Today’s Outline - November 07, 2016csrri.iit.edu/~segre/phys570/16F/lecture_21.pdfToday’s...
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Today’s Outline - November 07, 2016
• Lithiation of Sn-based anodes
• Photoemission
• Resonant Scattering
No class on Wednesday, November 09, 2016
Homework Assignment #06:Chapter 6: 1,6,7,8,9due Monday, November 14, 2016
Homework Assignment #7:Chapter 7: 2,3,9,10,11due Monday, November 28, 2018
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 1 / 19
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Today’s Outline - November 07, 2016
• Lithiation of Sn-based anodes
• Photoemission
• Resonant Scattering
No class on Wednesday, November 09, 2016
Homework Assignment #06:Chapter 6: 1,6,7,8,9due Monday, November 14, 2016
Homework Assignment #7:Chapter 7: 2,3,9,10,11due Monday, November 28, 2018
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 1 / 19
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Today’s Outline - November 07, 2016
• Lithiation of Sn-based anodes
• Photoemission
• Resonant Scattering
No class on Wednesday, November 09, 2016
Homework Assignment #06:Chapter 6: 1,6,7,8,9due Monday, November 14, 2016
Homework Assignment #7:Chapter 7: 2,3,9,10,11due Monday, November 28, 2018
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 1 / 19
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Today’s Outline - November 07, 2016
• Lithiation of Sn-based anodes
• Photoemission
• Resonant Scattering
No class on Wednesday, November 09, 2016
Homework Assignment #06:Chapter 6: 1,6,7,8,9due Monday, November 14, 2016
Homework Assignment #7:Chapter 7: 2,3,9,10,11due Monday, November 28, 2018
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 1 / 19
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Today’s Outline - November 07, 2016
• Lithiation of Sn-based anodes
• Photoemission
• Resonant Scattering
No class on Wednesday, November 09, 2016
Homework Assignment #06:Chapter 6: 1,6,7,8,9due Monday, November 14, 2016
Homework Assignment #7:Chapter 7: 2,3,9,10,11due Monday, November 28, 2018
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 1 / 19
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Today’s Outline - November 07, 2016
• Lithiation of Sn-based anodes
• Photoemission
• Resonant Scattering
No class on Wednesday, November 09, 2016
Homework Assignment #06:Chapter 6: 1,6,7,8,9due Monday, November 14, 2016
Homework Assignment #7:Chapter 7: 2,3,9,10,11due Monday, November 28, 2018
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 1 / 19
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Today’s Outline - November 07, 2016
• Lithiation of Sn-based anodes
• Photoemission
• Resonant Scattering
No class on Wednesday, November 09, 2016
Homework Assignment #06:Chapter 6: 1,6,7,8,9due Monday, November 14, 2016
Homework Assignment #7:Chapter 7: 2,3,9,10,11due Monday, November 28, 2018
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 1 / 19
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Synthesis of Sn-graphite nanocomposites
One-pot synthesisproduces evenly dis-tributed Sn3O2(OH)2nanoparticles on graphitenanoplatelets
XRD shows a smallamount of Sn metal inaddition to Sn3O2(OH)2
10 20 30 40 50 60 70
0
100
200
300
Inte
nsity
Scattering Angle (2 )
GnP
Sn3O2(OH)2
Sn3O2(OH)2/GnP
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 2 / 19
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In situ XAS studies of lithiation
1 2 3 4 50.0
0.3
0.6
0.9
1.2Sn-O
|x
(R)|
(Å-3
)
R (Å)
OCV 1st Charge 1st Discharge
Sn-Sn
1 2 3 4 50.0
0.1
0.2
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 3 / 19
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In situ battery box
Pouch cell clamped against front window in helium environment
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 4 / 19
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In situ battery box
Suitable for both transmission and fluorescence measurements
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 4 / 19
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In situ XAS studies of lithiation
0 5 10 15 20 25 300
100
200
300
400
500
600
700
2100
2800
In situ
C
apac
ity (m
Ah/
g)
Cycle Number
Charge Discharge
Ex situ
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 5 / 19
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In situ XAS studies of lithiation
-0.5
0.0
0.5
1.0
0 2 4 6 8 10 12
-0.5
0.0
0.5
1.0
0 1 2 3 4 50.0
0.5
1.0
k (Å-1)
k 2
(k) (Å-2
)
Re[
(R)]
(Å-3
)
|x(R
)| (Å
-3)
R (Å)
Fresh electrode can be fitwith Sn3O2(OH)2 struc-ture which is dominatedby the near neighbor Sn-O distances
Only a small amount ofmetallic Sn-Sn distancescan be seen
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 6 / 19
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In situ XAS studies of lithiation
-0.5
0.0
0.5
1.0
0 2 4 6 8 10 12
-0.5
0.0
0.5
1.0
0 1 2 3 4 50.0
0.5
1.0
k (Å-1)
k 2
(k) (Å-2
)
Re[
(R)]
(Å-3
)
|x(R
)| (Å
-3)
R (Å)
Fresh electrode can be fitwith Sn3O2(OH)2 struc-ture which is dominatedby the near neighbor Sn-O distances
Only a small amount ofmetallic Sn-Sn distancescan be seen
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 6 / 19
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In situ XAS studies of lithiation
-0.5
0.0
0.5
1.0
0 2 4 6 8 10 12
-0.5
0.0
0.5
1.0
0 1 2 3 4 50.0
0.5
1.0
k (Å-1)
k 2
(k) (Å-2
)
Re[
(R)]
(Å-3
)
|x(R
)| (Å
-3)
R (Å)
Fresh electrode can be fitwith Sn3O2(OH)2 struc-ture which is dominatedby the near neighbor Sn-O distances
Only a small amount ofmetallic Sn-Sn distancescan be seen
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 6 / 19
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In situ XAS studies of lithiation
-0.2
0.0
0.2
0 2 4 6 8 10 12
-0.1
0.0
0.1
0 1 2 3 4 50.0
0.1
0.2
k (Å-1)
k 2
(k) (Å-2
)
Re[
(R)]
(Å-3
)
|(R
)| (Å
-3)
R (Å)
Reduction of number ofSn-O near neighbors and3 Sn-Li paths characteris-tic of the Li22Sn5 struc-ture
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 7 / 19
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In situ XAS studies of lithiation
-0.2
0.0
0.2
0 2 4 6 8 10 12
-0.2
-0.1
0.0
0.1
0.2
0 1 2 3 4 50.0
0.1
0.2
k (Å-1)
k 2
(k) (Å-2
)
Re[
(R)]
(Å-3
)
|(R
)| (Å
-3)
R (Å)
Metallic Sn-Sn distancesappear but Sn-Li pathsare still present, furtherreduction in Sn-O nearneighbors.
