today’s outline - october 28, 2013csrri.iit.edu/~segre/phys570/13f/lecture_18.pdfphys 570 day at...
TRANSCRIPT
Today’s Outline - October 28, 2013
• PHYS 570 day at 10-BM
• Brief introduction to EXAFS
• Crystal Truncation Rods
• Lattice Vibrations
• Thermal Diffuse Scattering
• Debye Waller Factor
• Lorentz Factor, Extinction & Absorption
• Powder Diffraction
October 30, 2013 class in 111 Life Sciences(Chemistry Colloquium)
Homework Assignment #06:Chapter 6: 1,6,7,8,9due Wednesday, November 11, 2013
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 1 / 23
Today’s Outline - October 28, 2013
• PHYS 570 day at 10-BM
• Brief introduction to EXAFS
• Crystal Truncation Rods
• Lattice Vibrations
• Thermal Diffuse Scattering
• Debye Waller Factor
• Lorentz Factor, Extinction & Absorption
• Powder Diffraction
October 30, 2013 class in 111 Life Sciences(Chemistry Colloquium)
Homework Assignment #06:Chapter 6: 1,6,7,8,9due Wednesday, November 11, 2013
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 1 / 23
Today’s Outline - October 28, 2013
• PHYS 570 day at 10-BM
• Brief introduction to EXAFS
• Crystal Truncation Rods
• Lattice Vibrations
• Thermal Diffuse Scattering
• Debye Waller Factor
• Lorentz Factor, Extinction & Absorption
• Powder Diffraction
October 30, 2013 class in 111 Life Sciences(Chemistry Colloquium)
Homework Assignment #06:Chapter 6: 1,6,7,8,9due Wednesday, November 11, 2013
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 1 / 23
Today’s Outline - October 28, 2013
• PHYS 570 day at 10-BM
• Brief introduction to EXAFS
• Crystal Truncation Rods
• Lattice Vibrations
• Thermal Diffuse Scattering
• Debye Waller Factor
• Lorentz Factor, Extinction & Absorption
• Powder Diffraction
October 30, 2013 class in 111 Life Sciences(Chemistry Colloquium)
Homework Assignment #06:Chapter 6: 1,6,7,8,9due Wednesday, November 11, 2013
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 1 / 23
Today’s Outline - October 28, 2013
• PHYS 570 day at 10-BM
• Brief introduction to EXAFS
• Crystal Truncation Rods
• Lattice Vibrations
• Thermal Diffuse Scattering
• Debye Waller Factor
• Lorentz Factor, Extinction & Absorption
• Powder Diffraction
October 30, 2013 class in 111 Life Sciences(Chemistry Colloquium)
Homework Assignment #06:Chapter 6: 1,6,7,8,9due Wednesday, November 11, 2013
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 1 / 23
Today’s Outline - October 28, 2013
• PHYS 570 day at 10-BM
• Brief introduction to EXAFS
• Crystal Truncation Rods
• Lattice Vibrations
• Thermal Diffuse Scattering
• Debye Waller Factor
• Lorentz Factor, Extinction & Absorption
• Powder Diffraction
October 30, 2013 class in 111 Life Sciences(Chemistry Colloquium)
Homework Assignment #06:Chapter 6: 1,6,7,8,9due Wednesday, November 11, 2013
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 1 / 23
Today’s Outline - October 28, 2013
• PHYS 570 day at 10-BM
• Brief introduction to EXAFS
• Crystal Truncation Rods
• Lattice Vibrations
• Thermal Diffuse Scattering
• Debye Waller Factor
• Lorentz Factor, Extinction & Absorption
• Powder Diffraction
October 30, 2013 class in 111 Life Sciences(Chemistry Colloquium)
Homework Assignment #06:Chapter 6: 1,6,7,8,9due Wednesday, November 11, 2013
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 1 / 23
Today’s Outline - October 28, 2013
• PHYS 570 day at 10-BM
• Brief introduction to EXAFS
• Crystal Truncation Rods
• Lattice Vibrations
• Thermal Diffuse Scattering
• Debye Waller Factor
• Lorentz Factor, Extinction & Absorption
• Powder Diffraction
October 30, 2013 class in 111 Life Sciences(Chemistry Colloquium)
Homework Assignment #06:Chapter 6: 1,6,7,8,9due Wednesday, November 11, 2013
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 1 / 23
Today’s Outline - October 28, 2013
• PHYS 570 day at 10-BM
• Brief introduction to EXAFS
• Crystal Truncation Rods
• Lattice Vibrations
• Thermal Diffuse Scattering
• Debye Waller Factor
• Lorentz Factor, Extinction & Absorption
• Powder Diffraction
October 30, 2013 class in 111 Life Sciences(Chemistry Colloquium)
Homework Assignment #06:Chapter 6: 1,6,7,8,9due Wednesday, November 11, 2013
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 1 / 23
Today’s Outline - October 28, 2013
• PHYS 570 day at 10-BM
• Brief introduction to EXAFS
• Crystal Truncation Rods
• Lattice Vibrations
• Thermal Diffuse Scattering
• Debye Waller Factor
• Lorentz Factor, Extinction & Absorption
• Powder Diffraction
October 30, 2013 class in 111 Life Sciences(Chemistry Colloquium)
Homework Assignment #06:Chapter 6: 1,6,7,8,9due Wednesday, November 11, 2013
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 1 / 23
Today’s Outline - October 28, 2013
• PHYS 570 day at 10-BM
• Brief introduction to EXAFS
• Crystal Truncation Rods
• Lattice Vibrations
• Thermal Diffuse Scattering
• Debye Waller Factor
• Lorentz Factor, Extinction & Absorption
• Powder Diffraction
October 30, 2013 class in 111 Life Sciences(Chemistry Colloquium)
Homework Assignment #06:Chapter 6: 1,6,7,8,9due Wednesday, November 11, 2013
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 1 / 23
PHYS 570 day at 10-BM
1 3 sessions:• 09:00 – 12:00• 13:00 – 16:00• 17:00 – 20:00
2 Activities
• Absolute flux measurement• Reflectivity measurement• EXAFS measurement
3 Make sure your badge is ready
4 Leave plenty of time to get the badge
5 Let me know when you plan to come!
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 2 / 23
PHYS 570 day at 10-BM
1 3 sessions:• 09:00 – 12:00• 13:00 – 16:00• 17:00 – 20:00
2 Activities• Absolute flux measurement• Reflectivity measurement• EXAFS measurement
3 Make sure your badge is ready
4 Leave plenty of time to get the badge
5 Let me know when you plan to come!
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 2 / 23
PHYS 570 day at 10-BM
1 3 sessions:• 09:00 – 12:00• 13:00 – 16:00• 17:00 – 20:00
2 Activities• Absolute flux measurement• Reflectivity measurement• EXAFS measurement
3 Make sure your badge is ready
4 Leave plenty of time to get the badge
5 Let me know when you plan to come!
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 2 / 23
PHYS 570 day at 10-BM
1 3 sessions:• 09:00 – 12:00• 13:00 – 16:00• 17:00 – 20:00
2 Activities• Absolute flux measurement• Reflectivity measurement• EXAFS measurement
3 Make sure your badge is ready
4 Leave plenty of time to get the badge
5 Let me know when you plan to come!
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 2 / 23
PHYS 570 day at 10-BM
1 3 sessions:• 09:00 – 12:00• 13:00 – 16:00• 17:00 – 20:00
2 Activities• Absolute flux measurement• Reflectivity measurement• EXAFS measurement
3 Make sure your badge is ready
4 Leave plenty of time to get the badge
5 Let me know when you plan to come!
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 2 / 23
What is XAFS?
