Today’s Outline - November 04, 2013
• Bragg & Laue Geometries
• Reflection for a Single Layer
• Kinematical Approach for Many Layers
• Darwin Curve
• Dynamical Diffraction Theory
Homework Assignment #5:Chapter 5: 1, 3, 7, 9, 10due Monday, November 11, 2013
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 1 / 17
Today’s Outline - November 04, 2013
• Bragg & Laue Geometries
• Reflection for a Single Layer
• Kinematical Approach for Many Layers
• Darwin Curve
• Dynamical Diffraction Theory
Homework Assignment #5:Chapter 5: 1, 3, 7, 9, 10due Monday, November 11, 2013
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 1 / 17
Today’s Outline - November 04, 2013
• Bragg & Laue Geometries
• Reflection for a Single Layer
• Kinematical Approach for Many Layers
• Darwin Curve
• Dynamical Diffraction Theory
Homework Assignment #5:Chapter 5: 1, 3, 7, 9, 10due Monday, November 11, 2013
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 1 / 17
Today’s Outline - November 04, 2013
• Bragg & Laue Geometries
• Reflection for a Single Layer
• Kinematical Approach for Many Layers
• Darwin Curve
• Dynamical Diffraction Theory
Homework Assignment #5:Chapter 5: 1, 3, 7, 9, 10due Monday, November 11, 2013
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 1 / 17
Today’s Outline - November 04, 2013
• Bragg & Laue Geometries
• Reflection for a Single Layer
• Kinematical Approach for Many Layers
• Darwin Curve
• Dynamical Diffraction Theory
Homework Assignment #5:Chapter 5: 1, 3, 7, 9, 10due Monday, November 11, 2013
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 1 / 17
Today’s Outline - November 04, 2013
• Bragg & Laue Geometries
• Reflection for a Single Layer
• Kinematical Approach for Many Layers
• Darwin Curve
• Dynamical Diffraction Theory
Homework Assignment #5:Chapter 5: 1, 3, 7, 9, 10due Monday, November 11, 2013
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 1 / 17
Today’s Outline - November 04, 2013
• Bragg & Laue Geometries
• Reflection for a Single Layer
• Kinematical Approach for Many Layers
• Darwin Curve
• Dynamical Diffraction Theory
Homework Assignment #5:Chapter 5: 1, 3, 7, 9, 10due Monday, November 11, 2013
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 1 / 17
Lattice Vibrations Review
Recall that the inclusion of atomic vibration resulted in a scatteredintensity with two terms.
I =∑m
∑n
f (~Q)e−Me i~Q·~Rm f ∗(~Q)e−Me−i
~Q·~Rn
+∑m
∑n
f (~Q)e−Me i~Q·~Rm f ∗(~Q)e−Me−i
~Q·~Rn
[eQ
2〈uQmuQn〉 − 1]
The first term is just the elastic scattering from the lattice with the
addition of the term e−M = e−Q2〈u2
Q〉/2, called the Debye-Waller factor.
This second term is what we will focus on now.
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 2 / 17
Lattice Vibrations Review
Recall that the inclusion of atomic vibration resulted in a scatteredintensity with two terms.
I =∑m
∑n
f (~Q)e−Me i~Q·~Rm f ∗(~Q)e−Me−i
~Q·~Rn
+∑m
∑n
f (~Q)e−Me i~Q·~Rm f ∗(~Q)e−Me−i
~Q·~Rn
[eQ
2〈uQmuQn〉 − 1]
The first term is just the elastic scattering from the lattice with the
addition of the term e−M = e−Q2〈u2
Q〉/2, called the Debye-Waller factor.
This second term is what we will focus on now.
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 2 / 17
Lattice Vibrations Review
Recall that the inclusion of atomic vibration resulted in a scatteredintensity with two terms.
I =∑m
∑n
f (~Q)e−Me i~Q·~Rm f ∗(~Q)e−Me−i
~Q·~Rn
+∑m
∑n
f (~Q)e−Me i~Q·~Rm f ∗(~Q)e−Me−i
~Q·~Rn
[eQ
2〈uQmuQn〉 − 1]
The first term is just the elastic scattering from the lattice with the
addition of the term e−M = e−Q2〈u2
Q〉/2, called the Debye-Waller factor.
This second term is what we will focus on now.
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 2 / 17
Lattice Vibrations Review
Recall that the inclusion of atomic vibration resulted in a scatteredintensity with two terms.
I =∑m
∑n
f (~Q)e−Me i~Q·~Rm f ∗(~Q)e−Me−i
~Q·~Rn
+∑m
∑n
f (~Q)e−Me i~Q·~Rm f ∗(~Q)e−Me−i
~Q·~Rn
[eQ
2〈uQmuQn〉 − 1]
The first term is just the elastic scattering from the lattice with the
addition of the term e−M = e−Q2〈u2
Q〉/2, called the Debye-Waller factor.
