0 ) c 2
m
m0
1  v2 / c2

1
h
 
k0 p
h

eE

2m 0 eE1 
2
2
m
c
0




The potential V(r) is periodic and can be expressed as a
Fourier series
h2
V(r ) 
2me
2 i g.r
2 i g.r
U
e

V
e
 g
 g
g
g
h2
Vg  U g
2me
The wave function can take the general form of
Bloch waves
 (r )   
kj
(k j )
C
( k j ) 2 i ( k j  g ).r
g
e
g
The Schrödinger equation becomes
 e
(k j )
kj
g
2 i ( k j  g ).r



8 2 meE (k j )
2
(j)
2 (k j )
2
(k j )
(k j )

C g  4  U o C g   ' U h C g h   0
 4 (k  g) C g 
2
h



The basic eigenvalue equation
 e
(k j )
kj
2 i ( k j  g ).r
g



8 2 meE (k j )
2
(j)
2 (k j )
2
(k j )
(k j )

C g  4  U o C g   ' U h C g h   0
 4 (k  g) C g 
2
h



Since the plane waves are orthogonal to each other, each coefficient
in the last equation must be equated separately to zero
K
2

 (k  g) C
(j)
2
(k j )
g
  ' Uh C
(k j )
gh
0
h
2meE
K 
 Uo
2
h
2
K 2  k 0  Uo
2
The boundary conditions
At the surfaces of the crystal the wave function and its
gradient must be continuous
At the entrance surface
e
2 ik 0 .r
 
kj
(k j )
C
( k j ) 2 i ( k j  g ).r
g
e
g
At z = 0 there is no diffraction thus C

(k j )
C
(k j )
g
(k j )
g
0
g0
 0, g
j
The tangential components of all wave vectors must be equal to the
tangential component of the incident beam
k 0 t  k (t j)
k0t  Kt
Any wave vector inside the specimen can be written as the
incident wave vector plus a correction normal to the surface
k ( j)  K   ( j) n
n
K
K+g+Sg
k  Kn  
(j)
n
Sg
g
000
S< 0
Sg excitation error
( j)
After neglecting the back-reflected electrons
2KSg C   U gh C
( j)
g
h
( j)
h
g n ( j) ( j)
 2K n (1 
)  Cg
Kn
In matrix notation
1

g
2K n 1  n
 Kn



.. 0   0 U -g1 ... U -g n
0 0

 
0   U g1 0
U -g1
 0 2KSg1



 
 0 0 ... 2KSg   U U
n 

 g n g n -g1 ... 0
AC col   Ccol
( j)
(j)
C (j)
 C0
0
 (j)
(j)
C
 C g1
g1
( j)



(j)
 C (j)
C
gn
 gn
 (r )   
(k j )
kj
C
( k j ) 2 i ( k j  g ).r
g
e
g
The wave function at the exit surface
(r )   
( j)
j
C
j
g
e
2 i ( K   ( j ) n  g ).r
g

(r )     ( j) Cgj e 2 i
g  j
( j)
n .r
 2 i ( K g ).r
 e
The amplitude of the transmitted and diffracted beams
 g (r )    C e
( j)
j
j
g
2 i ( j ) n .r
The amplitude of the transmitted and diffracted beams
 g (r )    C e
( j)
j
g
2 i ( j ) n .r
j
In matrix notation
  (z)   C
  (z)   C


  (z)   C

 
0
g1
g n -1
(1)
0
(1)
g1
(1)
g n -1
C ... C   e
C ... C   0

C ... C   0
(2)
(j)
0
(2)
0
(j)
g1
g2
(2)
(j)
g n -1
g n -1

C e
2 i (1)n.r
0
e
2 i (j) n.r
2 i (2)n.r
0

D
.... 0   
.... 0   

 
e

2 i (j)n.r
α
(1)
(2)
(j)





