Transcript Document
Diffraction
T. Ishikawa
Part 2, Dynamical Diffraction
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Introduction
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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.
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Basic Idea
Kinematical Diffraction
Dynamical Diffraction
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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
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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
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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
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H
t
D
t
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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
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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
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Some Mathematic....
D
H
i D,
i H
t
t
E imo H
imo 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
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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 gh Eh exp iK g r 0 (*)
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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 106 Eg[ k ] Eg
g
the basic equation is well approximated by
K g2 K 2
K
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2
E g c g h Eh
h
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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
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Boundary Conditions (2/3)
Wavefield : Superposition of plane waves
E exp it 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
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m
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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
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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
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K gt K mt
Ema exp iK mz H
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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
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Polarization Factor
P 1
P cos 2 B
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for polarization
for polarization
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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
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o g
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1 2 2
K P cgcg
4
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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
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2
2
2
2 cos2 B
L2
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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
2oj
KPc g
KPc g
2 gj
j = 1, 2
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Diffraction Geometry
Symmetric Laue Case
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Symmetric Bragg Case
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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
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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
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cos B
W W 1
W W 1
2
2
upper sign: j=1, lower sign: j=2
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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
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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 LP 2 K B o sin B
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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
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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
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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.
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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
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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
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g
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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
1W
1 P
1
Egj
exp i g
2 P
1W 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
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Rocking Curve: Symmetric Laue Case (3/3)
I gW
Io
W
d
Ega
Eoa
a
d
a
o
I
E
Io
E
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2
2
sin 2 H W 2 1 L
W 2 1
W 2 cos 2 H W 2 1 L
W 2 1
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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
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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
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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
1W 2
i
i
Koz Im K oz K gz
L tan B
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W i 1 W 2
E
a
g
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P
P
exp i g W i 1 W 2 Eoa
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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
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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
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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
1W 2
B
Bloch Wave
small absorption
Pe
mH
exp
1
cos
1W 2
B
Bloch Wave b
large absorption
Anomalous Transmission (Borrman Effect)
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Symmetric Laue Case: Absorbing Crystal (2/2)
Forward Diffraction
2
Pe
I dW 1
W
mH
1
exp
1
I o 4
1W 2
1W 2
cos B
Pe
W
mH
exp
1
1
1W 2
1W 2
cos B
2
Thin Crystal
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Thick Crystal
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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
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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.
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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.
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