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DiffractionT. Ishikawa

Part 2, Dynamical Diffraction

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Introduction

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 DiffractionDynamical Diffraction

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Maxwell Equation (1/2)

E: electric fieldD: electric displacementH: magnetic fieldB: magnetic inductionρ: charge densityj: current densityP: polarizationM: magnetizationε0: permittivity of vacuumµ0: permeability of vacuumχ: electric susceptibilityc: speed of light in vacuum

0true

true

t

t

ρ∇ ⋅ =∇ ⋅ =

∂∇× = −

∂∂

∇× = +∂

DB

BE

DH j

o

o

ε

µ

= +

= −

D E PBH M

oε χ=P E

2

1o o cε µ =

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Maxwell Equation (2/2)For periodically oscillating electromagnetic field;

jtrue = 0, ρtrue = 0.

For non-magnetic materials, M=0 so that B = µ0H.

00

o t

t

µ

∇⋅ =∇ ⋅ =

∂∇× = −

∂∂

∇× =∂

DH

HE

DH

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PolarizationPolarization P = electric dipole moment in unit volume

( ) ( ) ( )2

2

o

er e r rm

ε χ

ρ ρ ρω

=

= = − = −

P E

P p x E

( ) ( ) ( ) ( )2 2 2 2

2 2 24 oo o

e e rm mc

λ λχ ρ ρ ρω ε π ε π

= − = − = −r r r r

χ(r) have the periodicity of crystal lattice

( ) ( )

( ) ( ) ( )

( )2

exp

1 exp

g

c

o

c

i

F iv

r Fv

χ χ

ρ

λχπ

= ⋅

= ⋅

⇒ = −

∑

∑

g

g

g

r g r

r g g r

g

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Electromagnetic Wave in Periodic Medium

Incident Plane Wave in Vacuum ( ){ }exp i tω⋅ −oK rBloch Theorem

Waves inside Periodic Medium ( ){ } ( )expE i t uω= ⋅ −oK r r

u(r) has periodicity of crystal lattice

u(r) can be expanded in a Fourier Series with reciprocal lattice vector, g.

( ){ } ( )

( )

exp exp

expi t

E i t E i

e E iω

ω

−

= ⋅ − ⋅

= ⋅

= +

∑

∑

o gg

g gg

g o

K r g r

K r

K K g

Bloch Wave

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Some Mathematic....

, i it t

ω ω∂ ∂= − = −

∂ ∂D HD H ( )

( )( )( )

( )( )( )( )

2

2

2

2

1

1

1

o

o

o o

i

i i

cK

ωµ

ωµ ω

ω µ ε χ

ω χ

χ

∇× ∇× = ∇×

= −

= +

= +

= +

E H

D

r E

r E

r E

2 ; X ray wavenumber in vacuumKcω π

λ= = −

( ){ } ( )( ){ } ( ) ( )

exp exp

exp exp

i i i

i i

∇× ⋅ = × ⋅

∇×∇× ⋅ = − × × ⋅

g g g g g

g g g g g g

E K r K E K r

E K r K K E K r

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Mathematics (cont’d)( )

( ){ } ( )

[ ] 2

2[ ]

1

exp expg

g g

K

E ig r K ig r

⊥

⊥

= − × ×

⇒∇×∇× ⋅ = ⋅

g

g

g K g g g

g K

E K K E

E

( ) ( ) ( ) ( )

( )

''exp exp

exp

E i i

i

χ χ

χ′

= ⋅ ⋅

= ⋅

∑∑

∑∑

h hhh h

g-h h gg h

r r h r E K r

E K r

, ′ ′+ = + =h gh h g h K K

( )2 2 2[ ] exp 0g

gK K K iχ⊥ −

− − ⋅ =

∑ ∑gg K g g h h g

hE E E K r (*)

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Basic Equations for Dynamical TheoryCondition for the equation (*) should be valid for arbitrary r gives the basic equation for dynamical diffraction theory:

2 2[ ]