This second term is what we will focus on now.
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 2 / 17
Properties of the Debye-Waller Factor
For crystals with several differenttypes of atoms, we generalize theunit cell scattering factor.
B jT = 8π2〈u2
Qj〉
for isotropic atomic vibrations
〈u2〉 = 〈u2x + u2
y + u2z 〉
= 3〈u2x 〉 = 3〈u2
Q〉
F u.c. =∑j
fj(~Q)e−Mj e i~Q·~rj
Mj =1
2Q2〈u2
Qj〉
=1
2
(4π
λ
)2
sin2 θ〈u2Qj〉
Mj = B jT
(sin θ
λ
)2
B isoT =
8π2
3〈u2〉
In general, Debye-Waller factors can be anisotropic
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 3 / 17
Properties of the Debye-Waller Factor
For crystals with several differenttypes of atoms, we generalize theunit cell scattering factor.
B jT = 8π2〈u2
Qj〉
for isotropic atomic vibrations
〈u2〉 = 〈u2x + u2
y + u2z 〉
= 3〈u2x 〉 = 3〈u2
Q〉
F u.c. =∑j
fj(~Q)e−Mj e i~Q·~rj
Mj =1
2Q2〈u2
Qj〉
=1
2
(4π
λ
)2
sin2 θ〈u2Qj〉
Mj = B jT
(sin θ
λ
)2
B isoT =
8π2
3〈u2〉
In general, Debye-Waller factors can be anisotropic
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 3 / 17
Properties of the Debye-Waller Factor
For crystals with several differenttypes of atoms, we generalize theunit cell scattering factor.
B jT = 8π2〈u2
Qj〉
for isotropic atomic vibrations
〈u2〉 = 〈u2x + u2
y + u2z 〉
= 3〈u2x 〉 = 3〈u2
Q〉
F u.c. =∑j
fj(~Q)e−Mj e i~Q·~rj
Mj =1
2Q2〈u2
Qj〉
=1
2
(4π
λ
)2
sin2 θ〈u2Qj〉
Mj = B jT
(sin θ
λ
)2
B isoT =
8π2
3〈u2〉
In general, Debye-Waller factors can be anisotropic
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 3 / 17
Properties of the Debye-Waller Factor
For crystals with several differenttypes of atoms, we generalize theunit cell scattering factor.
B jT = 8π2〈u2
Qj〉
for isotropic atomic vibrations
〈u2〉 = 〈u2x + u2
y + u2z 〉
= 3〈u2x 〉 = 3〈u2
Q〉
F u.c. =∑j
fj(~Q)e−Mj e i~Q·~rj
Mj =1
2Q2〈u2
Qj〉
=1
2
(4π
λ
)2
sin2 θ〈u2Qj〉
Mj = B jT
(sin θ
λ
)2
B isoT =
8π2
3〈u2〉
In general, Debye-Waller factors can be anisotropic
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 3 / 17
Properties of the Debye-Waller Factor
For crystals with several differenttypes of atoms, we generalize theunit cell scattering factor.
B jT = 8π2〈u2
Qj〉
for isotropic atomic vibrations
〈u2〉 = 〈u2x + u2
y + u2z 〉
= 3〈u2x 〉 = 3〈u2
Q〉
F u.c. =∑j
fj(~Q)e−Mj e i~Q·~rj
Mj =1
2Q2〈u2
Qj〉
=1
2
(4π
λ
)2
sin2 θ〈u2Qj〉
Mj = B jT
(sin θ
λ
)2
B isoT =
8π2
3〈u2〉
In general, Debye-Waller factors can be anisotropic
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 3 / 17
Properties of the Debye-Waller Factor
For crystals with several differenttypes of atoms, we generalize theunit cell scattering factor.
B jT = 8π2〈u2
Qj〉
for isotropic atomic vibrations
〈u2〉 = 〈u2x + u2
y + u2z 〉
= 3〈u2x 〉 = 3〈u2
Q〉
F u.c. =∑j
fj(~Q)e−Mj e i~Q·~rj
Mj =1
2Q2〈u2
Qj〉
=1
2
(4π
λ
)2
sin2 θ〈u2Qj〉
Mj = B jT
(sin θ
λ
)2
B isoT =
8π2
3〈u2〉
In general, Debye-Waller factors can be anisotropic
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 3 / 17
Properties of the Debye-Waller Factor
For crystals with several differenttypes of atoms, we generalize theunit cell scattering factor.
B jT = 8π2〈u2
Qj〉
for isotropic atomic vibrations
〈u2〉 = 〈u2x + u2
y + u2z 〉
= 3〈u2x 〉 = 3〈u2
Q〉
F u.c. =∑j
fj(~Q)e−Mj e i~Q·~rj
Mj =1
2Q2〈u2
Qj〉
=1
2
(4π
λ
)2
sin2 θ〈u2Qj〉
Mj = B jT
(sin θ
λ
)2
B isoT =
8π2
3〈u2〉
In general, Debye-Waller factors can be anisotropic
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 3 / 17
Properties of the Debye-Waller Factor
For crystals with several differenttypes of atoms, we generalize theunit cell scattering factor.