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 8 / 19
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In situ XAS studies of lithiation
OCV1s
t Cha
rge1s
t Disc
harge
2nd C
harge
2nd D
ischa
rge3rd
Cha
rge3rd
Disc
harge
4th C
harge
0
2
4
6
8
10
12
14
Med. Li
Long Li
Short Li
C/7.5C/2.5
Num
ber o
f Nei
ghbo
rs
Total Li
Number of Linear neighborsoscillates with thecharge/dischargecycles but neverreturns to zero
In situ cell pro-motes acceleratedaging because ofSn swelling andthe reduced pres-sure of the thinPEEK pouch cellassembly
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 9 / 19
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In situ XAS studies of lithiation
OCV1s
t Cha
rge1s
t Disc
harge
2nd C
harge
2nd D
ischa
rge3rd
Cha
rge3rd
Disc
harge
4th C
harge
0
2
4
6
8
10
12
14
Med. Li
Long Li
Short Li
C/7.5C/2.5
Num
ber o
f Nei
ghbo
rs
Total Li
Number of Linear neighborsoscillates with thecharge/dischargecycles but neverreturns to zero
In situ cell pro-motes acceleratedaging because ofSn swelling andthe reduced pres-sure of the thinPEEK pouch cellassembly
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 9 / 19
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In situ XAS studies of lithiation
OCV1s
t Cha
rge1s
t Disc
harge
2nd C
harge
2nd D
ischa
rge3rd
Cha
rge3rd
Disc
harge
4th C
harge
0
2
4
6
8
10
12
14
Med. Li
Long Li
Short Li
C/7.5C/2.5
Num
ber o
f Nei
ghbo
rs
Total Li
Number of Linear neighborsoscillates with thecharge/dischargecycles but neverreturns to zero
In situ cell pro-motes acceleratedaging because ofSn swelling andthe reduced pres-sure of the thinPEEK pouch cellassembly
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 9 / 19
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In situ XAS studies of lithiation
C. Pelliccione, E.V. Timofeeva, and C.U. Segre, “In situ XAS study of the capacity fading mechanism in hybridSn3O2(OH)2/graphite battery anode nanomaterials” Chem. Mater. 27, 574-580 (2015).
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 10 / 19
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The photoemission process
Photoemission is the comple-ment to XAFS. It probes thefilled states below the Fermi level
The dispersion relation of elec-trons in a solid, E(~q) can beprobed by angle resolved photoe-mission
Ekin, θ −→ E(~q)
Ekin =~2q2v2m
= ~ω − φ− EB
EB = EF − Ei
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 11 / 19
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The photoemission process
Photoemission is the comple-ment to XAFS. It probes thefilled states below the Fermi level
The dispersion relation of elec-trons in a solid, E(~q) can beprobed by angle resolved photoe-mission
Ekin, θ −→ E(~q)
Ekin =~2q2v2m
= ~ω − φ− EB
EB = EF − Ei
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 11 / 19
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The photoemission process
Photoemission is the comple-ment to XAFS. It probes thefilled states below the Fermi level
The dispersion relation of elec-trons in a solid, E(~q) can beprobed by angle resolved photoe-mission
Ekin, θ −→ E(~q)
Ekin =~2q2v2m
= ~ω − φ− EB
EB = EF − Ei
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 11 / 19
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The photoemission process
Photoemission is the comple-ment to XAFS. It probes thefilled states below the Fermi level
The dispersion relation of elec-trons in a solid, E(~q) can beprobed by angle resolved photoe-mission
Ekin, θ −→ E(~q)
Ekin =~2q2v2m
= ~ω − φ− EB
EB = EF − Ei
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 11 / 19
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Hemispherical mirror analyzer
The electric field between thetwo hemispheres has a R2 de-pendence from the center of thehemispheres
Electrons with E0, called the“pass energy”, will follow a cir-cular path of radiusR0 = (R1 + R2)/2
Electrons with lower energy willfall inside this circular path whilethose with higher enegy will falloutside
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 12 / 19
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Hemispherical mirror analyzer
The electric field between thetwo hemispheres has a R2 de-pendence from the center of thehemispheres
Electrons with E0, called the“pass energy”, will follow a cir-cular path of radiusR0 = (R1 + R2)/2
Electrons with lower energy willfall inside this circular path whilethose with higher enegy will falloutside
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 12 / 19
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Hemispherical mirror analyzer
The electric field between thetwo hemispheres has a R2 de-pendence from the center of thehemispheres
Electrons with E0, called the“pass energy”, will follow a cir-cular path of radiusR0 = (R1 + R2)/2
Electrons with lower energy willfall inside this circular path whilethose with higher enegy will falloutside
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 12 / 19
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Hemispherical mirror analyzer
The electric field between thetwo hemispheres has a R2 de-pendence from the center of thehemispheres
Electrons with E0, called the“pass energy”, will follow a cir-cular path of radiusR0 = (R1 + R2)/2
Electrons with lower energy willfall inside this circular path whilethose with higher enegy will falloutside
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 12 / 19
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A better scattering model
Up to now, scattering has been treated classically and the result ofradiation interaction with “free” electrons.
This is not a good approximation since we know:
f (~Q, ω) = f 0(~Q) + f ′(ω) + if ′′(ω)
The absorption cross section can bemodeled as a sum of forced, dissi-pative oscillators with distributiong(ωs).
This will produce the resonant scat-tering term but not the XANES andEXAFS, which are purely quantumeffects.
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 13 / 19
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A better scattering model
Up to now, scattering has been treated classically and the result ofradiation interaction with “free” electrons.
This is not a good approximation since we know:
f (~Q, ω) = f 0(~Q) + f ′(ω) + if ′′(ω)
The absorption cross section can bemodeled as a sum of forced, dissi-pative oscillators with distributiong(ωs).
This will produce the resonant scat-tering term but not the XANES andEXAFS, which are purely quantumeffects.
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 13 / 19
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A better scattering model
Up to now, scattering has been treated classically and the result ofradiation interaction with “free” electrons.
This is not a good approximation since we know:
f (~Q, ω) = f 0(~Q) + f ′(ω) + if ′′(ω)
The absorption cross section can bemodeled as a sum of forced, dissi-pative oscillators with distributiong(ωs).
This will produce the resonant scat-tering term but not the XANES andEXAFS, which are purely quantumeffects.
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 13 / 19
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A better scattering model
Up to now, scattering has been treated classically and the result ofradiation interaction with “free” electrons.
This is not a good approximation since we know:
f (~Q, ω) = f 0(~Q) + f ′(ω) + if ′′(ω)
The absorption cross section can bemodeled as a sum of forced, dissi-pative oscillators with distributiong(ωs).
This will produce the resonant scat-tering term but not the XANES andEXAFS, which are purely quantumeffects.
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 13 / 19
![Page 34: Today’s Outline - November 07, 2016csrri.iit.edu/~segre/phys570/16F/lecture_21.pdfToday’s Outline - November 07, 2016 Lithiation of Sn-based anodes Photoemission Resonant Scattering](https://reader034.fdocuments.us/reader034/viewer/2022042312/5edac7fcceb8760df365f2cb/html5/thumbnails/34.jpg)
A better scattering model
Up to now, scattering has been treated classically and the result ofradiation interaction with “free” electrons.