X-ray Absorption Fine-Structure (XAFS) is the modulation of the x-rayabsorption coefficient at energies near and above an x-ray absorption edge.XAFS is also referred to as X-ray Absorption Spectroscopy (XAS) and isbroken into 2 regimes:
XANES X-ray Absorption Near-Edge SpectroscopyEXAFS Extended X-ray Absorption Fine-Structure
which contain related, but slightly different information about an element’slocal coordination and chemical state.
EXAFS
XANES
E (eV)
µ
(
E
)
77007600750074007300720071007000
2.0
1.5
1.0
0.5
0.0
Fe K-edge XAFS for FeO
XAFS Characteristics:
• local atomic coordination
• chemical / oxidation state
• applies to any element
• works at low concentrations
• minimal sample requirements
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 3 / 23
The x-ray absorption process
An x-ray is absorbed by anatom when the energy of thex-ray is transferred to a core-level electron (K, L, or Mshell).
The atom is in an excitedstate with an empty elec-tronic level: a core hole.
Any excess energy fromthe x-ray is given to anejected photoelectron
, whichexpands as a spherical wave,reaches the neighboring elec-tron clouds, and scattersback to the core hole, cre-ating interference patternscalled XANES and EXAFS.
x−ray
L
K
M
Energy
photo−electron
Continuum
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 4 / 23
The x-ray absorption process
An x-ray is absorbed by anatom when the energy of thex-ray is transferred to a core-level electron (K, L, or Mshell).
The atom is in an excitedstate with an empty elec-tronic level: a core hole.
Any excess energy fromthe x-ray is given to anejected photoelectron
, whichexpands as a spherical wave,reaches the neighboring elec-tron clouds, and scattersback to the core hole, cre-ating interference patternscalled XANES and EXAFS.
x−ray
L
K
M
Energy
photo−electron
Continuum
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 4 / 23
The x-ray absorption process
An x-ray is absorbed by anatom when the energy of thex-ray is transferred to a core-level electron (K, L, or Mshell).
The atom is in an excitedstate with an empty elec-tronic level: a core hole.
Any excess energy fromthe x-ray is given to anejected photoelectron
, whichexpands as a spherical wave,reaches the neighboring elec-tron clouds, and scattersback to the core hole, cre-ating interference patternscalled XANES and EXAFS.
x−ray
L
K
M
Energy
photo−electron
Continuum
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 4 / 23
The x-ray absorption process
An x-ray is absorbed by anatom when the energy of thex-ray is transferred to a core-level electron (K, L, or Mshell).
The atom is in an excitedstate with an empty elec-tronic level: a core hole.
Any excess energy fromthe x-ray is given to anejected photoelectron, whichexpands as a spherical wave
,reaches the neighboring elec-tron clouds, and scattersback to the core hole, cre-ating interference patternscalled XANES and EXAFS.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 4 / 23
The x-ray absorption process
An x-ray is absorbed by anatom when the energy of thex-ray is transferred to a core-level electron (K, L, or Mshell).
The atom is in an excitedstate with an empty elec-tronic level: a core hole.
Any excess energy fromthe x-ray is given to anejected photoelectron, whichexpands as a spherical wave
,reaches the neighboring elec-tron clouds, and scattersback to the core hole, cre-ating interference patternscalled XANES and EXAFS.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 4 / 23
The x-ray absorption process
An x-ray is absorbed by anatom when the energy of thex-ray is transferred to a core-level electron (K, L, or Mshell).
The atom is in an excitedstate with an empty elec-tronic level: a core hole.
Any excess energy fromthe x-ray is given to anejected photoelectron, whichexpands as a spherical wave,reaches the neighboring elec-tron clouds
, and scattersback to the core hole, cre-ating interference patternscalled XANES and EXAFS.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 4 / 23
The x-ray absorption process
An x-ray is absorbed by anatom when the energy of thex-ray is transferred to a core-level electron (K, L, or Mshell).
The atom is in an excitedstate with an empty elec-tronic level: a core hole.
Any excess energy fromthe x-ray is given to anejected photoelectron, whichexpands as a spherical wave,reaches the neighboring elec-tron clouds, and scattersback to the core hole
, cre-ating interference patternscalled XANES and EXAFS.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 4 / 23
The x-ray absorption process
An x-ray is absorbed by anatom when the energy of thex-ray is transferred to a core-level electron (K, L, or Mshell).
The atom is in an excitedstate with an empty elec-tronic level: a core hole.
Any excess energy fromthe x-ray is given to anejected photoelectron, whichexpands as a spherical wave,reaches the neighboring elec-tron clouds, and scattersback to the core hole
, cre-ating interference patternscalled XANES and EXAFS.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 4 / 23
The x-ray absorption process
An x-ray is absorbed by anatom when the energy of thex-ray is transferred to a core-level electron (K, L, or Mshell).
The atom is in an excitedstate with an empty elec-tronic level: a core hole.
Any excess energy fromthe x-ray is given to anejected photoelectron, whichexpands as a spherical wave,reaches the neighboring elec-tron clouds, and scattersback to the core hole, cre-ating interference patternscalled XANES and EXAFS.
11500 12000 12500
E(eV)
-2
-1.5
-1
-0.5
0
0.5
ln(I
o/I
)
EXAFS
XA
NE
S /
NE
XA
FS
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 4 / 23
EXAFS data extraction
normalize by fitting pre-edgeand post-edge trends
remove “smooth” µ0 back-ground
convert to photoelectron mo-mentum space
k =2π
hc
√E − E0
weight by appropriate powerof k to obtain “good” enve-lope which clearly shows thatEXAFS is a sum of oscilla-tions with varying frequen-cies and phases
Fourier transform to get realspace EXAFS
11500 12000 12500
E(eV)
-2
-1.5
-1
-0.5
0
0.5
ln(I
o/I
)
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 5 / 23
EXAFS data extraction
normalize by fitting pre-edgeand post-edge trends
remove “smooth” µ0 back-ground
convert to photoelectron mo-mentum space
k =2π
hc
√E − E0
weight by appropriate powerof k to obtain “good” enve-lope which clearly shows thatEXAFS is a sum of oscilla-tions with varying frequen-cies and phases
Fourier transform to get realspace EXAFS
11500 12000 12500
E(eV)
0
0.5
1
ln(I
o/I
)
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 5 / 23
EXAFS data extraction
normalize by fitting pre-edgeand post-edge trends
remove “smooth” µ0 back-ground
convert to photoelectron mo-mentum space
k =2π
hc
√E − E0
weight by appropriate powerof k to obtain “good” enve-lope which clearly shows thatEXAFS is a sum of oscilla-tions with varying frequen-cies and phases
Fourier transform to get realspace EXAFS
11500 12000 12500
E(eV)
0
0.5
1
ln(I
o/I
)
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 5 / 23
EXAFS data extraction
normalize by fitting pre-edgeand post-edge trends
remove “smooth” µ0 back-ground
convert to photoelectron mo-mentum space
k =2π
hc
√E − E0
weight by appropriate powerof k to obtain “good” enve-lope which clearly shows thatEXAFS is a sum of oscilla-tions with varying frequen-cies and phases
Fourier transform to get realspace EXAFS
0 5 10 15
k(Å-1
)
-0.2
-0.1
0
0.1
0.2
χ
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 5 / 23
EXAFS data extraction
normalize by fitting pre-edgeand post-edge trends
remove “smooth” µ0 back-ground
convert to photoelectron mo-mentum space
k =2π
hc
√E − E0
weight by appropriate powerof k to obtain “good” enve-lope which clearly shows thatEXAFS is a sum of oscilla-tions with varying frequen-cies and phases
Fourier transform to get realspace EXAFS
0 5 10 15
k(Å-1
)
-0.2
-0.1
0
0.1
0.2
χ
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 5 / 23
EXAFS data extraction
normalize by fitting pre-edgeand post-edge trends
remove “smooth” µ0 back-ground
convert to photoelectron mo-mentum space
k =2π
hc
√E − E0
weight by appropriate powerof k to obtain “good” enve-lope which clearly shows thatEXAFS is a sum of oscilla-tions with varying frequen-cies and phases
Fourier transform to get realspace EXAFS
0 2 4 6
R(Å)
0
5
10
15
20
25
30
χ(R
)
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 5 / 23
The EXAFS equation
The EXAFS oscillations can be modelled and interpreted using aconceptually simple equation (the details are more subtle!):
χ(k) =∑j
NjS20 fj(k)e−2Rj/λ(k) e−2k2σ2
j
kRj2
sin [2kRj + δj(k)]
where the sum could be over shells of atoms (Fe-O, Fe-Fe) or . . .. . . over scattering paths for the photo-electron.