B jT = 8π2〈u2
Qj〉
for isotropic atomic vibrations
〈u2〉 = 〈u2x + u2
y + u2z 〉
= 3〈u2x 〉 = 3〈u2
Q〉
F u.c. =∑j
fj(~Q)e−Mj e i~Q·~rj
Mj =1
2Q2〈u2
Qj〉
=1
2
(4π
λ
)2
sin2 θ〈u2Qj〉
Mj = B jT
(sin θ
λ
)2
B isoT =
8π2
3〈u2〉
In general, Debye-Waller factors can be anisotropic
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 3 / 17
The Debye Model
The Debye model can be used tocompute BT by integrating a lin-ear phonon dispersion relation upto a cutoff frequency, ωD , calledthe Debye frequency.
BT is given as a function of theDebye temperature Θ.
BT =6h2
mAkBΘ
[φ(Θ/T )
Θ/T+
1
4
]φ(x) =
1
x
∫ Θ/T
0
ξ
eξ − 1dξ
BT [A2] =
11492T[K]
AΘ2[K2]φ(Θ/T) +
2873
AΘ[K]
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 4 / 17
The Debye Model
The Debye model can be used tocompute BT by integrating a lin-ear phonon dispersion relation upto a cutoff frequency, ωD , calledthe Debye frequency.
BT is given as a function of theDebye temperature Θ.
BT =6h2
mAkBΘ
[φ(Θ/T )
Θ/T+
1
4
]φ(x) =
1
x
∫ Θ/T
0
ξ
eξ − 1dξ
BT [A2] =
11492T[K]
AΘ2[K2]φ(Θ/T) +
2873
AΘ[K]
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 4 / 17
The Debye Model
The Debye model can be used tocompute BT by integrating a lin-ear phonon dispersion relation upto a cutoff frequency, ωD , calledthe Debye frequency.
BT is given as a function of theDebye temperature Θ.
BT =6h2
mAkBΘ
[φ(Θ/T )
Θ/T+
1
4
]
φ(x) =1
x
∫ Θ/T
0
ξ
eξ − 1dξ
BT [A2] =
11492T[K]
AΘ2[K2]φ(Θ/T) +
2873
AΘ[K]
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 4 / 17
The Debye Model
The Debye model can be used tocompute BT by integrating a lin-ear phonon dispersion relation upto a cutoff frequency, ωD , calledthe Debye frequency.
BT is given as a function of theDebye temperature Θ.
BT =6h2
mAkBΘ
[φ(Θ/T )
Θ/T+
1
4
]φ(x) =
1
x
∫ Θ/T
0
ξ
eξ − 1dξ
BT [A2] =
11492T[K]
AΘ2[K2]φ(Θ/T) +
2873
AΘ[K]
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 4 / 17
The Debye Model
The Debye model can be used tocompute BT by integrating a lin-ear phonon dispersion relation upto a cutoff frequency, ωD , calledthe Debye frequency.
BT is given as a function of theDebye temperature Θ.
BT =6h2
mAkBΘ
[φ(Θ/T )
Θ/T+
1
4
]φ(x) =
1
x
∫ Θ/T
0
ξ
eξ − 1dξ
BT [A2] =
11492T[K]
AΘ2[K2]φ(Θ/T) +
2873
AΘ[K]
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 4 / 17
Debye Temperatures
BT =11492T
AΘ2φ(Θ/T )
+2873
AΘ
diamond is very stiff and Θdoes not vary much withtemperature
copper has a much lowerDebye temperature and awider variation of thermalfactor with temperature
A Θ B4.2 B77 B293
(K) (A2)
C∗ 12 2230 0.11 0.11 0.12Al 27 428 0.25 0.30 0.72Cu 63.5 343 0.13 0.17 0.47∗diamond
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 5 / 17
Debye Temperatures
BT =11492T
AΘ2φ(Θ/T )
+2873
AΘ
diamond is very stiff and Θdoes not vary much withtemperature
copper has a much lowerDebye temperature and awider variation of thermalfactor with temperature
A Θ B4.2 B77 B293
(K) (A2)
C∗ 12 2230 0.11 0.11 0.12Al 27 428 0.25 0.30 0.72Cu 63.5 343 0.13 0.17 0.47∗diamond
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 5 / 17
Debye Temperatures
BT =11492T
AΘ2φ(Θ/T )
+2873
AΘ
diamond is very stiff and Θdoes not vary much withtemperature
copper has a much lowerDebye temperature and awider variation of thermalfactor with temperature
A Θ B4.2 B77 B293
(K) (A2)
C∗ 12 2230 0.11 0.11 0.12Al 27 428 0.25 0.30 0.72Cu 63.5 343 0.13 0.17 0.47∗diamond
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 5 / 17
Debye Temperatures
BT =11492T
AΘ2φ(Θ/T )
+2873
AΘ
diamond is very stiff and Θdoes not vary much withtemperature
copper has a much lowerDebye temperature and awider variation of thermalfactor with temperature
A Θ B4.2 B77 B293
(K) (A2)
C∗ 12 2230 0.11 0.11 0.12Al 27 428 0.25 0.30 0.72Cu 63.5 343 0.13 0.17 0.47∗diamond
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 5 / 17
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 6 / 17
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 7 / 17