This is not a good approximation since we know:
f (~Q, ω) = f 0(~Q) + f ′(ω) + if ′′(ω)
The absorption cross section can bemodeled as a sum of forced, dissi-pative oscillators with distributiong(ωs).
This will produce the resonant scat-tering term but not the XANES andEXAFS, which are purely quantumeffects.
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 13 / 19
![Page 35: Today’s Outline - November 07, 2016csrri.iit.edu/~segre/phys570/16F/lecture_21.pdfToday’s Outline - November 07, 2016 Lithiation of Sn-based anodes Photoemission Resonant Scattering](https://reader034.fdocuments.us/reader034/viewer/2022042312/5edac7fcceb8760df365f2cb/html5/thumbnails/35.jpg)
A better scattering model
Up to now, scattering has been treated classically and the result ofradiation interaction with “free” electrons.
This is not a good approximation since we know:
f (~Q, ω) = f 0(~Q) + f ′(ω) + if ′′(ω)
The absorption cross section can bemodeled as a sum of forced, dissi-pative oscillators with distributiong(ωs).
This will produce the resonant scat-tering term but not the XANES andEXAFS, which are purely quantumeffects.
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 13 / 19
![Page 36: Today’s Outline - November 07, 2016csrri.iit.edu/~segre/phys570/16F/lecture_21.pdfToday’s Outline - November 07, 2016 Lithiation of Sn-based anodes Photoemission Resonant Scattering](https://reader034.fdocuments.us/reader034/viewer/2022042312/5edac7fcceb8760df365f2cb/html5/thumbnails/36.jpg)
A better scattering model
Up to now, scattering has been treated classically and the result ofradiation interaction with “free” electrons.
This is not a good approximation since we know:
f (~Q, ω) = f 0(~Q) + f ′(ω) + if ′′(ω)
The absorption cross section can bemodeled as a sum of forced, dissi-pative oscillators with distributiong(ωs).
This will produce the resonant scat-tering term but not the XANES andEXAFS, which are purely quantumeffects.
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 13 / 19
![Page 37: Today’s Outline - November 07, 2016csrri.iit.edu/~segre/phys570/16F/lecture_21.pdfToday’s Outline - November 07, 2016 Lithiation of Sn-based anodes Photoemission Resonant Scattering](https://reader034.fdocuments.us/reader034/viewer/2022042312/5edac7fcceb8760df365f2cb/html5/thumbnails/37.jpg)
Forced charged oscillator
Consider an electron under the in-fluence of an oscillating electricfield ~Ein = xE0e
−iωt .
where Γ is the damping constant,ωs is the resonant frequency of theoscillator, and Γ� ωs .
assuming a solution of the form
x + Γx + ω2s x = −
(eE0
m
)e−iωt
x = x0e−iωt
x = −iωx0e−iωt
x = −ω2x0e−iωt
(−ω2 − iωΓ + ω2s )x0e
−iωt = −(eE0
m
)e−iωt
x0 = −(eE0
m
)1
(ω2s − ω2 − iωΓ)
The amplitude of the response has a resonance and dissipation
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 14 / 19
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Forced charged oscillator
Consider an electron under the in-fluence of an oscillating electricfield ~Ein = xE0e
−iωt .
where Γ is the damping constant,ωs is the resonant frequency of theoscillator, and Γ� ωs .
assuming a solution of the form
x + Γx + ω2s x = −
(eE0
m
)e−iωt
x = x0e−iωt
x = −iωx0e−iωt
x = −ω2x0e−iωt
(−ω2 − iωΓ + ω2s )x0e
−iωt = −(eE0
m
)e−iωt
x0 = −(eE0
m
)1
(ω2s − ω2 − iωΓ)
The amplitude of the response has a resonance and dissipation
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 14 / 19
![Page 39: Today’s Outline - November 07, 2016csrri.iit.edu/~segre/phys570/16F/lecture_21.pdfToday’s Outline - November 07, 2016 Lithiation of Sn-based anodes Photoemission Resonant Scattering](https://reader034.fdocuments.us/reader034/viewer/2022042312/5edac7fcceb8760df365f2cb/html5/thumbnails/39.jpg)
Forced charged oscillator
Consider an electron under the in-fluence of an oscillating electricfield ~Ein = xE0e
−iωt .
where Γ is the damping constant,ωs is the resonant frequency of theoscillator, and Γ� ωs .
assuming a solution of the form
x + Γx + ω2s x = −
(eE0
m
)e−iωt
x = x0e−iωt
x = −iωx0e−iωt
x = −ω2x0e−iωt
(−ω2 − iωΓ + ω2s )x0e
−iωt = −(eE0
m
)e−iωt
x0 = −(eE0
m
)1
(ω2s − ω2 − iωΓ)
The amplitude of the response has a resonance and dissipation
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 14 / 19
![Page 40: Today’s Outline - November 07, 2016csrri.iit.edu/~segre/phys570/16F/lecture_21.pdfToday’s Outline - November 07, 2016 Lithiation of Sn-based anodes Photoemission Resonant Scattering](https://reader034.fdocuments.us/reader034/viewer/2022042312/5edac7fcceb8760df365f2cb/html5/thumbnails/40.jpg)
Forced charged oscillator
Consider an electron under the in-fluence of an oscillating electricfield ~Ein = xE0e
−iωt .
where Γ is the damping constant,ωs is the resonant frequency of theoscillator, and Γ� ωs .
assuming a solution of the form
x + Γx + ω2s x = −
(eE0
m
)e−iωt
x = x0e−iωt
x = −iωx0e−iωt
x = −ω2x0e−iωt
(−ω2 − iωΓ + ω2s )x0e
−iωt = −(eE0
m
)e−iωt
x0 = −(eE0
m
)1
(ω2s − ω2 − iωΓ)
The amplitude of the response has a resonance and dissipation
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 14 / 19
![Page 41: Today’s Outline - November 07, 2016csrri.iit.edu/~segre/phys570/16F/lecture_21.pdfToday’s Outline - November 07, 2016 Lithiation of Sn-based anodes Photoemission Resonant Scattering](https://reader034.fdocuments.us/reader034/viewer/2022042312/5edac7fcceb8760df365f2cb/html5/thumbnails/41.jpg)
Forced charged oscillator
Consider an electron under the in-fluence of an oscillating electricfield ~Ein = xE0e
−iωt .
where Γ is the damping constant,ωs is the resonant frequency of theoscillator, and Γ� ωs .