2

gK KK

χ⊥−

−=∑gg K g

g h hh

E EE

6[ ]10χ −⊥⇒ ≅

gg g k gE ESince

the basic equation is well approximated by

2 2

2gK KK

χ −

−=∑g g h h

hE E

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Boundary Conditions (1/3)z

vacuum

crystal

z=H

Fields in vacuum: (Ea, Da)

Fields in crystal: (E, D)

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

( ) ( )exp expi t iω= − ⋅∑ g gg

E E K r

Km

Kg

Wave Vector in Crystal: KgWave Vector in Vacuum: Km

Kgt = Kmt

crystal wave

( ) ( )exp , expi i= ⋅ = ⋅∑ ∑g g g gg g

E E K r D D K r

vacuum wave

( ) ( )exp , expi i= ⋅ = ⋅∑ ∑a a a am m m m

m mE E K r D D K r

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Boundary Conditions (3/3)ˆunit vector normal to the surface:

ˆunit vector tangential to the surface: z

tˆˆˆˆ

z t

z t

K K

K K

= +

= +g g g

m m m

K z t

K z t

Boundary Condition at z=H (Crystal Surface)( ) ( )

( ) ( )

exp exp

exp expgt mt gt mt

gt mt gt mt

agt gz mt mz

K K K K

agz gz mz mz

K K K K

E iK H E iK H

D iK H D iK H

= =

= =

=

=

∑ ∑

∑ ∑

( ) ( )1 exp expgt mt gt mt

g gz mzK K K K

iK H iK Hχ= =

⇒ =∑ ∑ ag mE E

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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:

( ) ( )exp expi i= ⋅ + ⋅0 0 g gE E K r E K r

Basic Equation:

( )( )

2 2 2

2 2 2

0

0

K k E K P E

K P E K k E

χ

χ

− − =

− − =

o o g g

g o g g

Averaged refractive index of crystal:

1 12 2

o on k Kχ χ = + ⇒ = +

Polarization Factor

B

1 for polarizationcos 2 for polarization

PP

σθ π

= −= −

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Two-Wave Approximation (2/3)Condition for the basic equation,

( )( )

2 2 2

2 2 2

0

0

K k E K P E

K P E K k E

χ

χ

− − =

− − =

o o g g

g o g g

to have non-trivial solutions is2 2 2

2 2 2 0K k K PK P K k

χχ−

=−

o g

g g

= +g oK K g O G

dispersion sphere

dispersion surface

Ko Kg

g

By introducing new parameters:

o o

g g

K kK k

ξξ

= −= −

2 214

K Pξ ξ χ χ=o g g g

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Two Wave Approximation (3/3)For non-absorbing crystals,

* (complex conjugateof )χ χ χ=g g g

2χ χ χ=g g g

When we introduce a new parameter Λ as2 cos cosB B

K P Pπ θ λ θ

χ χΛ = =

g g,

2 2

2

cos Bπ θξ ξ =Λo g

x

y

Lo

To

Tg

Near the point Lo,sin cos

sin cosB B

B B

x yx y

ξ θ θξ θ θ

= − +

= +o

g

Dispersion surfaces formHyperbolla

2 22 2 2 2

2

cossin cos BB Bx y π θθ θ− + =

Λ

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Amplitude Ratio2 0

2 0o o g

o g g

E KP EKP E Eξ χ

χ ξ

− =

− =g

g

Amplitude Ratio

22

gj oj gj

oj g gj

E KPr

E KPξ χχ ξ

= = =

j = 1, 2

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Diffraction Geometry

Symmetric Laue Case 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

LPK

θ∆ =

( )

2sincos2 sin

g oo g B

B

B B

p p LP

K

ξ ξθ

θθ θ θ

−= = ⋅

= −

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Symmetric Laue Case: Deviation Parameter

Deviation Parameter W

( )

( )1 2

2 sin

sin 2

o g BB

B B

g

p pW

L L

P

θ θ θλ

θ θ θχ

Λ= = −

−=

cos 2 B

polarization Wpolarization W

W W

σ

π

σ π

σ

π

θ

−

−

=

2 2

2

2 cos

cos

g o B

Bg o

Wπξ ξ θ

π θξ ξ

− =Λ

=Λ

Solving above equations, we can get

( )( )