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 8 / 17
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 9 / 17
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 10 / 17
Bragg & Laue Geometries
Bragg
symmetric
asymmetric
Laue
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 11 / 17
Bragg & Laue Geometries
Bragg
symmetric
asymmetric
Laue
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 11 / 17
Bragg & Laue Geometries
Bragg
symmetric
asymmetric
Laue
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 11 / 17
Bragg & Laue Geometries
Bragg
symmetric
asymmetric
Laue
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 11 / 17
Bragg & Laue Geometries
Bragg
symmetric
asymmetric
Laue
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 11 / 17
Bragg & Laue Geometries
Bragg
symmetric
asymmetric
Laue
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 11 / 17
Scattering Geometry
Consider symmetric Bragg geometry
We expect the crystal to diffract in anenergy bandwidth defined by ∆k
This defines a spread of scattering vec-tors such that
ζ =∆Q
Q=
∆k
k
called the relative energy or wavelengthbandwidth
Q=mG
∆k
Q=mG(1+ζ)
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 12 / 17
Scattering Geometry
Consider symmetric Bragg geometry
We expect the crystal to diffract in anenergy bandwidth defined by ∆k
This defines a spread of scattering vec-tors such that
ζ =∆Q
Q=
∆k
k
called the relative energy or wavelengthbandwidth
Q=mG
Q=mG(1+ζ)
k k’
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 12 / 17
Scattering Geometry
Consider symmetric Bragg geometry
We expect the crystal to diffract in anenergy bandwidth defined by ∆k
This defines a spread of scattering vec-tors such that
ζ =∆Q
Q=
∆k
k
called the relative energy or wavelengthbandwidth
Q=mG
Q=mG(1+ζ)
k k’
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 12 / 17
Scattering Geometry
Consider symmetric Bragg geometry
We expect the crystal to diffract in anenergy bandwidth defined by ∆k
This defines a spread of scattering vec-tors such that
ζ =∆Q
Q=
∆k
k
called the relative energy or wavelengthbandwidth
Q=mG
∆k
Q=mG(1+ζ)
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 12 / 17
Dynamical Diffraction - Darwin Approach
The Darwin approach treats a perfectcrystal as an infinite stack of atomicplanes. This is fundamentally equivalentto the Ewald and von Laue approaches.
For a single thin slab of density ρ andthickness d � λ, the reflected and trans-mitted waves are
where
g =λroρd
sin θ
for a layer of unit cells ρ = |F |/vc and
d
T S
θ θ
S = −igT(1− igo)T ≈ e−igoT
g =[2d sin θ/m]ro(|F |/vc)d
sin θ=
1
m
2d2rovc|F |
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 13 / 17
Dynamical Diffraction - Darwin Approach
The Darwin approach treats a perfectcrystal as an infinite stack of atomicplanes. This is fundamentally equivalentto the Ewald and von Laue approaches.
For a single thin slab of density ρ andthickness d � λ, the reflected and trans-mitted waves are
where
g =λroρd
sin θ
for a layer of unit cells ρ = |F |/vc and
d
T S
θ θ
S = −igT(1− igo)T ≈ e−igoT
g =[2d sin θ/m]ro(|F |/vc)d
sin θ=
1
m
2d2rovc|F |
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 13 / 17
Dynamical Diffraction - Darwin Approach
The Darwin approach treats a perfectcrystal as an infinite stack of atomicplanes. This is fundamentally equivalentto the Ewald and von Laue approaches.
For a single thin slab of density ρ andthickness d � λ, the reflected and trans-mitted waves are
where
g =λroρd
sin θ
for a layer of unit cells ρ = |F |/vc and
d
T S
θ θ
S = −igT(1− igo)T ≈ e−igoT
g =[2d sin θ/m]ro(|F |/vc)d
sin θ=
1
m
2d2rovc|F |
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 13 / 17
Dynamical Diffraction - Darwin Approach
The Darwin approach treats a perfectcrystal as an infinite stack of atomicplanes. This is fundamentally equivalentto the Ewald and von Laue approaches.