Nj : path degeneracy
Rj : half path length
σ2j : path “disorder”
S20 : amplitude reduction factor
k is the photoelectron wave number
fj(k): scattering factor for the path
δj(k): phase shift for the path
λ(k): photoelectron mean free path
Because we can compute f (k) and δ(k), and λ(k) we can determine Z, R,N, and σ2 for scattering paths to neighboring atoms by fitting the data.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 6 / 23
The EXAFS equation
The EXAFS oscillations can be modelled and interpreted using aconceptually simple equation (the details are more subtle!):
χ(k) =∑j
NjS20 fj(k)e−2Rj/λ(k) e−2k2σ2
j
kRj2
sin [2kRj + δj(k)]
where the sum could be over shells of atoms (Fe-O, Fe-Fe) or . . .. . . over scattering paths for the photo-electron.
Nj : path degeneracy
Rj : half path length
σ2j : path “disorder”
S20 : amplitude reduction factor
k is the photoelectron wave number
fj(k): scattering factor for the path
δj(k): phase shift for the path
λ(k): photoelectron mean free path
Because we can compute f (k) and δ(k), and λ(k) we can determine Z, R,N, and σ2 for scattering paths to neighboring atoms by fitting the data.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 6 / 23
The EXAFS equation
The EXAFS oscillations can be modelled and interpreted using aconceptually simple equation (the details are more subtle!):
χ(k) =∑j
NjS20 fj(k)e−2Rj/λ(k) e−2k2σ2
j
kRj2
sin [2kRj + δj(k)]
where the sum could be over shells of atoms (Fe-O, Fe-Fe) or . . .
. . . over scattering paths for the photo-electron.
Nj : path degeneracy
Rj : half path length
σ2j : path “disorder”
S20 : amplitude reduction factor
k is the photoelectron wave number
fj(k): scattering factor for the path
δj(k): phase shift for the path
λ(k): photoelectron mean free path
Because we can compute f (k) and δ(k), and λ(k) we can determine Z, R,N, and σ2 for scattering paths to neighboring atoms by fitting the data.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 6 / 23
The EXAFS equation
The EXAFS oscillations can be modelled and interpreted using aconceptually simple equation (the details are more subtle!):
χ(k) =∑j
NjS20 fj(k)e−2Rj/λ(k) e−2k2σ2
j
kRj2
sin [2kRj + δj(k)]
where the sum could be over shells of atoms (Fe-O, Fe-Fe) or . . .. . . over scattering paths for the photo-electron.
Nj : path degeneracy
Rj : half path length
σ2j : path “disorder”
S20 : amplitude reduction factor
k is the photoelectron wave number
fj(k): scattering factor for the path
δj(k): phase shift for the path
λ(k): photoelectron mean free path
Because we can compute f (k) and δ(k), and λ(k) we can determine Z, R,N, and σ2 for scattering paths to neighboring atoms by fitting the data.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 6 / 23
The EXAFS equation
The EXAFS oscillations can be modelled and interpreted using aconceptually simple equation (the details are more subtle!):
χ(k) =∑j
NjS20 fj(k)e−2Rj/λ(k) e−2k2σ2
j
kRj2
sin [2kRj + δj(k)]
where the sum could be over shells of atoms (Fe-O, Fe-Fe) or . . .. . . over scattering paths for the photo-electron.
Nj : path degeneracy
Rj : half path length
σ2j : path “disorder”
S20 : amplitude reduction factor
k is the photoelectron wave number
fj(k): scattering factor for the path
δj(k): phase shift for the path
λ(k): photoelectron mean free path
Because we can compute f (k) and δ(k), and λ(k) we can determine Z, R,N, and σ2 for scattering paths to neighboring atoms by fitting the data.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 6 / 23
The EXAFS equation
The EXAFS oscillations can be modelled and interpreted using aconceptually simple equation (the details are more subtle!):
χ(k) =∑j
NjS20 fj(k)e−2Rj/λ(k) e−2k2σ2
j
kRj2
sin [2kRj + δj(k)]
where the sum could be over shells of atoms (Fe-O, Fe-Fe) or . . .. . . over scattering paths for the photo-electron.
Nj : path degeneracy
Rj : half path length
σ2j : path “disorder”
S20 : amplitude reduction factor
k is the photoelectron wave number
fj(k): scattering factor for the path
δj(k): phase shift for the path
λ(k): photoelectron mean free path
Because we can compute f (k) and δ(k), and λ(k) we can determine Z, R,N, and σ2 for scattering paths to neighboring atoms by fitting the data.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 6 / 23
The EXAFS equation
The EXAFS oscillations can be modelled and interpreted using aconceptually simple equation (the details are more subtle!):
χ(k) =∑j
NjS20 fj(k)e−2Rj/λ(k) e−2k2σ2
j
kRj2
sin [2kRj + δj(k)]
where the sum could be over shells of atoms (Fe-O, Fe-Fe) or . . .. . . over scattering paths for the photo-electron.
Nj : path degeneracy
Rj : half path length
σ2j : path “disorder”
S20 : amplitude reduction factor
k is the photoelectron wave number
fj(k): scattering factor for the path
δj(k): phase shift for the path
λ(k): photoelectron mean free path
Because we can compute f (k) and δ(k), and λ(k) we can determine Z, R,N, and σ2 for scattering paths to neighboring atoms by fitting the data.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 6 / 23
The EXAFS equation
The EXAFS oscillations can be modelled and interpreted using aconceptually simple equation (the details are more subtle!):
χ(k) =∑j
NjS20 fj(k)e−2Rj/λ(k) e−2k2σ2
j
kRj2
sin [2kRj + δj(k)]
where the sum could be over shells of atoms (Fe-O, Fe-Fe) or . . .. . . over scattering paths for the photo-electron.
Nj : path degeneracy
Rj : half path length
σ2j : path “disorder”
S20 : amplitude reduction factor
k is the photoelectron wave number
fj(k): scattering factor for the path
δj(k): phase shift for the path
λ(k): photoelectron mean free path
Because we can compute f (k) and δ(k), and λ(k) we can determine Z, R,N, and σ2 for scattering paths to neighboring atoms by fitting the data.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 6 / 23
The EXAFS equation
The EXAFS oscillations can be modelled and interpreted using aconceptually simple equation (the details are more subtle!):
χ(k) =∑j
NjS20 fj(k)e−2Rj/λ(k) e−2k2σ2
j
kRj2
sin [2kRj + δj(k)]
where the sum could be over shells of atoms (Fe-O, Fe-Fe) or . . .. . . over scattering paths for the photo-electron.