Using the boundary conditions at the entrance surface

C e
2 i (j)n.r
D
 C  
(j)
j
α
Cα  u
(j)
g
g (0) = 0,g
0 ,g
αC u
-1
1 
 0
u 
 0
 
For nearly normal incidence

C e
2 i (j) z

D
-1
C u
~*
C C
-1
Two beam dynamical expressions
AC col   ( j)Ccol
1

g
2K n 1  n
 Kn



.. 0   0 U -g1 ... U -g n
0 0

 
0   U g1 0
U -g1
 0 2KSg1


 

 
 0 0 ... 2KSg   U U
n 

 g n g n -g1 ... 0
(j)



C
 0 U -g  C(j)
g
0

    2K n (1  n )  ( j)  0 
(j)
 U 2KS  C(j)
Kn
g
 g 
 g
Cg 
g n ( j)



2
K
(
1

)
U -g

 (j)
n
K
n

 C 0   0
 

g n ( j)  C(j)
2KSg  2K n (1 
)   g 
 U g
Kn


2


g n ( j) 
g n ( j) 
2
K
(
1

)


2
KS
2
K
(
1

)    U g U g  0
g
n
 n

K
K




n
n
(j)
C (j)
 C0
0
 (j)
(j)
C
 C g1
g
  ( j) 1


 C (j)
C (j)
gn
 gn
Eigenvalues

(1,2)

1
g
2K n (1  n )
Kn
KS 
g
K 2Sg2  U g U g

Centrosymmetric crystal U  U
g
g

(1,2)

Ug
gn
2K n (1 
)
Kn
w 
w  Sg  g
w2 1

Extinction
distance
K
g 
Ug
(ksai)
Nearly normal incidence
Kn  K
g n  K

(1,2)


w
2
g
1
w2 1

Nearly normal incidence
k
K
(1)
k
( 2)
K
g
1
2
g
n
1
2
n
K
g 
Ug
k ( 2)  K 
w  Sg  g
S=0
1
g
k (1)  K 
S< 0
k(1)
k(1)+g
k(1)
K
K+g
K
(2)
(2)
k
k +g
k(2)
g
1
2
g
1
2
w 
w 2 1 n

w 
w2 1 n

Dispersion surfaces
k(1)+g+Sg
K+g+Sg
k(2)+g+Sg
Sg
000
g
000
g
Calculation of the coefficients Cg (centrosymmetric potential)
g n ( j)
 0 U-g  C (j)0 
)
 U 2KS  C (j)   2K n (1 
g   g 
Kn
 g

(1,2)

C (g1, 2)
C
(1, 2 )
0
Ug
g
2K n (1  n )
Kn
w 
w2 1
C (j)0 
C (j)g 

Normalization condition
 w  w 1
cot   w
C
2
(i ) 2
0
C
C(01)  C(g2)  sin(  / 2)
C
(1)
g
C
( 2)
0
 cos( / 2)
(i ) 2
g
1
Two beams wave functions

C e
 0 (r)    C
  (r)   C
 g  
(1)
0
(1)
g
C
C
2 i (j) z
(2)
0
(2)
g

  e 2 i
 
 0
General
expression
-1
C u
D
(1)
n.r
0
e
2 i (2)n.r
(1)
  C(1)
C
  0(2) g(2)
  C0 Cg
1 
 
0


Nearly normal incidence
0 (z)  C
C(01)  C(g2)  sin(  / 2)
C
(1)
g
C
( 2)
0
 cos( / 2)
(1) 2
0
g (z)  C C
(1)
0
e
2 i (1) z
(1)
g
e
C
2 i (1) z
(2) 2 2 i ( 2 ) z
0
C
e
(2)
0
(2) 2 i ( 2 ) z
g
C e
Nearly normal incidence
2

w
 1 
2
sin 
z



2
g


g (z) 
( w 2  1)
 (z)  1  g (z)
2
g
k (1)  K 
2
k ( 2)  K 
k  k
(1)
1
g
1
2
k
w 
w 2 1 n
w 
w2 1 n
( 2)


w 2 1

n
g
2
The diffracted intensity
vanishes when
z = 0,
g
w 2 1
, 2
g
w 2 1
w  Sg  g
Extinction contours in a bent foil
2