For a single thin slab of density ρ andthickness d � λ, the reflected and trans-mitted waves are
where
g =λroρd
sin θ
for a layer of unit cells ρ = |F |/vc and
d
T S
θ θ
S = −igT
(1− igo)T ≈ e−igoT
g =[2d sin θ/m]ro(|F |/vc)d
sin θ=
1
m
2d2rovc|F |
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 13 / 17
Dynamical Diffraction - Darwin Approach
The Darwin approach treats a perfectcrystal as an infinite stack of atomicplanes. This is fundamentally equivalentto the Ewald and von Laue approaches.
For a single thin slab of density ρ andthickness d � λ, the reflected and trans-mitted waves are
where
g =λroρd
sin θ
for a layer of unit cells ρ = |F |/vc and
d
T S
θ θ
S = −igT(1− igo)T ≈ e−igoT
g =[2d sin θ/m]ro(|F |/vc)d
sin θ=
1
m
2d2rovc|F |
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 13 / 17
Dynamical Diffraction - Darwin Approach
The Darwin approach treats a perfectcrystal as an infinite stack of atomicplanes. This is fundamentally equivalentto the Ewald and von Laue approaches.
For a single thin slab of density ρ andthickness d � λ, the reflected and trans-mitted waves are
where
g =λroρd
sin θ
for a layer of unit cells ρ = |F |/vc and
d
T S
θ θ
S = −igT(1− igo)T ≈ e−igoT
g =[2d sin θ/m]ro(|F |/vc)d
sin θ=
1
m
2d2rovc|F |
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 13 / 17
Dynamical Diffraction - Darwin Approach
The Darwin approach treats a perfectcrystal as an infinite stack of atomicplanes. This is fundamentally equivalentto the Ewald and von Laue approaches.
For a single thin slab of density ρ andthickness d � λ, the reflected and trans-mitted waves are
where
g =λroρd
sin θ
for a layer of unit cells ρ = |F |/vc and
d
T S
θ θ
S = −igT(1− igo)T ≈ e−igoT
g =[2d sin θ/m]ro(|F |/vc)d
sin θ=
1
m
2d2rovc|F |
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 13 / 17
Dynamical Diffraction - Darwin Approach
The Darwin approach treats a perfectcrystal as an infinite stack of atomicplanes. This is fundamentally equivalentto the Ewald and von Laue approaches.
For a single thin slab of density ρ andthickness d � λ, the reflected and trans-mitted waves are
where
g =λroρd
sin θ
for a layer of unit cells ρ = |F |/vc and
d
T S
θ θ
S = −igT(1− igo)T ≈ e−igoT
g =[2d sin θ/m]ro(|F |/vc)d
sin θ
=1
m
2d2rovc|F |
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 13 / 17
Dynamical Diffraction - Darwin Approach
The Darwin approach treats a perfectcrystal as an infinite stack of atomicplanes. This is fundamentally equivalentto the Ewald and von Laue approaches.
For a single thin slab of density ρ andthickness d � λ, the reflected and trans-mitted waves are
where
g =λroρd
sin θ
for a layer of unit cells ρ = |F |/vc and
d
T S
θ θ
S = −igT(1− igo)T ≈ e−igoT
g =[2d sin θ/m]ro(|F |/vc)d
sin θ=
1
m
2d2rovc|F |
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 13 / 17
Dynamical Diffraction - Darwin Approach
g =1
m
2d2rovc|F |
since vc ∼ d3 then g ∼ ro/d ≈ 10−5
the transmitted beam depends on
go =λρat f
0(0)ro∆
sin θ
which can be rewritten
go =|Fo ||F |
g
where Fo is the forward scattering factorat Q = θ = 0
d
T S
θ θ
S = −igT(1− igo)T ≈ e−igoT
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 14 / 17
Dynamical Diffraction - Darwin Approach
g =1
m
2d2rovc|F |
since vc ∼ d3 then g ∼ ro/d ≈ 10−5
the transmitted beam depends on
go =λρat f
0(0)ro∆
sin θ
which can be rewritten
go =|Fo ||F |
g
where Fo is the forward scattering factorat Q = θ = 0
d
T S
θ θ
S = −igT(1− igo)T ≈ e−igoT
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 14 / 17
Dynamical Diffraction - Darwin Approach
g =1
m
2d2rovc|F |
since vc ∼ d3 then g ∼ ro/d ≈ 10−5
the transmitted beam depends on
go =λρat f
0(0)ro∆
sin θ
which can be rewritten
go =|Fo ||F |
g
where Fo is the forward scattering factorat Q = θ = 0
d
T S
θ θ
S = −igT(1− igo)T ≈ e−igoT
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 14 / 17
Dynamical Diffraction - Darwin Approach
g =1
m
2d2rovc|F |
since vc ∼ d3 then g ∼ ro/d ≈ 10−5
the transmitted beam depends on
go =λρat f
0(0)ro∆
sin θ
which can be rewritten
go =|Fo ||F |
g
where Fo is the forward scattering factorat Q = θ = 0
d
T S
θ θ
S = −igT(1− igo)T ≈ e−igoT
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 14 / 17
Dynamical Diffraction - Darwin Approach
g =1
m
2d2rovc|F |
since vc ∼ d3 then g ∼ ro/d ≈ 10−5
the transmitted beam depends on
go =λρat f
0(0)ro∆
sin θ
which can be rewritten
go =|Fo ||F |
g
where Fo is the forward scattering factorat Q = θ = 0
d
T S
θ θ
S = −igT(1− igo)T ≈ e−igoT
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 14 / 17
Kinematical Reflection
If we now extend this model to N layers we can use this kinematicalapproximation if Ng � 1.