Nj : path degeneracy
Rj : half path length
σ2j : path “disorder”
S20 : amplitude reduction factor
k is the photoelectron wave number
fj(k): scattering factor for the path
δj(k): phase shift for the path
λ(k): photoelectron mean free path
Because we can compute f (k) and δ(k), and λ(k) we can determine Z, R,N, and σ2 for scattering paths to neighboring atoms by fitting the data.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 6 / 23
The EXAFS equation
The EXAFS oscillations can be modelled and interpreted using aconceptually simple equation (the details are more subtle!):
χ(k) =∑j
NjS20 fj(k)e−2Rj/λ(k) e−2k2σ2
j
kRj2
sin [2kRj + δj(k)]
where the sum could be over shells of atoms (Fe-O, Fe-Fe) or . . .. . . over scattering paths for the photo-electron.
Nj : path degeneracy
Rj : half path length
σ2j : path “disorder”
S20 : amplitude reduction factor
k is the photoelectron wave number
fj(k): scattering factor for the path
δj(k): phase shift for the path
λ(k): photoelectron mean free path
Because we can compute f (k) and δ(k), and λ(k) we can determine Z, R,N, and σ2 for scattering paths to neighboring atoms by fitting the data.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 6 / 23
The EXAFS equation
The EXAFS oscillations can be modelled and interpreted using aconceptually simple equation (the details are more subtle!):
χ(k) =∑j
NjS20 fj(k)e−2Rj/λ(k) e−2k2σ2
j
kRj2
sin [2kRj + δj(k)]
where the sum could be over shells of atoms (Fe-O, Fe-Fe) or . . .. . . over scattering paths for the photo-electron.
Nj : path degeneracy
Rj : half path length
σ2j : path “disorder”
S20 : amplitude reduction factor
k is the photoelectron wave number
fj(k): scattering factor for the path
δj(k): phase shift for the path
λ(k): photoelectron mean free path
Because we can compute f (k) and δ(k), and λ(k) we can determine Z, R,N, and σ2 for scattering paths to neighboring atoms by fitting the data.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 6 / 23
The EXAFS equation
The EXAFS oscillations can be modelled and interpreted using aconceptually simple equation (the details are more subtle!):
χ(k) =∑j
NjS20 fj(k)e−2Rj/λ(k) e−2k2σ2
j
kRj2
sin [2kRj + δj(k)]
where the sum could be over shells of atoms (Fe-O, Fe-Fe) or . . .. . . over scattering paths for the photo-electron.
Nj : path degeneracy
Rj : half path length
σ2j : path “disorder”
S20 : amplitude reduction factor
k is the photoelectron wave number
fj(k): scattering factor for the path
δj(k): phase shift for the path
λ(k): photoelectron mean free path
Because we can compute f (k) and δ(k), and λ(k) we can determine Z, R,N, and σ2 for scattering paths to neighboring atoms by fitting the data.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 6 / 23
The EXAFS equation
The EXAFS oscillations can be modelled and interpreted using aconceptually simple equation (the details are more subtle!):
χ(k) =∑j
NjS20 fj(k)e−2Rj/λ(k) e−2k2σ2
j
kRj2
sin [2kRj + δj(k)]
where the sum could be over shells of atoms (Fe-O, Fe-Fe) or . . .. . . over scattering paths for the photo-electron.
Nj : path degeneracy
Rj : half path length
σ2j : path “disorder”
S20 : amplitude reduction factor
k is the photoelectron wave number
fj(k): scattering factor for the path
δj(k): phase shift for the path
λ(k): photoelectron mean free path
Because we can compute f (k) and δ(k), and λ(k) we can determine Z, R,N, and σ2 for scattering paths to neighboring atoms by fitting the data.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 6 / 23
XANES edge shifts and pre-edge peaks
5460 5470 5480 5490 5500
E(eV)
0
0.2
0.4
0.6
0.8
1
1.2
ln(I
o/I)
V metal
V2O
3
V2O
5
LiVOPO4
The shift of the edge positioncan be used to determine thevalence state.
The heights and positions ofpre-edge peaks can also be re-liably used to determine ionicratios for many atomic species.
XANES can be used as a fin-gerprint of phases and XANESanalysis can be as simple asmaking linear combinations of“known” spectra to get com-position.
Modern codes, such as FEFF9,are able to accurately computeXANES features.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 7 / 23
XANES edge shifts and pre-edge peaks
5460 5470 5480 5490 5500
E(eV)
0
0.2
0.4
0.6
0.8
1
1.2
ln(I
o/I)
V metal
V2O
3
V2O
5
LiVOPO4
The shift of the edge positioncan be used to determine thevalence state.
The heights and positions ofpre-edge peaks can also be re-liably used to determine ionicratios for many atomic species.
XANES can be used as a fin-gerprint of phases and XANESanalysis can be as simple asmaking linear combinations of“known” spectra to get com-position.
Modern codes, such as FEFF9,are able to accurately computeXANES features.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 7 / 23
XANES edge shifts and pre-edge peaks
5460 5470 5480 5490 5500
E(eV)
0
0.2
0.4
0.6
0.8
1
1.2
ln(I
o/I)
V metal
V2O
3
V2O
5
LiVOPO4
The shift of the edge positioncan be used to determine thevalence state.
The heights and positions ofpre-edge peaks can also be re-liably used to determine ionicratios for many atomic species.
XANES can be used as a fin-gerprint of phases and XANESanalysis can be as simple asmaking linear combinations of“known” spectra to get com-position.
Modern codes, such as FEFF9,are able to accurately computeXANES features.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 7 / 23
XANES edge shifts and pre-edge peaks
5460 5470 5480 5490 5500
E(eV)
0
0.2
0.4
0.6
0.8
1
1.2
ln(I
o/I)
V metal
V2O
3
V2O
5
LiVOPO4
The shift of the edge positioncan be used to determine thevalence state.
The heights and positions ofpre-edge peaks can also be re-liably used to determine ionicratios for many atomic species.
XANES can be used as a fin-gerprint of phases and XANESanalysis can be as simple asmaking linear combinations of“known” spectra to get com-position.
Modern codes, such as FEFF9,are able to accurately computeXANES features.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 7 / 23
XANES edge shifts and pre-edge peaks
5460 5470 5480 5490 5500
E(eV)
0
0.2
0.4
0.6
0.8
1
1.2
ln(I
o/I)
V metal
V2O
3
V2O
5
LiVOPO4
The shift of the edge positioncan be used to determine thevalence state.
The heights and positions ofpre-edge peaks can also be re-liably used to determine ionicratios for many atomic species.
XANES can be used as a fin-gerprint of phases and XANESanalysis can be as simple asmaking linear combinations of“known” spectra to get com-position.
Modern codes, such as FEFF9,are able to accurately computeXANES features.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 7 / 23
Coordination chemistry
Cr6+
Cr3+
E (eV)
µ
(
E
)
60506040603060206010600059905980
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
The XANES of Cr3+ and Cr6+ shows a dramatic dependence on oxidationstate and coordination chemistry.
For ions with partially filled d shells, the p-d hybridization changesdramatically as regular octahedra distort, and is very large for tetrahedralcoordination.
This gives a dramatic pre-edge peak – absorption to a localized electronicstate.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 8 / 23
Coordination chemistry
Cr6+
Cr3+
E (eV)
µ
(
E
)
60506040603060206010600059905980
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
The XANES of Cr3+ and Cr6+ shows a dramatic dependence on oxidationstate and coordination chemistry.
For ions with partially filled d shells, the p-d hybridization changesdramatically as regular octahedra distort, and is very large for tetrahedralcoordination.
This gives a dramatic pre-edge peak – absorption to a localized electronicstate.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 8 / 23
Diffraction from a Truncated Surface
For an infinite sample, the diffractionspots are infinitesimally sharp.
With finite sample size, these spotsgrow in extent and become more dif-fuse.
If the sample is cleaved and left withflat surface, the diffraction will spreadinto rods perpendicular to the surface.
The scattering intensity can be ob-tained by treating the charge distri-bution as a convolution of an infinitesample with a step function in the z-direction.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 9 / 23
Diffraction from a Truncated Surface
For an infinite sample, the diffractionspots are infinitesimally sharp.