w
 1 
2
sin 
z



2
g


g (z) 
( w 2  1)
w  Sg  g
Two-beam convergent beam electron diffraction
2

w
 1 
2
sin 
z



2
g


g (z) 
( w 2  1)
000
400
w  Sg  g
Interpretation of the two Bloch waves

(1)
0

(2)
0
 (r )  C
 (r )  C

e
e
2 ik (1) .r
2 ik ( 2 ) .r
2
 (r )  1 
2
  (r )  1 
(1) 2 i(k (1) g ) .r
g
C e
(2) 2 i(k ( 2 ) g ) .r
g
C e
Taking the x direction normal
to the diffracting planes g,
then g.r = x/a
cos 2x / a
1 w2
cos 2x / a
1 w2

The wave  (r ) has it maximum intensity on the planes
x = na, this is along the atoms. On the other hand the wave
  (r ) has it maximum intensity in between the atoms.
This is related to the kinetic energy. The k(1) vector is longer


)
that k(2), thus electron represented by
will(rspend
more time in areas with a lower potential energy (more
negative), i.e., in the vicinity of the atom cores. These
electrons may suffer preferentially inelastic scattering with
phonons, excitation of inner shells, etc., giving rise to the
anomalous absorption.
x
Anomalous absorption effect
Electrons that are scattered outside the objective aperture
contribute to the attenuation of the transmitted and diffracted
waves. It looks like absorption from the point of view of the
image, but they are not truly absorbed by the crystal.
Attenuation of the wave functions
complex component:
k
( j)
k
( j)
iq
(j)
For normal incidence

( j)
abs

( j)
 i qz
wave vectors with a
This can be achieved by
introducing a complex
potential of the form:
V(r )  V(r )  i V (r )
Anomalous absorption effect
V(r )  V(r )  i V (r )
Since the imaginary part of is associated with the crystal lattice, it can
also be expanded as a Fourier series based on the reciprocal lattice, with
coefficients
U g  U g  i U g
K
0 
U 0
K
g 
Ug
Imaginary component of
the extinction distance
absorption length
The basic eigenvalue equation
 iU0

 U  iU 
g
 g
U
(j)



i
U


C

( j)
g
g
g
0
n
    2K n (1 
)  abs
(j)

2KSg  iU0
Kn
 Cg 
A Ccol   Ccol
abs
A abs
( j)
abs
 iU0

1



gn  

  U g  iUg 
2 K n 1 
 Kn 
U
 iUg  



2KSg  iU0 
g
C(j)

0
 (j) 
Cg 
Complex eigenvalues for nearly normal incidence
1  U0
2K  U g  iUg 

( j)
abs

( j)
iq
( j)
z
U
(j)



i
U


C

( j)
g
g
0
 
abs
2KSg  iU0  C(j)
g 

q (z1, 2 )
C(j)

0
 (j) 
Cg 
1 1
1
  
2  0 g 1  w 2





The imaginary component of the wave vector corresponding to
j = 1 has a higher value, therefore the Bloch waves corresponding
to k(1) will attenuate faster with thickness. For thicker specimen
only the solution corresponding to j = 2 will contribute significantly
to the image.
Two-beams wave functions with absorption
 0   C C   e
    (1) (2) 
  C C  
0  0
 g  g
(1)
0
0 (t)  e
2
2
q z  q
(1)
z


2 t
0
q
e
0
 2 i (  ( 2 )  iq(z1) ) z
(1)
  C(1)
 1 
C
0
g

 
  C(2) C(2)   0 
0 
 0


 1  w 2  w 2 e -2  q z t  1  w 2  w 2 e 2  q z t cos 2k t 
z 


4(1  w 2 )
2(1  w 2 ) 



 2 t
0
g (t)  e
2 i (  (1)  iq(z1) ) z
(2)
0
cosh( 2k z t) - cos(2 k z t) 
2(1  w 2 )
( 2)
z

1
g 1  w
2
k z 
w2 1
g
w  Sg  g
Dependence on thickness
0 (t)  e
2
2
 2 t
0
g (t)  e




 1  w 2  w 2 e -2  q z t  1  w 2  w 2 e 2  q z t cos 2k t 
z 


4(1  w 2 )
2(1  w 2 ) 