Proceed by adding reflectivity from each layer with the usual phase factor
rN(Q) = −igN−1∑j=0
e iQdje−igo je−igo j = −igN−1∑j=0
e i(Qd−2go)j
Q=mG
∆k
Q=mG(1+ζ)where the x-rays pass through each layertwice
these N unit cell layers will give a recip-rocal lattice with points at multiples ofG = 2π/d we are interested in small de-viations from the Bragg condition:
ζ =∆Q
Q=
∆k
k=
∆EE
=∆λ
λ
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 15 / 17
Kinematical Reflection
If we now extend this model to N layers we can use this kinematicalapproximation if Ng � 1.
Proceed by adding reflectivity from each layer with the usual phase factor
rN(Q) = −igN−1∑j=0
e iQdje−igo je−igo j = −igN−1∑j=0
e i(Qd−2go)j
Q=mG
∆k
Q=mG(1+ζ)where the x-rays pass through each layertwice
these N unit cell layers will give a recip-rocal lattice with points at multiples ofG = 2π/d we are interested in small de-viations from the Bragg condition:
ζ =∆Q
Q=
∆k
k=
∆EE
=∆λ
λ
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 15 / 17
Kinematical Reflection
If we now extend this model to N layers we can use this kinematicalapproximation if Ng � 1.
Proceed by adding reflectivity from each layer with the usual phase factor
rN(Q) = −igN−1∑j=0
e iQdje−igo je−igo j
= −igN−1∑j=0
e i(Qd−2go)j
Q=mG
∆k
Q=mG(1+ζ)where the x-rays pass through each layertwice
these N unit cell layers will give a recip-rocal lattice with points at multiples ofG = 2π/d we are interested in small de-viations from the Bragg condition:
ζ =∆Q
Q=
∆k
k=
∆EE
=∆λ
λ
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 15 / 17
Kinematical Reflection
If we now extend this model to N layers we can use this kinematicalapproximation if Ng � 1.
Proceed by adding reflectivity from each layer with the usual phase factor
rN(Q) = −igN−1∑j=0
e iQdje−igo je−igo j
= −igN−1∑j=0
e i(Qd−2go)j
Q=mG
∆k
Q=mG(1+ζ)
where the x-rays pass through each layertwice
these N unit cell layers will give a recip-rocal lattice with points at multiples ofG = 2π/d we are interested in small de-viations from the Bragg condition:
ζ =∆Q
Q=
∆k
k=
∆EE
=∆λ
λ
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 15 / 17
Kinematical Reflection
If we now extend this model to N layers we can use this kinematicalapproximation if Ng � 1.
Proceed by adding reflectivity from each layer with the usual phase factor
rN(Q) = −igN−1∑j=0
e iQdje−igo je−igo j = −igN−1∑j=0
e i(Qd−2go)j
Q=mG
∆k
Q=mG(1+ζ)
where the x-rays pass through each layertwice
these N unit cell layers will give a recip-rocal lattice with points at multiples ofG = 2π/d we are interested in small de-viations from the Bragg condition:
ζ =∆Q
Q=
∆k
k=
∆EE
=∆λ
λ
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 15 / 17
Kinematical Reflection
If we now extend this model to N layers we can use this kinematicalapproximation if Ng � 1.
Proceed by adding reflectivity from each layer with the usual phase factor
rN(Q) = −igN−1∑j=0
e iQdje−igo je−igo j = −igN−1∑j=0
e i(Qd−2go)j
Q=mG
∆k
Q=mG(1+ζ)
where the x-rays pass through each layertwice
these N unit cell layers will give a recip-rocal lattice with points at multiples ofG = 2π/d
we are interested in small de-viations from the Bragg condition:
ζ =∆Q
Q=
∆k
k=
∆EE
=∆λ
λ
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 15 / 17
Kinematical Reflection
If we now extend this model to N layers we can use this kinematicalapproximation if Ng � 1.