assuming a solution of the form
x + Γx + ω2s x = −
(eE0
m
)e−iωt
x = x0e−iωt
x = −iωx0e−iωt
x = −ω2x0e−iωt
(−ω2 − iωΓ + ω2s )x0e
−iωt = −(eE0
m
)e−iωt
x0 = −(eE0
m
)1
(ω2s − ω2 − iωΓ)
The amplitude of the response has a resonance and dissipation
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 14 / 19
![Page 42: Today’s Outline - November 07, 2016csrri.iit.edu/~segre/phys570/16F/lecture_21.pdfToday’s Outline - November 07, 2016 Lithiation of Sn-based anodes Photoemission Resonant Scattering](https://reader034.fdocuments.us/reader034/viewer/2022042312/5edac7fcceb8760df365f2cb/html5/thumbnails/42.jpg)
Forced charged oscillator
Consider an electron under the in-fluence of an oscillating electricfield ~Ein = xE0e
−iωt .
where Γ is the damping constant,ωs is the resonant frequency of theoscillator, and Γ� ωs .
assuming a solution of the form
x + Γx + ω2s x = −
(eE0
m
)e−iωt
x = x0e−iωt
x = −iωx0e−iωt
x = −ω2x0e−iωt
(−ω2 − iωΓ + ω2s )x0e
−iωt = −(eE0
m
)e−iωt
x0 = −(eE0
m
)1
(ω2s − ω2 − iωΓ)
The amplitude of the response has a resonance and dissipation
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 14 / 19
![Page 43: Today’s Outline - November 07, 2016csrri.iit.edu/~segre/phys570/16F/lecture_21.pdfToday’s Outline - November 07, 2016 Lithiation of Sn-based anodes Photoemission Resonant Scattering](https://reader034.fdocuments.us/reader034/viewer/2022042312/5edac7fcceb8760df365f2cb/html5/thumbnails/43.jpg)
Forced charged oscillator
Consider an electron under the in-fluence of an oscillating electricfield ~Ein = xE0e
−iωt .
where Γ is the damping constant,ωs is the resonant frequency of theoscillator, and Γ� ωs .
assuming a solution of the form
x + Γx + ω2s x = −
(eE0
m
)e−iωt
x = x0e−iωt
x = −iωx0e−iωt
x = −ω2x0e−iωt
(−ω2 − iωΓ + ω2s )x0e
−iωt = −(eE0
m
)e−iωt
x0 = −(eE0
m
)1
(ω2s − ω2 − iωΓ)
The amplitude of the response has a resonance and dissipation
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 14 / 19
![Page 44: Today’s Outline - November 07, 2016csrri.iit.edu/~segre/phys570/16F/lecture_21.pdfToday’s Outline - November 07, 2016 Lithiation of Sn-based anodes Photoemission Resonant Scattering](https://reader034.fdocuments.us/reader034/viewer/2022042312/5edac7fcceb8760df365f2cb/html5/thumbnails/44.jpg)
Forced charged oscillator
Consider an electron under the in-fluence of an oscillating electricfield ~Ein = xE0e
−iωt .
where Γ is the damping constant,ωs is the resonant frequency of theoscillator, and Γ� ωs .
assuming a solution of the form
x + Γx + ω2s x = −
(eE0
m
)e−iωt
x = x0e−iωt
x = −iωx0e−iωt
x = −ω2x0e−iωt
(−ω2 − iωΓ + ω2s )x0e
−iωt = −(eE0
m
)e−iωt
x0 = −(eE0
m
)1
(ω2s − ω2 − iωΓ)
The amplitude of the response has a resonance and dissipation
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 14 / 19
![Page 45: Today’s Outline - November 07, 2016csrri.iit.edu/~segre/phys570/16F/lecture_21.pdfToday’s Outline - November 07, 2016 Lithiation of Sn-based anodes Photoemission Resonant Scattering](https://reader034.fdocuments.us/reader034/viewer/2022042312/5edac7fcceb8760df365f2cb/html5/thumbnails/45.jpg)
Forced charged oscillator
Consider an electron under the in-fluence of an oscillating electricfield ~Ein = xE0e
−iωt .
where Γ is the damping constant,ωs is the resonant frequency of theoscillator, and Γ� ωs .
assuming a solution of the form
x + Γx + ω2s x = −
(eE0
m
)e−iωt
x = x0e−iωt
x = −iωx0e−iωt
x = −ω2x0e−iωt
(−ω2 − iωΓ + ω2s )x0e
−iωt = −(eE0
m
)e−iωt
x0 = −(eE0
m
)1
(ω2s − ω2 − iωΓ)
The amplitude of the response has a resonance and dissipation
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 14 / 19
![Page 46: Today’s Outline - November 07, 2016csrri.iit.edu/~segre/phys570/16F/lecture_21.pdfToday’s Outline - November 07, 2016 Lithiation of Sn-based anodes Photoemission Resonant Scattering](https://reader034.fdocuments.us/reader034/viewer/2022042312/5edac7fcceb8760df365f2cb/html5/thumbnails/46.jpg)
Forced charged oscillator
Consider an electron under the in-fluence of an oscillating electricfield ~Ein = xE0e
−iωt .
where Γ is the damping constant,ωs is the resonant frequency of theoscillator, and Γ� ωs .
assuming a solution of the form
x + Γx + ω2s x = −
(eE0
m
)e−iωt
x = x0e−iωt
x = −iωx0e−iωt
x = −ω2x0e−iωt
(−ω2 − iωΓ + ω2s )x0e
−iωt = −(eE0
m
)e−iωt
x0 = −(eE0
m
)1
(ω2s − ω2 − iωΓ)
The amplitude of the response has a resonance and dissipation
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 14 / 19
![Page 47: Today’s Outline - November 07, 2016csrri.iit.edu/~segre/phys570/16F/lecture_21.pdfToday’s Outline - November 07, 2016 Lithiation of Sn-based anodes Photoemission Resonant Scattering](https://reader034.fdocuments.us/reader034/viewer/2022042312/5edac7fcceb8760df365f2cb/html5/thumbnails/47.jpg)
Radiated field
The radiated (scattered) electric field at a distance R from the electron isdirectly proportional to the electron’s acceleration with a retarded timet ′ = t − R/c (allowing for the travel time to the detector).
Erad(R, t) =
(e
4πε0Rc2
)x(t − R/c) =
(e
4πε0Rc2
)(−ω2)x0e
−iωte iωR/c
=ω2
(ω2s − ω2 − iωΓ)
(e2
4πε0mc2
)E0e−iωt
(e ikR
R
)Erad(R, t)
Ein= −r0
ω2
(ω2 − ω2s + iωΓ)
(e ikR
R
)= −r0fs
(e ikR
R
)
which is an outgoing spherical wavewith scattering amplitude fs =
ω2
(ω2 − ω2s + iωΓ)
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 15 / 19
![Page 48: Today’s Outline - November 07, 2016csrri.iit.edu/~segre/phys570/16F/lecture_21.pdfToday’s Outline - November 07, 2016 Lithiation of Sn-based anodes Photoemission Resonant Scattering](https://reader034.fdocuments.us/reader034/viewer/2022042312/5edac7fcceb8760df365f2cb/html5/thumbnails/48.jpg)
Radiated field
The radiated (scattered) electric field at a distance R from the electron isdirectly proportional to the electron’s acceleration with a retarded timet ′ = t − R/c (allowing for the travel time to the detector).