2 t
0
cosh( 2k z t) - cos(2 k z t) 
2(1  w 2 )
The first exponential factor represents a uniform attenuation with
thickness of both transmitted and diffracted beams.
The first two terms into the brackets in the transmitted beam and the first
term in the diffracted beam gives a background of intensity, while the
last term, in both expressions, gives an oscillation of the intensity as
1
1
t

function of thickness with a period k S  (1/  )  1  w
Thus the oscillations show the same period as in the case without
absorption.
g
z
2
g
2
g
2
Dependence with the excitation error Sg
0 (t)  e
2
2
 2 t
0
g (t)  e




 1  w 2  w 2 e -2  q z t  1  w 2  w 2 e 2  q z t cos 2k t 
z 


4(1  w 2 )
2(1  w 2 ) 



2 t
0
cosh( 2k z t) - cos(2 k z t) 
2(1  w )
2
k z 
w2 1
g
w  Sg  g
The transmitted intensity is strongly absorbed for Sg< 0, this is
because the first term, containing the exponential with negative
argument dominates. This term comes from the j=1 solution
(having a longer wave vector). On the contrary a higher
intensity is obtained for Sg > 0. The image is asymmetric with
respect to Sg= 0.
The diffracted beam is symmetric respect to Sg= 0
Dependence with the excitation error Sg
0 (t)  e
2
2
 2 t
0
g (t)  e




 1  w 2  w 2 e -2  q z t  1  w 2  w 2 e 2  q z t cos 2k t 
z 


2
2
4(1  w )
2(1  w ) 



2 t
0
cosh( 2k z t) - cos(2 k z t) 
2(1  w 2 )
S>0
000
S<0
k z 
w2 1
2
g
S>0
S<0
400
w  Sg  g
The images of defects are clearer when observed under the
Sg > 0 condition, because the background of the perfect
crystal is brighter.
Determination of the excitation error
P
R
P
D D
2
-g
R
Q
E
1
1
0
g
x
E
Q
D D
2
2
2g
-g
E
1
E
1
2
g
0
2g
x1 x2
S=0
S<
K
K
K+g
K+g
+S
000
g
000
q g

S
0
An inelastic scattering takes place at the
point P in the figure. Electrons will
scattered at different directions, those
arriving at the planes R and Q with the
Bragg angle will be diffracted according
to the Bragg low. The inelastically
scattered and diffracted electrons having
the Bragg angle with the planes R and Q
will, in general, lye on the surface of a
cone. The intersection of the cones with
the screen gives hyperbolic lines, which
look like straight lines in the diffraction
pattern. The line E1 will be more intense
than the background because of the
higher probability for forward inelastic
scattering at point P, this is called the
excess line. On the other hand the line D
is called the deficient lines because the
electron diffracted to E1 are absent from
the background around the transmitted
spot.
P
R
After tilting the specimen an angle 2q B ,
the Kikuchi lines E1 and D1 will move a
distance x away from the associated spot at
S=0.
P
R
Q
Q
After tilting an angle q1they will move a
distance
D D
2
E
1
1
E
D D
2
2
E
1
x 1. Thus
q1 x 1

2q B
x
E
1
2
From the figure we have:
-g
0
g
x
2g
-g
g
0
2g
S  g q1
x1 x2
S=0
S<
K
K
K+g
K+g
+S
000
g
000
q g

S
0
Combining
x 1
S  2q B g
x
Thickness and extinction distance measurements
S>0
S<0
S>0
000
400
The diffracted intensity has a minimum when
2
 Si 
 1   1   1 

  
    2 
n i
 n  i   g   t 
2
S<0
2
S2  (1 /  g ) 2 t  n
Thickness and extinction distance measurements
2
 Si 
 1   1   1 

  
    2 
n i
 n  i   g   t 
2
2
n is the largest integer
n  t / g
X i 2
Si 
g 
X
Calculated
 g  49nm
Calculation of the extinction distance
The potential
V(r )   Vg e
2 i g.r
g
2me
U g  2 Vg
h
Inverse Fourier transform
1
1 -2 i ( g g )..r
- 2 ig.r
V
(
r
)
e
d
r