Proceed by adding reflectivity from each layer with the usual phase factor
rN(Q) = −igN−1∑j=0
e iQdje−igo je−igo j = −igN−1∑j=0
e i(Qd−2go)j
Q=mG
∆k
Q=mG(1+ζ)where the x-rays pass through each layertwice
these N unit cell layers will give a recip-rocal lattice with points at multiples ofG = 2π/d we are interested in small de-viations from the Bragg condition:
ζ =∆Q
Q=
∆k
k=
∆EE
=∆λ
λ
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 15 / 17
Multiple Layer Reflection
rN(Q) = −igN−1∑j=0
e i(Qd−2go)j
Q=mG
∆k
Q=mG(1+ζ)
The term in the phase factor now be-comes
Qd − 2go = mG (1 + ζ)2π
G− 2go
= 2π(m + mζ − goπ
)
rN(Q) = −igN−1∑j=0
e i2π(m+mζ−go/π
= −igN−1∑j=0
e i2πmje i2π(mζ−go/π)
= −igN−1∑j=0
1 · e i2π(mζ−go/π)
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 16 / 17
Multiple Layer Reflection
rN(Q) = −igN−1∑j=0
e i(Qd−2go)j
Q=mG
∆k
Q=mG(1+ζ)
The term in the phase factor now be-comes
Qd − 2go
= mG (1 + ζ)2π
G− 2go
= 2π(m + mζ − goπ
)
rN(Q) = −igN−1∑j=0
e i2π(m+mζ−go/π
= −igN−1∑j=0
e i2πmje i2π(mζ−go/π)
= −igN−1∑j=0
1 · e i2π(mζ−go/π)
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 16 / 17
Multiple Layer Reflection
rN(Q) = −igN−1∑j=0
e i(Qd−2go)j
Q=mG
∆k
Q=mG(1+ζ)
The term in the phase factor now be-comes
Qd − 2go = mG (1 + ζ)2π
G− 2go
= 2π(m + mζ − goπ
)
rN(Q) = −igN−1∑j=0
e i2π(m+mζ−go/π
= −igN−1∑j=0
e i2πmje i2π(mζ−go/π)
= −igN−1∑j=0
1 · e i2π(mζ−go/π)
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 16 / 17
Multiple Layer Reflection
rN(Q) = −igN−1∑j=0
e i(Qd−2go)j
Q=mG
∆k
Q=mG(1+ζ)
The term in the phase factor now be-comes
Qd − 2go = mG (1 + ζ)2π
G− 2go
= 2π(m + mζ − goπ
)
rN(Q) = −igN−1∑j=0
e i2π(m+mζ−go/π
= −igN−1∑j=0
e i2πmje i2π(mζ−go/π)
= −igN−1∑j=0
1 · e i2π(mζ−go/π)
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 16 / 17
Multiple Layer Reflection
rN(Q) = −igN−1∑j=0
e i(Qd−2go)j
Q=mG
∆k
Q=mG(1+ζ)
The term in the phase factor now be-comes
Qd − 2go = mG (1 + ζ)2π
G− 2go
= 2π(m + mζ − goπ
)
rN(Q) = −igN−1∑j=0
e i2π(m+mζ−go/π
= −igN−1∑j=0
e i2πmje i2π(mζ−go/π)
= −igN−1∑j=0
1 · e i2π(mζ−go/π)
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 16 / 17
Multiple Layer Reflection
rN(Q) = −igN−1∑j=0
e i(Qd−2go)j
Q=mG
∆k
Q=mG(1+ζ)
The term in the phase factor now be-comes
Qd − 2go = mG (1 + ζ)2π
G− 2go
= 2π(m + mζ − goπ
)
rN(Q) = −igN−1∑j=0
e i2π(m+mζ−go/π
= −igN−1∑j=0
e i2πmje i2π(mζ−go/π)
= −igN−1∑j=0
1 · e i2π(mζ−go/π)
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 16 / 17
Multiple Layer Reflection
rN(Q) = −igN−1∑j=0
e i(Qd−2go)j
Q=mG
∆k
Q=mG(1+ζ)
The term in the phase factor now be-comes
Qd − 2go = mG (1 + ζ)2π
G− 2go
= 2π(m + mζ − goπ
)
rN(Q) = −igN−1∑j=0
e i2π(m+mζ−go/π
= −igN−1∑j=0
e i2πmje i2π(mζ−go/π)
= −igN−1∑j=0
1 · e i2π(mζ−go/π)
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 16 / 17
Multiple Layer Reflection
This geometric series can besummed as usual
where
ζo =goπ
=2d2|Fo |πmvc
ro
rN(Q) = −igN−1∑j=0
e i2π(mζ−go/π)
|rN(ζ)| =
[sin(πN[mζ − ζo ])
sin(π[mζ − ζo ])
]|rN(ζ)|2 → g2
2(π[mζ − ζo ])2
This describes a shift of the Bragg peak away from the reciprocal latticepoint, the maximum being at ζ = ζo/m
The kinematical approach now breaks down and we need to develop a newtheory for dynamical diffraction.