Erad(R, t) =
(e
4πε0Rc2
)x(t − R/c)
=
(e
4πε0Rc2
)(−ω2)x0e
−iωte iωR/c
=ω2
(ω2s − ω2 − iωΓ)
(e2
4πε0mc2
)E0e−iωt
(e ikR
R
)Erad(R, t)
Ein= −r0
ω2
(ω2 − ω2s + iωΓ)
(e ikR
R
)= −r0fs
(e ikR
R
)
which is an outgoing spherical wavewith scattering amplitude fs =
ω2
(ω2 − ω2s + iωΓ)
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 15 / 19
![Page 49: Today’s Outline - November 07, 2016csrri.iit.edu/~segre/phys570/16F/lecture_21.pdfToday’s Outline - November 07, 2016 Lithiation of Sn-based anodes Photoemission Resonant Scattering](https://reader034.fdocuments.us/reader034/viewer/2022042312/5edac7fcceb8760df365f2cb/html5/thumbnails/49.jpg)
Radiated field
The radiated (scattered) electric field at a distance R from the electron isdirectly proportional to the electron’s acceleration with a retarded timet ′ = t − R/c (allowing for the travel time to the detector).
Erad(R, t) =
(e
4πε0Rc2
)x(t − R/c) =
(e
4πε0Rc2
)(−ω2)x0e
−iωte iωR/c
=ω2
(ω2s − ω2 − iωΓ)
(e2
4πε0mc2
)E0e−iωt
(e ikR
R
)Erad(R, t)
Ein= −r0
ω2
(ω2 − ω2s + iωΓ)
(e ikR
R
)= −r0fs
(e ikR
R
)
which is an outgoing spherical wavewith scattering amplitude fs =
ω2
(ω2 − ω2s + iωΓ)
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 15 / 19
![Page 50: Today’s Outline - November 07, 2016csrri.iit.edu/~segre/phys570/16F/lecture_21.pdfToday’s Outline - November 07, 2016 Lithiation of Sn-based anodes Photoemission Resonant Scattering](https://reader034.fdocuments.us/reader034/viewer/2022042312/5edac7fcceb8760df365f2cb/html5/thumbnails/50.jpg)
Radiated field
The radiated (scattered) electric field at a distance R from the electron isdirectly proportional to the electron’s acceleration with a retarded timet ′ = t − R/c (allowing for the travel time to the detector).
Erad(R, t) =
(e
4πε0Rc2
)x(t − R/c) =
(e
4πε0Rc2
)(−ω2)x0e
−iωte iωR/c
=ω2
(ω2s − ω2 − iωΓ)
(e2
4πε0mc2
)E0e−iωt
(e ikR
R
)
Erad(R, t)
Ein= −r0
ω2
(ω2 − ω2s + iωΓ)
(e ikR
R
)= −r0fs
(e ikR
R
)
which is an outgoing spherical wavewith scattering amplitude fs =
ω2
(ω2 − ω2s + iωΓ)
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 15 / 19
![Page 51: Today’s Outline - November 07, 2016csrri.iit.edu/~segre/phys570/16F/lecture_21.pdfToday’s Outline - November 07, 2016 Lithiation of Sn-based anodes Photoemission Resonant Scattering](https://reader034.fdocuments.us/reader034/viewer/2022042312/5edac7fcceb8760df365f2cb/html5/thumbnails/51.jpg)
Radiated field
The radiated (scattered) electric field at a distance R from the electron isdirectly proportional to the electron’s acceleration with a retarded timet ′ = t − R/c (allowing for the travel time to the detector).
Erad(R, t) =
(e
4πε0Rc2
)x(t − R/c) =
(e
4πε0Rc2
)(−ω2)x0e
−iωte iωR/c
=ω2
(ω2s − ω2 − iωΓ)
(e2
4πε0mc2
)E0e−iωt
(e ikR
R
)Erad(R, t)
Ein= −r0
ω2
(ω2 − ω2s + iωΓ)
(e ikR
R
)
= −r0fs(e ikR
R
)
which is an outgoing spherical wavewith scattering amplitude fs =
ω2
(ω2 − ω2s + iωΓ)
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 15 / 19
![Page 52: Today’s Outline - November 07, 2016csrri.iit.edu/~segre/phys570/16F/lecture_21.pdfToday’s Outline - November 07, 2016 Lithiation of Sn-based anodes Photoemission Resonant Scattering](https://reader034.fdocuments.us/reader034/viewer/2022042312/5edac7fcceb8760df365f2cb/html5/thumbnails/52.jpg)
Radiated field
The radiated (scattered) electric field at a distance R from the electron isdirectly proportional to the electron’s acceleration with a retarded timet ′ = t − R/c (allowing for the travel time to the detector).
Erad(R, t) =
(e
4πε0Rc2
)x(t − R/c) =
(e
4πε0Rc2
)(−ω2)x0e
−iωte iωR/c
=ω2
(ω2s − ω2 − iωΓ)
(e2
4πε0mc2
)E0e−iωt
(e ikR
R
)Erad(R, t)
Ein= −r0
ω2
(ω2 − ω2s + iωΓ)
(e ikR
R
)= −r0fs
(e ikR
R
)
which is an outgoing spherical wavewith scattering amplitude fs =
ω2
(ω2 − ω2s + iωΓ)
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 15 / 19
![Page 53: Today’s Outline - November 07, 2016csrri.iit.edu/~segre/phys570/16F/lecture_21.pdfToday’s Outline - November 07, 2016 Lithiation of Sn-based anodes Photoemission Resonant Scattering](https://reader034.fdocuments.us/reader034/viewer/2022042312/5edac7fcceb8760df365f2cb/html5/thumbnails/53.jpg)
Radiated field
The radiated (scattered) electric field at a distance R from the electron isdirectly proportional to the electron’s acceleration with a retarded timet ′ = t − R/c (allowing for the travel time to the detector).
Erad(R, t) =
(e
4πε0Rc2
)x(t − R/c) =
(e
4πε0Rc2
)(−ω2)x0e
−iωte iωR/c
=ω2
(ω2s − ω2 − iωΓ)
(e2
4πε0mc2
)E0e−iωt
(e ikR
R
)Erad(R, t)
Ein= −r0
ω2
(ω2 − ω2s + iωΓ)
(e ikR
R
)= −r0fs
(e ikR
R
)
which is an outgoing spherical wavewith scattering amplitude
fs =ω2
(ω2 − ω2s + iωΓ)
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 15 / 19
![Page 54: Today’s Outline - November 07, 2016csrri.iit.edu/~segre/phys570/16F/lecture_21.pdfToday’s Outline - November 07, 2016 Lithiation of Sn-based anodes Photoemission Resonant Scattering](https://reader034.fdocuments.us/reader034/viewer/2022042312/5edac7fcceb8760df365f2cb/html5/thumbnails/54.jpg)
Radiated field
The radiated (scattered) electric field at a distance R from the electron isdirectly proportional to the electron’s acceleration with a retarded timet ′ = t − R/c (allowing for the travel time to the detector).