With finite sample size, these spotsgrow in extent and become more dif-fuse.
If the sample is cleaved and left withflat surface, the diffraction will spreadinto rods perpendicular to the surface.
The scattering intensity can be ob-tained by treating the charge distri-bution as a convolution of an infinitesample with a step function in the z-direction.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 9 / 23
Diffraction from a Truncated Surface
For an infinite sample, the diffractionspots are infinitesimally sharp.
With finite sample size, these spotsgrow in extent and become more dif-fuse.
If the sample is cleaved and left withflat surface, the diffraction will spreadinto rods perpendicular to the surface.
The scattering intensity can be ob-tained by treating the charge distri-bution as a convolution of an infinitesample with a step function in the z-direction.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 9 / 23
Diffraction from a Truncated Surface
For an infinite sample, the diffractionspots are infinitesimally sharp.
With finite sample size, these spotsgrow in extent and become more dif-fuse.
If the sample is cleaved and left withflat surface, the diffraction will spreadinto rods perpendicular to the surface.
The scattering intensity can be ob-tained by treating the charge distri-bution as a convolution of an infinitesample with a step function in the z-direction.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 9 / 23
CTR Scattering Factor
The scattering amplitude FCTR along a crystal truncation rod is given bysumming an infinite stack of atomic layers, each with scattering amplitudeA(~Q).
FCTR = A(~Q)∞∑j=0
e iQza3j
=A(~Q)
1− e iQza3=
A(~Q)
1− e i2πl
this sum has been discussed previ-ously and gives
or, in terms of the momentumtransfer along the z-axis,Qz = 2πl/a3
since the intensity is the square of the scattering factor
ICTR =∣∣∣FCTR
∣∣∣2 =
∣∣∣A(~Q)∣∣∣2
(1− e i2πl) (1− e−i2πl)=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl)
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 10 / 23
CTR Scattering Factor
The scattering amplitude FCTR along a crystal truncation rod is given bysumming an infinite stack of atomic layers, each with scattering amplitudeA(~Q).
FCTR = A(~Q)∞∑j=0
e iQza3j
=A(~Q)
1− e iQza3=
A(~Q)
1− e i2πl
this sum has been discussed previ-ously and gives
or, in terms of the momentumtransfer along the z-axis,Qz = 2πl/a3
since the intensity is the square of the scattering factor
ICTR =∣∣∣FCTR
∣∣∣2 =
∣∣∣A(~Q)∣∣∣2
(1− e i2πl) (1− e−i2πl)=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl)
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 10 / 23
CTR Scattering Factor
The scattering amplitude FCTR along a crystal truncation rod is given bysumming an infinite stack of atomic layers, each with scattering amplitudeA(~Q).
FCTR = A(~Q)∞∑j=0
e iQza3j
=A(~Q)
1− e iQza3=
A(~Q)
1− e i2πl
this sum has been discussed previ-ously and gives
or, in terms of the momentumtransfer along the z-axis,Qz = 2πl/a3
since the intensity is the square of the scattering factor
ICTR =∣∣∣FCTR
∣∣∣2 =
∣∣∣A(~Q)∣∣∣2
(1− e i2πl) (1− e−i2πl)=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl)
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 10 / 23
CTR Scattering Factor
The scattering amplitude FCTR along a crystal truncation rod is given bysumming an infinite stack of atomic layers, each with scattering amplitudeA(~Q).
FCTR = A(~Q)∞∑j=0
e iQza3j
=A(~Q)
1− e iQza3
=A(~Q)
1− e i2πl
this sum has been discussed previ-ously and gives
or, in terms of the momentumtransfer along the z-axis,Qz = 2πl/a3
since the intensity is the square of the scattering factor
ICTR =∣∣∣FCTR
∣∣∣2 =
∣∣∣A(~Q)∣∣∣2
(1− e i2πl) (1− e−i2πl)=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl)
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 10 / 23
CTR Scattering Factor
The scattering amplitude FCTR along a crystal truncation rod is given bysumming an infinite stack of atomic layers, each with scattering amplitudeA(~Q).
FCTR = A(~Q)∞∑j=0
e iQza3j
=A(~Q)
1− e iQza3=
A(~Q)
1− e i2πl
this sum has been discussed previ-ously and gives
or, in terms of the momentumtransfer along the z-axis,Qz = 2πl/a3
since the intensity is the square of the scattering factor
ICTR =∣∣∣FCTR
∣∣∣2 =
∣∣∣A(~Q)∣∣∣2
(1− e i2πl) (1− e−i2πl)=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl)
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 10 / 23
CTR Scattering Factor
The scattering amplitude FCTR along a crystal truncation rod is given bysumming an infinite stack of atomic layers, each with scattering amplitudeA(~Q).
FCTR = A(~Q)∞∑j=0
e iQza3j
=A(~Q)
1− e iQza3=
A(~Q)
1− e i2πl
this sum has been discussed previ-ously and gives
or, in terms of the momentumtransfer along the z-axis,Qz = 2πl/a3
since the intensity is the square of the scattering factor
ICTR =∣∣∣FCTR
∣∣∣2 =
∣∣∣A(~Q)∣∣∣2
(1− e i2πl) (1− e−i2πl)=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl)
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 10 / 23
CTR Scattering Factor
The scattering amplitude FCTR along a crystal truncation rod is given bysumming an infinite stack of atomic layers, each with scattering amplitudeA(~Q).
FCTR = A(~Q)∞∑j=0
e iQza3j
=A(~Q)
1− e iQza3=
A(~Q)
1− e i2πl
this sum has been discussed previ-ously and gives
or, in terms of the momentumtransfer along the z-axis,Qz = 2πl/a3
since the intensity is the square of the scattering factor
ICTR =∣∣∣FCTR
∣∣∣2 =
∣∣∣A(~Q)∣∣∣2
(1− e i2πl) (1− e−i2πl)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl)
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 10 / 23
CTR Scattering Factor
The scattering amplitude FCTR along a crystal truncation rod is given bysumming an infinite stack of atomic layers, each with scattering amplitudeA(~Q).
FCTR = A(~Q)∞∑j=0
e iQza3j
=A(~Q)
1− e iQza3=
A(~Q)
1− e i2πl
this sum has been discussed previ-ously and gives
or, in terms of the momentumtransfer along the z-axis,Qz = 2πl/a3
since the intensity is the square of the scattering factor
ICTR =∣∣∣FCTR
∣∣∣2 =
∣∣∣A(~Q)∣∣∣2
(1− e i2πl) (1− e−i2πl)=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl)
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 10 / 23
Dependence on Q
When l is an integer (meeting the Laue condition), the scattering factor isinfinite but just off this value, the scattering factor can be computed byletting Qz = qz + 2π/a3, with qz small.
ICTR =
∣∣∣A(~Q)∣∣∣2
4 sin2 (Qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl + qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (qza3/2)
≈
∣∣∣A(~Q)∣∣∣2
4(qza3/2)2=
∣∣∣A(~Q)∣∣∣2
q2za
23
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 11 / 23
Dependence on Q
When l is an integer (meeting the Laue condition), the scattering factor isinfinite but just off this value, the scattering factor can be computed byletting Qz = qz + 2π/a3, with qz small.
ICTR =
∣∣∣A(~Q)∣∣∣2
4 sin2 (Qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl + qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (qza3/2)
≈
∣∣∣A(~Q)∣∣∣2
4(qza3/2)2=
∣∣∣A(~Q)∣∣∣2
q2za
23
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 11 / 23
Dependence on Q
When l is an integer (meeting the Laue condition), the scattering factor isinfinite but just off this value, the scattering factor can be computed byletting Qz = qz + 2π/a3, with qz small.