First, let’s explore how the intensity would vary using the kinematicalexpression
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 17 / 17
Multiple Layer Reflection
This geometric series can besummed as usual
where
ζo =goπ
=2d2|Fo |πmvc
ro
rN(Q) = −igN−1∑j=0
e i2π(mζ−go/π)
|rN(ζ)| =
[sin(πN[mζ − ζo ])
sin(π[mζ − ζo ])
]
|rN(ζ)|2 → g2
2(π[mζ − ζo ])2
This describes a shift of the Bragg peak away from the reciprocal latticepoint, the maximum being at ζ = ζo/m
The kinematical approach now breaks down and we need to develop a newtheory for dynamical diffraction.
First, let’s explore how the intensity would vary using the kinematicalexpression
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 17 / 17
Multiple Layer Reflection
This geometric series can besummed as usual
where
ζo =goπ
=2d2|Fo |πmvc
ro
rN(Q) = −igN−1∑j=0
e i2π(mζ−go/π)
|rN(ζ)| =
[sin(πN[mζ − ζo ])
sin(π[mζ − ζo ])
]
|rN(ζ)|2 → g2
2(π[mζ − ζo ])2
This describes a shift of the Bragg peak away from the reciprocal latticepoint, the maximum being at ζ = ζo/m
The kinematical approach now breaks down and we need to develop a newtheory for dynamical diffraction.
First, let’s explore how the intensity would vary using the kinematicalexpression
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 17 / 17
Multiple Layer Reflection
This geometric series can besummed as usual
where
ζo =goπ
=2d2|Fo |πmvc
ro
rN(Q) = −igN−1∑j=0
e i2π(mζ−go/π)
|rN(ζ)| =
[sin(πN[mζ − ζo ])
sin(π[mζ − ζo ])
]
|rN(ζ)|2 → g2
2(π[mζ − ζo ])2
This describes a shift of the Bragg peak away from the reciprocal latticepoint, the maximum being at ζ = ζo/m
The kinematical approach now breaks down and we need to develop a newtheory for dynamical diffraction.
First, let’s explore how the intensity would vary using the kinematicalexpression
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 17 / 17
Multiple Layer Reflection
This geometric series can besummed as usual
where
ζo =goπ
=2d2|Fo |πmvc
ro
rN(Q) = −igN−1∑j=0
e i2π(mζ−go/π)
|rN(ζ)| =
[sin(πN[mζ − ζo ])
sin(π[mζ − ζo ])
]
|rN(ζ)|2 → g2
2(π[mζ − ζo ])2
This describes a shift of the Bragg peak away from the reciprocal latticepoint, the maximum being at ζ = ζo/m
The kinematical approach now breaks down and we need to develop a newtheory for dynamical diffraction.
First, let’s explore how the intensity would vary using the kinematicalexpression
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 17 / 17
Multiple Layer Reflection
This geometric series can besummed as usual
where
ζo =goπ
=2d2|Fo |πmvc
ro
rN(Q) = −igN−1∑j=0
e i2π(mζ−go/π)
|rN(ζ)| =
[sin(πN[mζ − ζo ])
sin(π[mζ − ζo ])
]
|rN(ζ)|2 → g2
2(π[mζ − ζo ])2
This describes a shift of the Bragg peak away from the reciprocal latticepoint, the maximum being at ζ = ζo/m
The kinematical approach now breaks down and we need to develop a newtheory for dynamical diffraction.
First, let’s explore how the intensity would vary using the kinematicalexpression
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 17 / 17
Multiple Layer Reflection
This geometric series can besummed as usual
where
ζo =goπ
=2d2|Fo |πmvc
ro
rN(Q) = −igN−1∑j=0
e i2π(mζ−go/π)
|rN(ζ)| =
[sin(πN[mζ − ζo ])
sin(π[mζ − ζo ])
]|rN(ζ)|2 → g2
2 sin2(π[mζ − ζo ])
This describes a shift of the Bragg peak away from the reciprocal latticepoint, the maximum being at ζ = ζo/m
The kinematical approach now breaks down and we need to develop a newtheory for dynamical diffraction.
First, let’s explore how the intensity would vary using the kinematicalexpression
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 17 / 17
Multiple Layer Reflection
This geometric series can besummed as usual
where
ζo =goπ
=2d2|Fo |πmvc
ro
rN(Q) = −igN−1∑j=0
e i2π(mζ−go/π)
|rN(ζ)| =
[sin(πN[mζ − ζo ])
sin(π[mζ − ζo ])
]|rN(ζ)|2 → g2
2(π[mζ − ζo ])2
This describes a shift of the Bragg peak away from the reciprocal latticepoint, the maximum being at ζ = ζo/m
The kinematical approach now breaks down and we need to develop a newtheory for dynamical diffraction.
First, let’s explore how the intensity would vary using the kinematicalexpression
C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 17 / 17