Erad(R, t) =
(e
4πε0Rc2
)x(t − R/c) =
(e
4πε0Rc2
)(−ω2)x0e
−iωte iωR/c
=ω2
(ω2s − ω2 − iωΓ)
(e2
4πε0mc2
)E0e−iωt
(e ikR
R
)Erad(R, t)
Ein= −r0
ω2
(ω2 − ω2s + iωΓ)
(e ikR
R
)= −r0fs
(e ikR
R
)
which is an outgoing spherical wavewith scattering amplitude fs =
ω2
(ω2 − ω2s + iωΓ)
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 15 / 19
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Dispersion corrections
The scattering factor can berewritten
and since Γ� ωs
the second term being thedispersion correction whosereal and imaginary compo-nents can be extracted
fs =ω2
(ω2 − ω2s + iωΓ)
= 1 +ω2s − iωΓ
(ω2 − ω2s + iωΓ)
≈ 1 +ω2s
(ω2 − ω2s + iωΓ)
χ(ω) = f ′s + if ′′s
=ω2s
(ω2 − ω2s + iωΓ)
χ(ω) =ω2s
(ω2 − ω2s + iωΓ)
· (ω2 − ω2s − iωΓ)
(ω2 − ω2s − iωΓ)
=ω2s (ω2 − ω2
s − iωΓ)
(ω2 − ω2s )2 + (ωΓ)2
f ′s =ω2s (ω2 − ω2
s )
(ω2 − ω2s )2 + (ωΓ)2
f ′′s = − ω2sωΓ
(ω2 − ω2s )2 + (ωΓ)2
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 16 / 19
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Dispersion corrections
The scattering factor can berewritten
and since Γ� ωs
the second term being thedispersion correction whosereal and imaginary compo-nents can be extracted
fs =ω2 + (−ω2
s + iωΓ)− (−ω2s + iωΓ)
(ω2 − ω2s + iωΓ)
= 1 +ω2s − iωΓ
(ω2 − ω2s + iωΓ)
≈ 1 +ω2s
(ω2 − ω2s + iωΓ)
χ(ω) = f ′s + if ′′s
=ω2s
(ω2 − ω2s + iωΓ)
χ(ω) =ω2s
(ω2 − ω2s + iωΓ)
· (ω2 − ω2s − iωΓ)
(ω2 − ω2s − iωΓ)
=ω2s (ω2 − ω2
s − iωΓ)
(ω2 − ω2s )2 + (ωΓ)2
f ′s =ω2s (ω2 − ω2
s )
(ω2 − ω2s )2 + (ωΓ)2
f ′′s = − ω2sωΓ
(ω2 − ω2s )2 + (ωΓ)2
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 16 / 19
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Dispersion corrections
The scattering factor can berewritten
and since Γ� ωs
the second term being thedispersion correction whosereal and imaginary compo-nents can be extracted
fs =ω2 + (−ω2
s + iωΓ)− (−ω2s + iωΓ)
(ω2 − ω2s + iωΓ)
= 1 +ω2s − iωΓ
(ω2 − ω2s + iωΓ)
≈ 1 +ω2s
(ω2 − ω2s + iωΓ)
χ(ω) = f ′s + if ′′s
=ω2s
(ω2 − ω2s + iωΓ)
χ(ω) =ω2s
(ω2 − ω2s + iωΓ)
· (ω2 − ω2s − iωΓ)
(ω2 − ω2s − iωΓ)
=ω2s (ω2 − ω2
s − iωΓ)
(ω2 − ω2s )2 + (ωΓ)2
f ′s =ω2s (ω2 − ω2
s )
(ω2 − ω2s )2 + (ωΓ)2
f ′′s = − ω2sωΓ
(ω2 − ω2s )2 + (ωΓ)2
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 16 / 19
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Dispersion corrections
The scattering factor can berewritten
and since Γ� ωs
the second term being thedispersion correction whosereal and imaginary compo-nents can be extracted
fs =ω2 + (−ω2
s + iωΓ)− (−ω2s + iωΓ)
(ω2 − ω2s + iωΓ)
= 1 +ω2s − iωΓ
(ω2 − ω2s + iωΓ)
≈ 1 +ω2s
(ω2 − ω2s + iωΓ)
χ(ω) = f ′s + if ′′s
=ω2s
(ω2 − ω2s + iωΓ)
χ(ω) =ω2s
(ω2 − ω2s + iωΓ)
· (ω2 − ω2s − iωΓ)
(ω2 − ω2s − iωΓ)
=ω2s (ω2 − ω2
s − iωΓ)
(ω2 − ω2s )2 + (ωΓ)2
f ′s =ω2s (ω2 − ω2
s )
(ω2 − ω2s )2 + (ωΓ)2
f ′′s = − ω2sωΓ
(ω2 − ω2s )2 + (ωΓ)2
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 16 / 19
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Dispersion corrections
The scattering factor can berewritten
and since Γ� ωs
the second term being thedispersion correction whosereal and imaginary compo-nents can be extracted
fs =ω2 + (−ω2
s + iωΓ)− (−ω2s + iωΓ)
(ω2 − ω2s + iωΓ)
= 1 +ω2s − iωΓ
(ω2 − ω2s + iωΓ)
≈ 1 +ω2s
(ω2 − ω2s + iωΓ)
χ(ω) = f ′s + if ′′s
=ω2s
(ω2 − ω2s + iωΓ)
χ(ω) =ω2s
(ω2 − ω2s + iωΓ)
· (ω2 − ω2s − iωΓ)
(ω2 − ω2s − iωΓ)
=ω2s (ω2 − ω2
s − iωΓ)
(ω2 − ω2s )2 + (ωΓ)2
f ′s =ω2s (ω2 − ω2
s )
(ω2 − ω2s )2 + (ωΓ)2
f ′′s = − ω2sωΓ
(ω2 − ω2s )2 + (ωΓ)2
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 16 / 19
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Dispersion corrections
The scattering factor can berewritten
and since Γ� ωs
the second term being thedispersion correction
whosereal and imaginary compo-nents can be extracted
fs =ω2 + (−ω2
s + iωΓ)− (−ω2s + iωΓ)
(ω2 − ω2s + iωΓ)
= 1 +ω2s − iωΓ
(ω2 − ω2s + iωΓ)
≈ 1 +ω2s
(ω2 − ω2s + iωΓ)
χ(ω) = f ′s + if ′′s
=ω2s
(ω2 − ω2s + iωΓ)
χ(ω) =ω2s
(ω2 − ω2s + iωΓ)
· (ω2 − ω2s − iωΓ)
(ω2 − ω2s − iωΓ)
=ω2s (ω2 − ω2
s − iωΓ)
(ω2 − ω2s )2 + (ωΓ)2
f ′s =ω2s (ω2 − ω2
s )
(ω2 − ω2s )2 + (ωΓ)2
f ′′s = − ω2sωΓ
(ω2 − ω2s )2 + (ωΓ)2
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 16 / 19
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Dispersion corrections
The scattering factor can berewritten
and since Γ� ωs
the second term being thedispersion correction
whosereal and imaginary compo-nents can be extracted
fs =ω2 + (−ω2
s + iωΓ)− (−ω2s + iωΓ)
(ω2 − ω2s + iωΓ)
= 1 +ω2s − iωΓ
(ω2 − ω2s + iωΓ)
≈ 1 +ω2s
(ω2 − ω2s + iωΓ)
χ(ω) = f ′s + if ′′s
=ω2s
(ω2 − ω2s + iωΓ)
χ(ω) =ω2s
(ω2 − ω2s + iωΓ)
· (ω2 − ω2s − iωΓ)
(ω2 − ω2s − iωΓ)