V
e
dr

g


V
V
g
1 -2 i(g-g).r
e
dr   g ,g

V
The Fourier
coefficients
1
- 2 ig..r
Vg   V(r ) e
dr
V
Calculation of the extinction distance
The crystal potential in term of the single atomic potentials
V(r )   Vj (r  r j  rmnp )
j mnp
r
r
b
rj
Vg 
1
V(r ) e -2 ig..r dr
V
r  r   r j  rmnp
rmnp  ma  nb  pc
rmnp
a
1
- 2 ig.( r   rj  rmnp )
Vg    Vj (r) e
dr
V j mnp
1
-2 ig.rmnp
-2 ig.rj
-2 ig.r

Vg   e
e
Vj (r ) e
dr 


V mnp
j
Calculation of the extinction distance
1
-2 ig.rmnp
-2 ig.rj
-2 ig.r

Vg   e
e
V
(
r
)
e
dr 

j

V mnp
j
g.rmnp  int eger
e
-2 ig.rmnp
 MNP
mnp
MNP
-2 ig.rj
-2 ig.r

Vg 
e
Vj (r ) e
dr 


V j
Atomic scattering amplitude
2me
- 2 ig.r 

f j (g ) 
V
(
r
)
e
dr 
j
2

h
number of
unit cells
Atomic scattering amplitude
2me
- 2 ig.r 

f j (g ) 
V
(
r
)
e
dr 
j
2

h
Poisson equation for the atomic potential
 V(r )  4e  n (r )   e (r )
2
2

2me
f j (g)  2 2 Z  f jx (g)
h g

X-ray scattering factor
f jx (g)   e e -2 ig.rdr
Fourier coefficient
MNP
-2 ig.rj
-2 ig.r

Vg 
e
Vj (r ) e
dr 


V j
2me
- 2 ig.r 

f j (g ) 
Vj (r ) e
dr 
2

h
F(g)   f j (g)e
-2 ig.rj
v cell
V

MNP
j
Structure factor
2
1 h
Vg 
F(g)
v cell 2me
The extinction distance
1 h2
Vg 
F(g)
v cell 2me
2me
U g  2 Vg
h
K
g 
Ug
 v cell K
g 
F(g )
Atomic scattering amplitude
Structure factor
F(g)   f j (g)e
j
-2 ig.rj

2me 2
f j (g)  2 2 Z  f jx (g)
h g
f (g )    e e
x
j
X-ray scattering factor
-2 ig.r 

dr
The x-ray scattering factor for the hydrogen atom
f jx (g)   e e -2 ig.rdr
For spherical symmetry
sin2 gr 2
f (g )  4   e
r dr
2gr
x
j
1 2r / a 0
 0 (r )  3 e
a 0
e (r )   0 (r )
2
Ground state of H
Bohr radius
1
x
f j (g ) 
(1   2 a 02 g 2 ) 2

1
a0  2
 0.529 A
2
4 m 0 e
Atomic scattering amplitude of H

2me 2
f j (g)  2 2 Z  f jx (g)
h g
2me 2
f j (g )  2 2
hg



1
1 
2 2 2 2
(
1


a 0g ) 

1
f (g ) 
2 2 2 2
(1   a 0 g )
x
j
For g  0
f j (0) 
a0
1  2

a 0  0.529 A
Debye-Waller factor
Due to atom vibrations
Structure factor
F(g)   f j (g)e
-2 ig.rj
f j (g) e
-M g
j
Mg  2  u  g
2
6h 2
Mg 
m at k D T
2
2
u 
2
 () 1  g 2

 
4 4
 
Mean-square displacement
of the atom
DT

T

1  d
()   
 0 e 1
DT
Debye temperature
of the crystal
(kai)
Atomic scattering amplitude in ordered alloys
I
I
P
P
Site I:
A
B
b
II
a
II
I
I
A
P Probability to find an A atom at the site I
fI (g)  PAI fA  PBI fB
fII (g)  PAIIfA  PBIIfB
II
Site I: PA PB