ICTR =
∣∣∣A(~Q)∣∣∣2
4 sin2 (Qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl + qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (qza3/2)
≈
∣∣∣A(~Q)∣∣∣2
4(qza3/2)2=
∣∣∣A(~Q)∣∣∣2
q2za
23
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 11 / 23
Dependence on Q
When l is an integer (meeting the Laue condition), the scattering factor isinfinite but just off this value, the scattering factor can be computed byletting Qz = qz + 2π/a3, with qz small.
ICTR =
∣∣∣A(~Q)∣∣∣2
4 sin2 (Qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl + qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (qza3/2)
≈
∣∣∣A(~Q)∣∣∣2
4(qza3/2)2=
∣∣∣A(~Q)∣∣∣2
q2za
23
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 11 / 23
Dependence on Q
When l is an integer (meeting the Laue condition), the scattering factor isinfinite but just off this value, the scattering factor can be computed byletting Qz = qz + 2π/a3, with qz small.
ICTR =
∣∣∣A(~Q)∣∣∣2
4 sin2 (Qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl + qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (qza3/2)
≈
∣∣∣A(~Q)∣∣∣2
4(qza3/2)2
=
∣∣∣A(~Q)∣∣∣2
q2za
23
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 11 / 23
Dependence on Q
When l is an integer (meeting the Laue condition), the scattering factor isinfinite but just off this value, the scattering factor can be computed byletting Qz = qz + 2π/a3, with qz small.
ICTR =
∣∣∣A(~Q)∣∣∣2
4 sin2 (Qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl + qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (qza3/2)
≈
∣∣∣A(~Q)∣∣∣2
4(qza3/2)2=
∣∣∣A(~Q)∣∣∣2
q2za
23
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 11 / 23
Dependence on Q
When l is an integer (meeting the Laue condition), the scattering factor isinfinite but just off this value, the scattering factor can be computed byletting Qz = qz + 2π/a3, with qz small.
ICTR =
∣∣∣A(~Q)∣∣∣2
4 sin2 (Qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl + qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (qza3/2)
≈
∣∣∣A(~Q)∣∣∣2
4(qza3/2)2=
∣∣∣A(~Q)∣∣∣2
q2za
23
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 11 / 23
Absorption Effect
Absorption effects can be in-cluded as well
FCTR = A(~Q)∞∑j=0
e iQza3je−βj
=A(~Q)
1− e iQza3e−βj
This removes the infinity andincreases the scattering pro-file of the crystal truncationrod
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 12 / 23
Absorption Effect
Absorption effects can be in-cluded as well
FCTR = A(~Q)∞∑j=0
e iQza3j
e−βj
=A(~Q)
1− e iQza3e−βj
This removes the infinity andincreases the scattering pro-file of the crystal truncationrod
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 12 / 23
Absorption Effect
Absorption effects can be in-cluded as well
FCTR = A(~Q)∞∑j=0
e iQza3je−βj
=A(~Q)
1− e iQza3e−βj
This removes the infinity andincreases the scattering pro-file of the crystal truncationrod
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 12 / 23
Absorption Effect
Absorption effects can be in-cluded as well
FCTR = A(~Q)∞∑j=0
e iQza3je−βj
=A(~Q)
1− e iQza3e−βj
This removes the infinity andincreases the scattering pro-file of the crystal truncationrod
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 12 / 23
Absorption Effect
Absorption effects can be in-cluded as well
FCTR = A(~Q)∞∑j=0
e iQza3je−βj
=A(~Q)
1− e iQza3e−βj
This removes the infinity andincreases the scattering pro-file of the crystal truncationrod
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 12 / 23
Density Effect
The CTR profile is sensitive to the termination of the surface. This makesit an ideal probe of electron density of adsorbed species or single atomoverlayers.
F total = FCTR + F top layer
=A(~Q)
1− e i2πl
+ A(~Q)e−i2π(1+z0)l
where z0 is the relative dis-placement of the top layerfrom the bulk lattice spacinga3
This effect gets larger forlarger momentum transfers
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 13 / 23
Density Effect
The CTR profile is sensitive to the termination of the surface. This makesit an ideal probe of electron density of adsorbed species or single atomoverlayers.
F total = FCTR + F top layer
=A(~Q)
1− e i2πl
+ A(~Q)e−i2π(1+z0)l
where z0 is the relative dis-placement of the top layerfrom the bulk lattice spacinga3
This effect gets larger forlarger momentum transfers
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 13 / 23
Density Effect
The CTR profile is sensitive to the termination of the surface. This makesit an ideal probe of electron density of adsorbed species or single atomoverlayers.
F total = FCTR + F top layer
=A(~Q)
1− e i2πl
+ A(~Q)e−i2π(1+z0)l
where z0 is the relative dis-placement of the top layerfrom the bulk lattice spacinga3
This effect gets larger forlarger momentum transfers
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 13 / 23
Density Effect
The CTR profile is sensitive to the termination of the surface. This makesit an ideal probe of electron density of adsorbed species or single atomoverlayers.
F total = FCTR + F top layer
=A(~Q)
1− e i2πl
+ A(~Q)e−i2π(1+z0)l
where z0 is the relative dis-placement of the top layerfrom the bulk lattice spacinga3
This effect gets larger forlarger momentum transfers
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 13 / 23
Density Effect
The CTR profile is sensitive to the termination of the surface. This makesit an ideal probe of electron density of adsorbed species or single atomoverlayers.
F total = FCTR + F top layer
=A(~Q)
1− e i2πl
+ A(~Q)e−i2π(1+z0)l
where z0 is the relative dis-placement of the top layerfrom the bulk lattice spacinga3
This effect gets larger forlarger momentum transfers
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 13 / 23
Density Effect
The CTR profile is sensitive to the termination of the surface. This makesit an ideal probe of electron density of adsorbed species or single atomoverlayers.
F total = FCTR + F top layer
=A(~Q)
1− e i2πl
+ A(~Q)e−i2π(1+z0)l
where z0 is the relative dis-placement of the top layerfrom the bulk lattice spacinga3
This effect gets larger forlarger momentum transfers
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 13 / 23
Lattice Vibrations
Atoms on a lattice are not rigid but vibrate. There is zero-point motion aswell as thermal motion. These vibrations influence the x-ray scattering.
For a 1D lattice, we replace the position of the atom with itsinstantaneous position, ~Rn + ~un where ~un is the displacement from theequilibrium position, ~Rn. Computing the intensity:
I =
⟨∑m
f (~Q)e i~Q·(~Rm+~um)
∑n
f ∗(~Q)e−i~Q·(~Rn+~un)
⟩=∑m
∑n
f (~Q)f ∗(~Q)e i~Q·(~Rm−~Rn)
⟨e i~Q·(~um−~un)
⟩The last term is a time average which can be simplified using theBaker-Hausdorff theorem,
⟨e ix⟩
= e−〈x2〉/2⟨
e i~Q·(~um−~un)
⟩=⟨e iQ(uQm−uQn)
⟩= e−〈Q
2(uQm−uQn)2〉/2
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 14 / 23
Lattice Vibrations
Atoms on a lattice are not rigid but vibrate. There is zero-point motion aswell as thermal motion. These vibrations influence the x-ray scattering.
For a 1D lattice, we replace the position of the atom with itsinstantaneous position, ~Rn + ~un where ~un is the displacement from theequilibrium position, ~Rn.