2

2

cos 1

cos 1

Boj

Bgj

W W

W W

π θξ

π θξ

= − ± +Λ

= ± +Λ

upper sign: j=1, lower sign: j=2Usually W=Wσ

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Symmetric Laue Case: Amplitude Ratio

( )( )2exp 1

upper sign 1, lower sign 2

gjj g

oj

E Pr i W W

E Pj j

α= = − ± +

= =

Here, ( )expg g giχ χ α=

For non-absorbing crystals,

( )

* , ,

exp

g g g g g g

gg

g

i

χ χ α α χ χ

χα

χ

= = − =

=

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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

( )cos

2sin 2 sin

B

g oo g

B

B B o B

LPK

p p

L P K

θ θ

ξ ξθ

θ θ θ θ θ

− =

− −=

′= ⋅ = − − ∆

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Symmetric Bragg Case (2/2)∆θo : Deviation from geometrical Bragg angle by refraction

( )2 1sin 2 sin 2

oo

B B

nχθθ θ

−−∆ = =

Deviation parameter, W

( )1 2

2 sino g BB o

p pW

L Lθ θ θ θ

λΛ

= = − −∆

2 cosg o BWπξ ξ θ+ = −Λ

( )( )

2

2

cos 1

cos 1

Boj

Bgj

W W

W W

π θξ

π θξ

= − −Λ

= ± −Λ

m

upper sign: j=1, lower sign: j=2

Amplitude Ratio

( )( )2exp 1gjj g

oj

E Pr i W W

E Pα= = − ± −

upper sign: j=1, lower sign: j=2

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Rocking CurvesUse 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)

Ka,Eoa

Ko2, E02

Ko1, Eo1

Ka,Eda

Kg2, Eg2

Kg1, Eg1

Kga,Eg

a

Incident Wave

z=0

z=H

( )expaoE i ⋅aK r

Crystal Waveo-wave ( )

( )1

2

exp

expo

o

E i

E i

⋅

⋅o1

o2

K r

K r

g-wave ( )( )

1

2

exp

exp

g

g

E i

E i

⋅

⋅

g1

g2

K r

K r

Outgoing Wave

o-wave ( )expadE i ⋅aK r

g-wave ( )expagE i ⋅a

gK r

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Rocking Curve: Symmetric Laue Case (2/3)Boundary condition at z = 0 (incident surface)

1 2

1 20

ao o o

g g

E E EE E

= +

= +

( )

2

2

1 12 11 1exp2 1

aoj o

gj g

WE EW

PE iP W

α

= ±

+ ±

=+

upper sign: j=1, lower sign: j=2

At W=0 (exact Bragg condition),

oj gjE E=

Boundary condition at z = H(exit surface)

( ) ( ) ( )( ) ( ) ( )

1 1 2 2

1 1 2 2

exp exp exp

exp exp exp

ao o o o d z

a ag g g g g gz

E iK H E iK H E iK H

E iK H E iK H E iK H

+ =

+ =

j

j

ˆPP

ˆPPˆ:inner normal vector at the incident surface

= −

= − +

aoj

agj

K z + K

K z + K gz

oj

j

-2PP

cos

o

B

Kχξ

θ

+ =

jPP aojz gjz zK K K= = − +

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Rocking Curve: Symmetric Laue Case (3/3)

( )

( )

2 22

2

2 2 22

2

sin 1

1

cos 1

1

W ag g

ao o

W ad d

ao o

H WI EI E W

W H WI EI E W

π

π

+ Λ= =

+

+ + Λ= =

+

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Rocking Curve: Symmetric Bragg Case (1/3)

Ka,Eoa Kg

a,Ega

Ko1, Eo1: W<-1

Ko2, Eo2: W>1

Kg1, Eg1: W<-1

Kg2, Eg2: W>1

Boundary condition at z = 0

z = 0

1 2

1 2

( 1), ( 1)