=ω2s (ω2 − ω2
s − iωΓ)
(ω2 − ω2s )2 + (ωΓ)2
f ′s =ω2s (ω2 − ω2
s )
(ω2 − ω2s )2 + (ωΓ)2
f ′′s = − ω2sωΓ
(ω2 − ω2s )2 + (ωΓ)2
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 16 / 19
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Dispersion corrections
The scattering factor can berewritten
and since Γ� ωs
the second term being thedispersion correction
whosereal and imaginary compo-nents can be extracted
fs =ω2 + (−ω2
s + iωΓ)− (−ω2s + iωΓ)
(ω2 − ω2s + iωΓ)
= 1 +ω2s − iωΓ
(ω2 − ω2s + iωΓ)
≈ 1 +ω2s
(ω2 − ω2s + iωΓ)
χ(ω) = f ′s + if ′′s =ω2s
(ω2 − ω2s + iωΓ)
χ(ω) =ω2s
(ω2 − ω2s + iωΓ)
· (ω2 − ω2s − iωΓ)
(ω2 − ω2s − iωΓ)
=ω2s (ω2 − ω2
s − iωΓ)
(ω2 − ω2s )2 + (ωΓ)2
f ′s =ω2s (ω2 − ω2
s )
(ω2 − ω2s )2 + (ωΓ)2
f ′′s = − ω2sωΓ
(ω2 − ω2s )2 + (ωΓ)2
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 16 / 19
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Dispersion corrections
The scattering factor can berewritten
and since Γ� ωs
the second term being thedispersion correction whosereal and imaginary compo-nents can be extracted
fs =ω2 + (−ω2
s + iωΓ)− (−ω2s + iωΓ)
(ω2 − ω2s + iωΓ)
= 1 +ω2s − iωΓ
(ω2 − ω2s + iωΓ)
≈ 1 +ω2s
(ω2 − ω2s + iωΓ)
χ(ω) = f ′s + if ′′s =ω2s
(ω2 − ω2s + iωΓ)
χ(ω) =ω2s
(ω2 − ω2s + iωΓ)
· (ω2 − ω2s − iωΓ)
(ω2 − ω2s − iωΓ)
=ω2s (ω2 − ω2
s − iωΓ)
(ω2 − ω2s )2 + (ωΓ)2
f ′s =ω2s (ω2 − ω2
s )
(ω2 − ω2s )2 + (ωΓ)2
f ′′s = − ω2sωΓ
(ω2 − ω2s )2 + (ωΓ)2
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 16 / 19
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Dispersion corrections
The scattering factor can berewritten
and since Γ� ωs
the second term being thedispersion correction whosereal and imaginary compo-nents can be extracted
fs =ω2 + (−ω2
s + iωΓ)− (−ω2s + iωΓ)
(ω2 − ω2s + iωΓ)
= 1 +ω2s − iωΓ
(ω2 − ω2s + iωΓ)
≈ 1 +ω2s
(ω2 − ω2s + iωΓ)
χ(ω) = f ′s + if ′′s =ω2s
(ω2 − ω2s + iωΓ)
χ(ω) =ω2s
(ω2 − ω2s + iωΓ)
· (ω2 − ω2s − iωΓ)
(ω2 − ω2s − iωΓ)
=ω2s (ω2 − ω2
s − iωΓ)
(ω2 − ω2s )2 + (ωΓ)2
f ′s =ω2s (ω2 − ω2
s )
(ω2 − ω2s )2 + (ωΓ)2
f ′′s = − ω2sωΓ
(ω2 − ω2s )2 + (ωΓ)2
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 16 / 19
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Dispersion corrections
The scattering factor can berewritten
and since Γ� ωs
the second term being thedispersion correction whosereal and imaginary compo-nents can be extracted
fs =ω2 + (−ω2
s + iωΓ)− (−ω2s + iωΓ)
(ω2 − ω2s + iωΓ)
= 1 +ω2s − iωΓ
(ω2 − ω2s + iωΓ)
≈ 1 +ω2s
(ω2 − ω2s + iωΓ)
χ(ω) = f ′s + if ′′s =ω2s
(ω2 − ω2s + iωΓ)
χ(ω) =ω2s
(ω2 − ω2s + iωΓ)
· (ω2 − ω2s − iωΓ)
(ω2 − ω2s − iωΓ)
=ω2s (ω2 − ω2
s − iωΓ)
(ω2 − ω2s )2 + (ωΓ)2
f ′s =ω2s (ω2 − ω2
s )
(ω2 − ω2s )2 + (ωΓ)2
f ′′s = − ω2sωΓ
(ω2 − ω2s )2 + (ωΓ)2
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 16 / 19
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Dispersion corrections
The scattering factor can berewritten
and since Γ� ωs
the second term being thedispersion correction whosereal and imaginary compo-nents can be extracted
fs =ω2 + (−ω2
s + iωΓ)− (−ω2s + iωΓ)
(ω2 − ω2s + iωΓ)
= 1 +ω2s − iωΓ
(ω2 − ω2s + iωΓ)
≈ 1 +ω2s
(ω2 − ω2s + iωΓ)
χ(ω) = f ′s + if ′′s =ω2s
(ω2 − ω2s + iωΓ)
χ(ω) =ω2s
(ω2 − ω2s + iωΓ)
· (ω2 − ω2s − iωΓ)
(ω2 − ω2s − iωΓ)
=ω2s (ω2 − ω2
s − iωΓ)
(ω2 − ω2s )2 + (ωΓ)2
f ′s =ω2s (ω2 − ω2
s )
(ω2 − ω2s )2 + (ωΓ)2
f ′′s = − ω2sωΓ
(ω2 − ω2s )2 + (ωΓ)2
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 16 / 19
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Dispersion corrections
The scattering factor can berewritten
and since Γ� ωs
the second term being thedispersion correction whosereal and imaginary compo-nents can be extracted
fs =ω2 + (−ω2
s + iωΓ)− (−ω2s + iωΓ)
(ω2 − ω2s + iωΓ)
= 1 +ω2s − iωΓ
(ω2 − ω2s + iωΓ)
≈ 1 +ω2s
(ω2 − ω2s + iωΓ)
χ(ω) = f ′s + if ′′s =ω2s
(ω2 − ω2s + iωΓ)
χ(ω) =ω2s
(ω2 − ω2s + iωΓ)
· (ω2 − ω2s − iωΓ)
(ω2 − ω2s − iωΓ)
=ω2s (ω2 − ω2
s − iωΓ)
(ω2 − ω2s )2 + (ωΓ)2
f ′s =ω2s (ω2 − ω2
s )
(ω2 − ω2s )2 + (ωΓ)2
f ′′s = − ω2sωΓ
(ω2 − ω2s )2 + (ωΓ)2
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 16 / 19
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Single oscillator dispersion terms
These dispersion terms giveresonant corrections to thescattering factor
f ′s =ω2s (ω2 + ω2
s )
(ω2 − ω2s )2 + (ωΓ)2
f ′′s = − ω2sωΓ
(ω2 − ω2s )2 + (ωΓ)2
-10
-8
-6
-4
-2
0
2
4
6
0.6 0.8 1 1.2 1.4
ω/ωs
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Single oscillator dispersion terms
These dispersion terms giveresonant corrections to thescattering factor
f ′s =ω2s (ω2 + ω2
s )
(ω2 − ω2s )2 + (ωΓ)2
f ′′s = − ω2sωΓ
(ω2 − ω2s )2 + (ωΓ)2
-10
-8
-6
-4
-2
0
2
4
6