Computing the intensity:
I =
⟨∑m
f (~Q)e i~Q·(~Rm+~um)
∑n
f ∗(~Q)e−i~Q·(~Rn+~un)
⟩=∑m
∑n
f (~Q)f ∗(~Q)e i~Q·(~Rm−~Rn)
⟨e i~Q·(~um−~un)
⟩The last term is a time average which can be simplified using theBaker-Hausdorff theorem,
⟨e ix⟩
= e−〈x2〉/2⟨
e i~Q·(~um−~un)
⟩=⟨e iQ(uQm−uQn)
⟩= e−〈Q
2(uQm−uQn)2〉/2
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 14 / 23
Lattice Vibrations
Atoms on a lattice are not rigid but vibrate. There is zero-point motion aswell as thermal motion. These vibrations influence the x-ray scattering.
For a 1D lattice, we replace the position of the atom with itsinstantaneous position, ~Rn + ~un where ~un is the displacement from theequilibrium position, ~Rn. Computing the intensity:
I =
⟨∑m
f (~Q)e i~Q·(~Rm+~um)
∑n
f ∗(~Q)e−i~Q·(~Rn+~un)
⟩=∑m
∑n
f (~Q)f ∗(~Q)e i~Q·(~Rm−~Rn)
⟨e i~Q·(~um−~un)
⟩The last term is a time average which can be simplified using theBaker-Hausdorff theorem,
⟨e ix⟩
= e−〈x2〉/2⟨
e i~Q·(~um−~un)
⟩=⟨e iQ(uQm−uQn)
⟩= e−〈Q
2(uQm−uQn)2〉/2
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 14 / 23
Lattice Vibrations
Atoms on a lattice are not rigid but vibrate. There is zero-point motion aswell as thermal motion. These vibrations influence the x-ray scattering.
For a 1D lattice, we replace the position of the atom with itsinstantaneous position, ~Rn + ~un where ~un is the displacement from theequilibrium position, ~Rn. Computing the intensity:
I =
⟨∑m
f (~Q)e i~Q·(~Rm+~um)
∑n
f ∗(~Q)e−i~Q·(~Rn+~un)
⟩
=∑m
∑n
f (~Q)f ∗(~Q)e i~Q·(~Rm−~Rn)
⟨e i~Q·(~um−~un)
⟩The last term is a time average which can be simplified using theBaker-Hausdorff theorem,
⟨e ix⟩
= e−〈x2〉/2⟨
e i~Q·(~um−~un)
⟩=⟨e iQ(uQm−uQn)
⟩= e−〈Q
2(uQm−uQn)2〉/2
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 14 / 23
Lattice Vibrations
Atoms on a lattice are not rigid but vibrate. There is zero-point motion aswell as thermal motion. These vibrations influence the x-ray scattering.
For a 1D lattice, we replace the position of the atom with itsinstantaneous position, ~Rn + ~un where ~un is the displacement from theequilibrium position, ~Rn. Computing the intensity:
I =
⟨∑m
f (~Q)e i~Q·(~Rm+~um)
∑n
f ∗(~Q)e−i~Q·(~Rn+~un)
⟩=∑m
∑n
f (~Q)f ∗(~Q)e i~Q·(~Rm−~Rn)
⟨e i~Q·(~um−~un)
⟩
The last term is a time average which can be simplified using theBaker-Hausdorff theorem,
⟨e ix⟩
= e−〈x2〉/2⟨
e i~Q·(~um−~un)
⟩=⟨e iQ(uQm−uQn)
⟩= e−〈Q
2(uQm−uQn)2〉/2
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 14 / 23
Lattice Vibrations
Atoms on a lattice are not rigid but vibrate. There is zero-point motion aswell as thermal motion. These vibrations influence the x-ray scattering.
For a 1D lattice, we replace the position of the atom with itsinstantaneous position, ~Rn + ~un where ~un is the displacement from theequilibrium position, ~Rn. Computing the intensity:
I =
⟨∑m
f (~Q)e i~Q·(~Rm+~um)
∑n
f ∗(~Q)e−i~Q·(~Rn+~un)
⟩=∑m
∑n
f (~Q)f ∗(~Q)e i~Q·(~Rm−~Rn)
⟨e i~Q·(~um−~un)
⟩The last term is a time average which can be simplified using theBaker-Hausdorff theorem,
⟨e ix⟩
= e−〈x2〉/2
⟨e i~Q·(~um−~un)
⟩=⟨e iQ(uQm−uQn)
⟩= e−〈Q
2(uQm−uQn)2〉/2
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 14 / 23
Lattice Vibrations
Atoms on a lattice are not rigid but vibrate. There is zero-point motion aswell as thermal motion. These vibrations influence the x-ray scattering.
For a 1D lattice, we replace the position of the atom with itsinstantaneous position, ~Rn + ~un where ~un is the displacement from theequilibrium position, ~Rn. Computing the intensity:
I =
⟨∑m
f (~Q)e i~Q·(~Rm+~um)
∑n
f ∗(~Q)e−i~Q·(~Rn+~un)
⟩=∑m
∑n
f (~Q)f ∗(~Q)e i~Q·(~Rm−~Rn)
⟨e i~Q·(~um−~un)
⟩The last term is a time average which can be simplified using theBaker-Hausdorff theorem,
⟨e ix⟩
= e−〈x2〉/2⟨
e i~Q·(~um−~un)
⟩=⟨e iQ(uQm−uQn)
⟩
= e−〈Q2(uQm−uQn)2〉/2
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 14 / 23
Lattice Vibrations
Atoms on a lattice are not rigid but vibrate. There is zero-point motion aswell as thermal motion. These vibrations influence the x-ray scattering.
For a 1D lattice, we replace the position of the atom with itsinstantaneous position, ~Rn + ~un where ~un is the displacement from theequilibrium position, ~Rn. Computing the intensity:
I =
⟨∑m
f (~Q)e i~Q·(~Rm+~um)
∑n
f ∗(~Q)e−i~Q·(~Rn+~un)
⟩=∑m
∑n
f (~Q)f ∗(~Q)e i~Q·(~Rm−~Rn)
⟨e i~Q·(~um−~un)
⟩The last term is a time average which can be simplified using theBaker-Hausdorff theorem,
⟨e ix⟩
= e−〈x2〉/2⟨
e i~Q·(~um−~un)
⟩=⟨e iQ(uQm−uQn)
⟩= e−〈Q
2(uQm−uQn)2〉/2
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 14 / 23
Lattice Vibrations
⟨e iQ(uQm−uQn)
⟩= e−Q
2〈u2Qm〉/2e−Q
2〈u2Qn〉/2eQ
2〈uQmuQn〉
= e−Q2〈u2
Q〉eQ2〈uQmuQn〉 = e−MeQ
2〈uQmuQn〉
= e−M[1 + eQ
2〈uQmuQn〉 − 1]
Substituting into the expression for intensity
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.
The second term is the Thermal Diffuse Scattering and actually increaseswith mean squared displacement.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 15 / 23
Lattice Vibrations
⟨e iQ(uQm−uQn)
⟩= e−Q
2〈u2Qm〉/2e−Q
2〈u2Qn〉/2eQ
2〈uQmuQn〉
= e−Q2〈u2
Q〉eQ2〈uQmuQn〉
= e−MeQ2〈uQmuQn〉
= e−M[1 + eQ
2〈uQmuQn〉 − 1]
Substituting into the expression for intensity
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.
The second term is the Thermal Diffuse Scattering and actually increaseswith mean squared displacement.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 15 / 23
Lattice Vibrations
⟨e iQ(uQm−uQn)
⟩= e−Q
2〈u2Qm〉/2e−Q
2〈u2Qn〉/2eQ
2〈uQmuQn〉
= e−Q2〈u2
Q〉eQ2〈uQmuQn〉 = e−MeQ
2〈uQmuQn〉
= e−M[1 + eQ
2〈uQmuQn〉 − 1]
Substituting into the expression for intensity
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.
