Transcript Document

Diffraction
T. Ishikawa
Part 2, Dynamical Diffraction
7/17/2015
JASS02
1
Introduction


7/17/2015
In the 1st part, we dealt with “Kinematical
Theory” where the scattered x-rays suffer no
additional scattering.
The 2nd part is designed to give basic ideas of
“Dynamical Diffraction” observed with perfect
crystals as a result of multiple scattering.
JASS02
2
Basic Idea
Kinematical Diffraction
Dynamical Diffraction
7/17/2015
JASS02
3
Maxwell Equation (1/2)
  D  rtrue
B  0
 E  
B
t
  H  jtrue 
D
t
D  eo E  P
H
B
mo
M
P  eo c E
7/17/2015
E: electric field
D: electric displacement
H: magnetic field
B: magnetic induction
r: charge density
j: current density
P: polarization
M: magnetization
e0: permittivity of vacuum
m0: permeability of vacuum
c: electric susceptibility
c: speed of light in vacuum
JASS02
e o mo 
1
c2
4
Maxwell Equation (2/2)
For periodically oscillating electromagnetic field;
jtrue = 0, rtrue = 0.
For non-magnetic materials, M=0 so that B = m0H.
 D  0
 H  0
  E   mo
 H 
7/17/2015
H
t
D
t
JASS02
5
Polarization
Polarization P = electric dipole moment in unit volume
P  eo c E
e2
P  p r  r   ex r  r   
r r  E
m 2
e2  2
2
c r   
r r    2
r  r   ro
r r 
2
2
m e o
4 e o mc

e2
c(r) have the periodicity of crystal lattice
c  r    c g exp  ig  r 
g
r r  
1
vc
 F  g  exp  ig  r 
g
ro  2
 cg  
F  g
 vc
7/17/2015
JASS02
6
Electromagnetic Wave in Periodic Medium
Bloch Theorem
Incident Plane Wave in Vacuum
exp i  K o  r   t 
Waves inside Periodic Medium E  exp i  K o  r   t  u  r 
u(r) has periodicity of crystal lattice
u(r) can be expanded in a Fourier Series with
reciprocal lattice vector, g.
E  exp i  K o  r   t   Eg exp  ig  r 
g
e
 i t
E
g
exp  iK g  r 
Bloch Wave
g
K g  Ko  g
7/17/2015
JASS02
7
Some Mathematic....
D
H
 i D,
 i H
t
t
     E   imo  H
 imo  i  D
  2 m oe o 1  c  r   E

2
2
c
 K2
K


c

2

1  c  r   E
1  c  r   E
; X  ray wavenumber in vacuum

  E g exp  iK g  r   iK g  E g exp  iK g  r 


   E g exp  iK g  r    K g   K g  E g  exp  iK g  r 
7/17/2015
JASS02
8
Mathematics (cont’d)
E g[  K g ]  
1
K g   K g  Eg 
2
Kg
    Eg exp  ig  r   K g2 E g[ K g ] exp  ig  r 
c  r  E  r    c h exp  ih'  r Eh exp  iK h  r 
'
h
h
g
h
  c g -h Eh exp  iK g  r 
h  h  g, h  Kh  K g
 2

2
2
g K g Eg[ K g ]  K Eg  K h c gh Eh  exp iK g  r   0 (*)
7/17/2015
JASS02
9
Basic Equations for Dynamical Theory
Condition for the equation (*) should be valid for
arbitrary r gives the basic equation for dynamical
diffraction theory:
K g2 E g[ K g ]  K 2 E g
K
  c g  h Eh
2
h
Since
c g 106  Eg[ k ]  Eg
g
the basic equation is well approximated by
K g2  K 2
K
7/17/2015
2
E g   c g h Eh
h
JASS02
10
Boundary Conditions (1/3)
z
vacuum
Fields in vacuum: (Ea, Da)
Fields in crystal:
(E, D)
z=H
crystal
Boundary conditions from Maxwell Equations:
Continuity of tangential components of Electric
Fields
Et =Eat
Continuity of normal components of Electric
Displacements
Dz =Daz
t: tangential component, z: z(=normal) component
7/17/2015
JASS02
11
Boundary Conditions (2/3)
Wavefield : Superposition of plane waves
E  exp  it   Eg exp iK g  r 
g
Wave Vector in Crystal: Kg
Wave Vector in Vacuum: Km
Kgt = Kmt
Km
crystal wave
E   Eg exp iK g  r , D   Dg exp iK g  r 
g
Kg
g
vacuum wave
E a   Ema exp  iK m  r , Da   Dma exp  iK m  r 
m
7/17/2015
m
JASS02
12
Boundary Conditions (3/3)
unit vector normal to the surface: zˆ
unit vector tangential to the surface: tˆ
K g  K gz zˆ  K gt tˆ
K m  K mz zˆ  K mt tˆ
Boundary Condition at z=H (Crystal Surface)