( 1), ( 1)

ao o oag g g

E E W E WE E W E W

= ≤ − ≥

= ≤ − ≥

( )( )2exp 1a ag g o

PE i W W E

Pα= − −m

upper sign: W<-1, lower sign: W>1

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Rocking Curve: Symmetric Bragg Case (2/3)1 z-componentsof and arecomplexW ≤ ⇒ o gK K

ˆPP

ˆPP

2PP sin

j

j

ooj

jB

Kχξ

θ

= −

= −

+ = −

oj

gj

K z + K

K z + K + g

( )21Im

tani ioz oz gz

B

WK K K πθ

−= = =

Λ

( )2cos 1Bo W i Wπ θξ = − + −

Λ

Another solution will give a divergent solution

( )( )2exp 1a ag g o

PE i W i W E

Pα= − + −

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Rocking Curve: Symmetric Bragg Case (3/3)Rocking curve (Darwin Curve)

( )222

2

1 ( 1)

1 ( 1)

aWgg

ao o

EI W W WI E W

− − ≥= = ≤

|W|<1: All incident energies are reflected back.

Total Reflection

sin 2 sin 2g o

BB B

PW

χ χθ θ

θ θ− = +

Center of total reflectiuon, W=0, is deviated from geometrical Bragg angle θBby

sin 2o

B

χθ

Range of total reflection (-1<W<1)

2

2sin 2 sin

2sin 2

g

B B

og g g

c B

P

Pr F or Fv

χ λωθ θ

λ χπ θ

= =Λ

= ∝

Darwin Width, ~microradian order

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Absorbing CrystalAbsorption: Anomalous dispersion term into atomic scattering factor

( ) ( )

( )

2 2

2 2

exp

exp

g g g

oo og g j j j

jc c

o og g j j

jc c

i

r rF f f iv v

r rF f iv v

χ χ χ

λ λχπ π

λ λχπ π

′ ′′= +

′ ′ ′= − = − + − ⋅

′′ ′′ ′′= − = − − ⋅

∑

∑

g r

g r

, :complexg gχ χ′ ′′

* *,g g g gχ χ χ χ′ ′ ′′ ′= =

Centrosymmetric Crsyatls

, :realg gχ χ′ ′′

,g g g gχ χ χ χ′ ′ ′′ ′′= =

g gχ χ=

2 2g g g g g g giχ χ χ χ χ χ χ′′ ′ ′ ′ ′′⇒ ≈ +

A new parameter κ is defined as

g

g

χκ

χ′′

=′

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Symmetric Laue Case: Absorbing Crystal (1/2)

2 22 2

2

exp cos 1 1sin sinh1

Wg B

o

HI H W H WI W

µθ π κπ

− + + = + + Λ Λ

( )sin 2cos , B BB

g g

WP P

θ θ θλ θχ χ

−Λ = =

′ ′

22 1sin H Wπ + Λ

Oscillating Term, Hardly to be observed experimentally without very good plane wave

averaging( )2 2 2

1 exp 1 exp 1cos cos4 1 1 1

Wg

o B B

I P PH HI W W W

ε εµ µθ θ

= − − + − + + + +

g

o

χε

χ′′

=′′

Bloch Wave αsmall absorption

Bloch Wave βlarge absorption

Anomalous Transmission (Borrman Effect)

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Symmetric Laue Case: Absorbing Crystal (2/2)Forward Diffraction

2 2

2 2 2 2

1 1 exp 1 1 exp 14 cos cos1 1 1 1

Wd

o B B

P PI W H W HI W W W W

ε εµ µθ θ

= + − − + − − + + + + +

Thin Crystal 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)

( ) ( )

2

2 22 2 2 2 2

2

1

1 4

1

Wg

o

o

g

IL L

I

W g W g gWL

gP

κ κ

κχχ

= − −

+ + − − + + −=

+′′

=′

κ = 0

κ = 0.1

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SummaryVery 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, 2001Handbook on Synchrotron Radiation Vol. 3, North-Holland, 1991.

11/15/2002 JASS02 36

Thank you for your Thank you for your attention.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.