The second term is the Thermal Diffuse Scattering and actually increaseswith mean squared displacement.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 15 / 23
Lattice Vibrations
⟨e iQ(uQm−uQn)
⟩= e−Q
2〈u2Qm〉/2e−Q
2〈u2Qn〉/2eQ
2〈uQmuQn〉
= e−Q2〈u2
Q〉eQ2〈uQmuQn〉 = e−MeQ
2〈uQmuQn〉
= e−M[1 + eQ
2〈uQmuQn〉 − 1]
Substituting into the expression for intensity
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.
The second term is the Thermal Diffuse Scattering and actually increaseswith mean squared displacement.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 15 / 23
Lattice Vibrations
⟨e iQ(uQm−uQn)
⟩= e−Q
2〈u2Qm〉/2e−Q
2〈u2Qn〉/2eQ
2〈uQmuQn〉
= e−Q2〈u2
Q〉eQ2〈uQmuQn〉 = e−MeQ
2〈uQmuQn〉
= e−M[1 + eQ
2〈uQmuQn〉 − 1]
Substituting into the expression for intensity
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.
The second term is the Thermal Diffuse Scattering and actually increaseswith mean squared displacement.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 15 / 23
Lattice Vibrations
⟨e iQ(uQm−uQn)
⟩= e−Q
2〈u2Qm〉/2e−Q
2〈u2Qn〉/2eQ
2〈uQmuQn〉
= e−Q2〈u2
Q〉eQ2〈uQmuQn〉 = e−MeQ
2〈uQmuQn〉
= e−M[1 + eQ
2〈uQmuQn〉 − 1]
Substituting into the expression for intensity
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.
The second term is the Thermal Diffuse Scattering and actually increaseswith mean squared displacement.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 15 / 23
Lattice Vibrations
⟨e iQ(uQm−uQn)
⟩= e−Q
2〈u2Qm〉/2e−Q
2〈u2Qn〉/2eQ
2〈uQmuQn〉
= e−Q2〈u2
Q〉eQ2〈uQmuQn〉 = e−MeQ
2〈uQmuQn〉
= e−M[1 + eQ
2〈uQmuQn〉 − 1]
Substituting into the expression for intensity
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.
The second term is the Thermal Diffuse Scattering and actually increaseswith mean squared displacement.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 15 / 23
Lattice Vibrations
⟨e iQ(uQm−uQn)
⟩= e−Q
2〈u2Qm〉/2e−Q
2〈u2Qn〉/2eQ
2〈uQmuQn〉
= e−Q2〈u2
Q〉eQ2〈uQmuQn〉 = e−MeQ
2〈uQmuQn〉
= e−M[1 + eQ
2〈uQmuQn〉 − 1]
Substituting into the expression for intensity
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.
The second term is the Thermal Diffuse Scattering and actually increaseswith mean squared displacement.
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 15 / 23
Thermal Diffuse Scattering
ITDS =∑m
∑n
f (~Q)e−Me i~Q·~Rm f ∗(~Q)e−Me−i
~Q·~Rn
[eQ
2〈uQmuQn〉 − 1]
The TDS has a width deter-mined by the correlated dis-placement of atoms which ismuch broader than a Braggpeak.
These correlated motions arejust phonons.
A 0.5mm Si wafer illumi-nated by 28keV x-rays froman APS undulator were usedto measure the phonon dis-persion curves of silicon
M. Holt, et al. Phys. Rev. Lett. 83, 3317 (1999).
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 16 / 23
Thermal Diffuse Scattering
ITDS =∑m
∑n
f (~Q)e−Me i~Q·~Rm f ∗(~Q)e−Me−i
~Q·~Rn
[eQ
2〈uQmuQn〉 − 1]
The TDS has a width deter-mined by the correlated dis-placement of atoms which ismuch broader than a Braggpeak.
These correlated motions arejust phonons.
A 0.5mm Si wafer illumi-nated by 28keV x-rays froman APS undulator were usedto measure the phonon dis-persion curves of silicon
M. Holt, et al. Phys. Rev. Lett. 83, 3317 (1999).
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 16 / 23
Thermal Diffuse Scattering
ITDS =∑m
∑n
f (~Q)e−Me i~Q·~Rm f ∗(~Q)e−Me−i
~Q·~Rn
[eQ
2〈uQmuQn〉 − 1]
The TDS has a width deter-mined by the correlated dis-placement of atoms which ismuch broader than a Braggpeak.
These correlated motions arejust phonons.
A 0.5mm Si wafer illumi-nated by 28keV x-rays froman APS undulator were usedto measure the phonon dis-persion curves of silicon
M. Holt, et al. Phys. Rev. Lett. 83, 3317 (1999).
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 16 / 23
Thermal Diffuse Scattering
ITDS =∑m
∑n
f (~Q)e−Me i~Q·~Rm f ∗(~Q)e−Me−i
~Q·~Rn
[eQ
2〈uQmuQn〉 − 1]
The TDS has a width deter-mined by the correlated dis-placement of atoms which ismuch broader than a Braggpeak.
These correlated motions arejust phonons.
A 0.5mm Si wafer illumi-nated by 28keV x-rays froman APS undulator were usedto measure the phonon dis-persion curves of silicon
M. Holt, et al. Phys. Rev. Lett. 83, 3317 (1999).
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 16 / 23
Thermal Diffuse Scattering
ITDS =∑m
∑n
f (~Q)e−Me i~Q·~Rm f ∗(~Q)e−Me−i
~Q·~Rn
[eQ
2〈uQmuQn〉 − 1]
The TDS has a width deter-mined by the correlated dis-placement of atoms which ismuch broader than a Braggpeak.
These correlated motions arejust phonons.
A 0.5mm Si wafer illumi-nated by 28keV x-rays froman APS undulator were usedto measure the phonon dis-persion curves of silicon
M. Holt, et al. Phys. Rev. Lett. 83, 3317 (1999).
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 16 / 23
Thermal Diffuse Scattering
ITDS =∑m
∑n
f (~Q)e−Me i~Q·~Rm f ∗(~Q)e−Me−i
~Q·~Rn
[eQ
2〈uQmuQn〉 − 1]
The TDS has a width deter-mined by the correlated dis-placement of atoms which ismuch broader than a Braggpeak.
These correlated motions arejust phonons.
A 0.5mm Si wafer illumi-nated by 28keV x-rays froman APS undulator were usedto measure the phonon dis-persion curves of silicon
M. Holt, et al. Phys. Rev. Lett. 83, 3317 (1999).
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 16 / 23
Thermal Diffuse Scattering
ITDS =∑m
∑n
f (~Q)e−Me i~Q·~Rm f ∗(~Q)e−Me−i
~Q·~Rn
[eQ
2〈uQmuQn〉 − 1]
The TDS has a width deter-mined by the correlated dis-placement of atoms which ismuch broader than a Braggpeak.
These correlated motions arejust phonons.
A 0.5mm Si wafer illumi-nated by 28keV x-rays froman APS undulator were usedto measure the phonon dis-persion curves of silicon
M. Holt, et al. Phys. Rev. Lett. 83, 3317 (1999).
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 16 / 23
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 October 28, 2013 17 / 23
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 October 28, 2013 17 / 23
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 October 28, 2013 17 / 23
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 October 28, 2013 17 / 23
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 October 28, 2013 17 / 23
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 October 28, 2013 17 / 23
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 October 28, 2013 17 / 23
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 October 28, 2013 17 / 23
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 October 28, 2013 18 / 23
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 October 28, 2013 18 / 23
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 October 28, 2013 18 / 23
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 October 28, 2013 18 / 23
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 October 28, 2013 18 / 23
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 October 28, 2013 19 / 23
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 October 28, 2013 19 / 23
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 October 28, 2013 19 / 23
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 October 28, 2013 19 / 23
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 20 / 23
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 21 / 23
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 22 / 23
C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 23 / 23