Egt exp  iK gz H  

Dgz exp  iK gz H  
K gt  Kmt
K gt  Kmt
cg
7/17/2015
1

K gt  K mt

Emta exp  iK mz H 

a
Dmz
exp  iK mz H 
K gt  K mt
K gt  Kmt
E g exp  iK gz H  
JASS02

K gt  K mt
Ema exp  iK mz H 
13
Two-Wave Approximation (1/3)
Under usual experimental conditions, only two waves with K0 (incident
direction) and Kg (diffracted direction) are strong inside the crystal.
Wavefield in crystal:
E  E0 exp  iK 0  r   E g exp  iK g  r 
Basic Equation:
K
2
o

 k 2 Eo  K 2 Pc g Eg  0


K 2 Pc g Eo  K g2  k 2 Eg  0
Averaged refractive index of crystal:
co
 co 
n  1
 k  K 1  
2
2 

7/17/2015
Polarization Factor
P 1
P  cos 2 B
JASS02
for   polarization
for   polarization
14
Two-Wave Approximation (2/3)
Condition for the basic equation,
K
2
o

 k 2 Eo  K 2 Pc g Eg  0


dispersion surface
dispersion sphere
K 2 Pc g Eo  K g2  k 2 Eg  0
to have non-trivial solutions is
K o2  k 2
K 2 Pc g
K 2 Pc g
0
K g2  k 2
Ko
Kg
g
K g  Ko  g
O
G
By introducing new parameters:
o  K o  k
g  Kg  k
7/17/2015
o g 
JASS02
1 2 2
K P cgcg
4
15
Two Wave Approximation (3/3)
y
For non-absorbing crystals,
c g  c g* (complex conjugateof c g )
cgcg  cg
Tg
2
Lo
When we introduce a new parameter L as
2 cos B  cos B
L

,
K P cg
P cg
o g 
x
To
Near the point Lo,
o   x sin  B  y cos  B
 g  x sin  B  y cos  B
 2 cos 2  B
L2
Dispersion surfaces form
Hyperbolla
 x sin  B  y cos  B 
2
7/17/2015
JASS02
2
2
2
 2 cos2  B
L2
16
Amplitude Ratio
2 o Eo  KP c g E g  0
KP c g Eo  2 g E g  0
Amplitude Ratio
rj 
Egj
Eoj

2oj
KPc g

KPc g
2 gj
j = 1, 2
7/17/2015
JASS02
17
Diffraction Geometry
Symmetric Laue Case
7/17/2015
Symmetric Bragg Case
JASS02
18
Symmetric Laue Case
Dispersion sphere of
vacuum wave (radius K)
Starting point of wave
vector Ko: P
Laue point: L
Deviation from Bragg
Condition
LP
 
K
 g  o
po pg 
 2sin  B  LP
cos  B
 2 K  B    sin  B
7/17/2015
JASS02
19
Symmetric Laue Case: Deviation Parameter
Deviation Parameter W
W

po pg
L1 L2

2L sin  B

2 W
cos  B
L
 2 cos 2  B
 g o 
L2
 B   
 g  o 
 B    sin 2 B
P cg
  polarization
W
  polarization

W
Solving above equations, we can get
 oj 
L
 cos  B
 gj 
L
W   cos 2 B W 
Usually W=W
7/17/2015
 cos  B
 W  W  1 
W  W  1 
2
2
upper sign: j=1, lower sign: j=2
JASS02
20
Symmetric Laue Case: Amplitude Ratio
rj 
Egj
Eoj

P
P

exp  i g  W  W 2  1

upper sign j  1, lower sign j  2
Here,
c g  c g exp  i g 
For non-absorbing crystals,
c g  c g* , g   g , c g  c g
cg
 exp  i g 
cg
7/17/2015
JASS02
21
Symmetric Bragg Case (1/2)
Between L1 and L2, z has no
intersections with dispersion surfaces.
Total Reflection Region
Deviation from Bragg Condition
 B 
po pg 
LP
K
 g  o
cos  B
 2sin  B  LP  2 K    B   o  sin  B
7/17/2015
JASS02
22
Symmetric Bragg Case (2/2)
o : Deviation from geometrical
Bragg angle by refraction
 o 
2 1  n 
co

sin 2 B sin 2 B
 oj 
L
 cos  B
 gj 
L
po pg
L1L2

2L sin  B

  B  o 
7/17/2015
W 2 1
W 2 1


Amplitude Ratio
rj 
 g  o  
 W
W 
upper sign: j=1, lower sign: j=2
Deviation parameter, W
W
 cos  B
2 W
cos  B
L
Egj
Eoj

P
P

exp  i g  W  W 2 1

upper sign: j=1, lower sign: j=2
JASS02
23
Rocking Curves
Use monochromatic plane wave as an incident beam;
Rocking the sample crystal around the Bragg angle;
We can observe so-called rocking curve.
7/17/2015
JASS02
24
Rocking Curve: Symmetric Laue Case (1/3)
Incident Wave
Ka,Eoa

Eoa exp iK a  r

Crystal Wave
z=0
o-wave
Eo 2 exp  iK o2  r 
Kg2, Eg2
Ko2, E02
z=HK , E
g1
g1
Kga,Ega
7/17/2015
Eo1 exp  iK o1  r 
Ko1, Eo1
Ka,Eda
g-wave
Eg1 exp  iK g1  r 
Eg 2 exp  iK g2  r 
Outgoing Wave

exp  iK

r
o-wave
Eda exp iK a  r
g-wave
Ega
JASS02
a
g
25
Rocking Curve: Symmetric Laue Case (2/3)
Boundary condition at z = 0
(incident surface)
Eoa  Eo1  Eo 2
0  Eg 1  Eg 2
 a
1
W
Eoj  1 
 Eo
2
2
1W 
1 P
1
Egj 
exp  i g 
2 P
1W 2
Boundary condition at z = H
(exit surface)
Eo1 exp  iKo1H   Eo 2 exp  iKo 2 H   Eda exp  iK z H 

Eg1 exp  iK g1H   Eg 2 exp  iK g 2 H   Ega exp iK gza H
K oj  PPj zˆ + K a
K gj  PPj zˆ + K a  g
zˆ:inner normal vector at the incident surface
K co 

-  oj 
2 

PPj 
cos  B
upper sign: j=1, lower sign: j=2
At W=0 (exact Bragg condition),
Kojz  K gjz  PPj  K za
Eoj  E gj
7/17/2015

JASS02
26
Rocking Curve: Symmetric Laue Case (3/3)
I gW
Io
W
d

Ega

Eoa
a
d
a
o
I
E

Io
E
7/17/2015
2
2


sin 2  H W 2  1 L

W 2 1

W 2  cos 2  H W 2  1 L

W 2 1
JASS02
27
Rocking Curve: Symmetric Bragg Case (1/3)
Boundary condition at z = 0
Eoa  Eo1 (W  1), Eo 2 (W  1)
Ka,Eoa
Ega  Eg1 (W  1), Eg 2 (W  1)
Kga,Ega
z=0
Ko1, Eo1: W<-1
Ko2, Eo2: W>1
7/17/2015
Ega 
Kg1, Eg1: W<-1
P
P

exp  i g  W

W 2  1 Eoa
upper sign: W<-1, lower sign: W>1
Kg2, Eg2: W>1
JASS02
28
Rocking Curve: Symmetric Bragg Case (2/3)
W  1  z-componentsof Ko and K g arecomplex
K oj   PP j zˆ + K
o 
 cos  B
L
K gj   PP j zˆ + K + g
K co 



 oj

2


PP j  

Another solution will give a divergent
solution
sin  B
 1W 2
i
i
Koz  Im  K oz   K gz 
L tan  B
7/17/2015

W  i 1  W 2
E 
a
g
JASS02
P
P


exp  i g  W  i 1  W 2 Eoa
29
Rocking Curve: Symmetric Bragg Case (3/3)
Rocking curve (Darwin Curve)
I gW
Io

Ega
2
Eoa
2


2
 W  W 1


 1

2
|W|<1: All incident energies are reflected back.
Total Reflection
( W  1)
( W  1)
 B 
P cg
sin 2 B
W
co
sin 2 B
Center of total reflectiuon, W=0, is
deviated from geometrical Bragg angle B
by
c
o
sin 2 B
Range of total reflection (-1<W<1)


2 P cg
sin 2 B


L sin  B
P
2 ro
Fg  2
 c g or Fg
 vc
sin 2 B
Darwin Width, ~microradian order
7/17/2015
JASS02
30
Absorbing Crystal
Absorption: Anomalous dispersion term
into atomic scattering factor
Centrosymmetric Crsyatls
c g , c g :real
c g  c g  i c g
c g  
ro 
r
Fg   o
 vc
 vc
2
2
ro  2
ro  2
c g  
F   
 vc g
 vc
 f
o
j
c g  c g , c g  c g

 f j exp  ig  rj 
j
cg  cg
 f exp  ig  r 
j
j
j
c g
c g , c g :complex
c g  c g*, c g  c g*
c g  c g c g  c g 2  2ic g c g
A new parameter k is
defined as
c g
k
c g
7/17/2015
JASS02
31
Symmetric Laue Case: Absorbing Crystal (1/2)
 m H

exp


2
2




I
cos

B

2  H W 1
2 k H W  1 

  sinh 

sin 



Io
W 2 1
L
L







W
g
L
  H W 2 1 
sin 



L


Oscillating Term, Hardly to be observed
experimentally without very good plane
wave
2
averaging
c g
e
c o
I gW
1

Io 4 W 2  1

    sin 2 B
 cos B
,W  B
P c g
P c g



Pe
 mH 
1

exp 

cos

1W 2


B


Bloch Wave 
small absorption


Pe

 mH 

exp

1




cos

1W 2

B






 


Bloch Wave b
large absorption
Anomalous Transmission (Borrman Effect)
7/17/2015
JASS02
32
Symmetric Laue Case: Absorbing Crystal (2/2)
Forward Diffraction
2

Pe
I dW 1 
W 
 mH 

 1 
exp 
1


I o 4 
1W 2 
1W 2

 cos B 



Pe
W 
 
 mH 
exp 
1
   1 


1W 2 
1W 2


 
 cos B 
2
Thin Crystal
7/17/2015


 


Thick Crystal
JASS02
33
Symmetric Bragg Case: Absorbing Crystal
Rocking curve for a symmetric Bragg case
diffraction from a semi-infinite absorbing crystal
(with centrosymmetry)
I gW
Io
L
g
 L  L2  1
W 2  g2 
c o
P c g
W
2
 g 2 1 k 2

2
 4  gW  k 
k=0
2
1 k 2
k = 0.1
7/17/2015
JASS02
34
Summary



Very quick scan of x-ray diffraction
theory was attempted.
You may need reference text books.
References
Dynamical Theory of X-Ray Diffraction,
A. Authie, Oxford University Press,
2001
 Handbook on Synchrotron Radiation
Vol. 3, North-Holland, 1991.

7/17/2015
JASS02
35
Thank you for your
attention.
Acknowledgement
Some materials presented here are originally prepared by Prof. Seishi Kikuta
for his textbook written in Japanese. Some ppt materials have been prepared
by Dr. Shunji Goto. Discussion in preparing the lecture with Drs. Shunji Goto,
Kenji Tamasaku and Makina Yabashi is appreciated.
7/17/2015
JASS02
36