0.6 0.8 1 1.2 1.4
ω/ωs
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 17 / 19
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Single oscillator dispersion terms
These dispersion terms giveresonant corrections to thescattering factor
f ′s =ω2s (ω2 + ω2
s )
(ω2 − ω2s )2 + (ωΓ)2
f ′′s = − ω2sωΓ
(ω2 − ω2s )2 + (ωΓ)2
-10
-8
-6
-4
-2
0
2
4
6
0.6 0.8 1 1.2 1.4
ω/ωs
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 17 / 19
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Total cross-section
The total cross-section for scatter-ing from a free electron is
for an electron bound to an atom,we can now generalize
this shows a frequency dependencewith a peak at ω ≈ ωs
if ω � ωs and when Γ → 0, thecross-section becomes
σT =
(8π
3
)(ω
ωs
)4
r20
when ω � ωs , σT → σfree
σfree =
(8π
3
)r20
σT =
(8π
3
)ω4
(ω2 − ω2s )2 + (ωΓ)2
r20
0
1
2
3
4
5
6
7
0.1 1 10
σT/σ
fre
e
ω/ωs
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Total cross-section
The total cross-section for scatter-ing from a free electron is
for an electron bound to an atom,we can now generalize
this shows a frequency dependencewith a peak at ω ≈ ωs
if ω � ωs and when Γ → 0, thecross-section becomes
σT =
(8π
3
)(ω
ωs
)4
r20
when ω � ωs , σT → σfree
σfree =
(8π
3
)r20
σT =
(8π
3
)ω4
(ω2 − ω2s )2 + (ωΓ)2
r20
0
1
2
3
4
5
6
7
0.1 1 10
σT/σ
fre
e
ω/ωs
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 18 / 19
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Total cross-section
The total cross-section for scatter-ing from a free electron is
for an electron bound to an atom,we can now generalize
this shows a frequency dependencewith a peak at ω ≈ ωs
if ω � ωs and when Γ → 0, thecross-section becomes
σT =
(8π
3
)(ω
ωs
)4
r20
when ω � ωs , σT → σfree
σfree =
(8π
3
)r20
σT =
(8π
3
)ω4
(ω2 − ω2s )2 + (ωΓ)2
r20
0
1
2
3
4
5
6
7
0.1 1 10
σT/σ
fre
e
ω/ωs
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 18 / 19
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Total cross-section
The total cross-section for scatter-ing from a free electron is
for an electron bound to an atom,we can now generalize
this shows a frequency dependencewith a peak at ω ≈ ωs
if ω � ωs and when Γ → 0, thecross-section becomes
σT =
(8π
3
)(ω
ωs
)4
r20
when ω � ωs , σT → σfree
σfree =
(8π
3
)r20
σT =
(8π
3
)ω4
(ω2 − ω2s )2 + (ωΓ)2
r20
0
1
2
3
4
5
6
7
0.1 1 10
σT/σ
fre
e
ω/ωs
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 18 / 19
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Total cross-section
The total cross-section for scatter-ing from a free electron is
for an electron bound to an atom,we can now generalize
this shows a frequency dependencewith a peak at ω ≈ ωs
if ω � ωs and when Γ → 0, thecross-section becomes
σT =
(8π
3
)(ω
ωs
)4
r20
when ω � ωs , σT → σfree
σfree =
(8π
3
)r20
σT =
(8π
3
)ω4
(ω2 − ω2s )2 + (ωΓ)2
r20
0
1
2
3
4
5
6
7
0.1 1 10
σT/σ
fre
e
ω/ωs
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 18 / 19
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Total cross-section
The total cross-section for scatter-ing from a free electron is
for an electron bound to an atom,we can now generalize
this shows a frequency dependencewith a peak at ω ≈ ωs
if ω � ωs and when Γ → 0, thecross-section becomes
σT =
(8π
3
)(ω
ωs
)4
r20
when ω � ωs , σT → σfree
σfree =
(8π
3
)r20
σT =
(8π
3
)ω4
(ω2 − ω2s )2 + (ωΓ)2
r20
0
1
2
3
4
5
6
7
0.1 1 10
σT/σ
fre
e
ω/ωs
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 18 / 19
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Total cross-section
The total cross-section for scatter-ing from a free electron is
for an electron bound to an atom,we can now generalize
this shows a frequency dependencewith a peak at ω ≈ ωs
if ω � ωs and when Γ → 0, thecross-section becomes
σT =
(8π
3
)(ω
ωs
)4
r20
when ω � ωs , σT → σfree
σfree =
(8π
3
)r20
σT =
(8π
3
)ω4
(ω2 − ω2s )2 + (ωΓ)2
r20
0
1
2
3
4
5
6
7
0.1 1 10
σT/σ
fre
e
ω/ωs
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 18 / 19
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Total cross-section
The total cross-section for scatter-ing from a free electron is
for an electron bound to an atom,we can now generalize
this shows a frequency dependencewith a peak at ω ≈ ωs
if ω � ωs and when Γ → 0, thecross-section becomes
σT =
(8π
3
)(ω
ωs
)4
r20
when ω � ωs , σT → σfree
σfree =
(8π
3
)r20
σT =
(8π
3
)ω4
(ω2 − ω2s )2 + (ωΓ)2
r20
0
1
2
3
4
5
6
7
0.1 1 10
σT/σ
fre
e
ω/ωs
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 18 / 19
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Refractive index
n2 = 1 +
(e2ρ
ε0m
)1
(ω2s − ω2 − iωΓ)
= 1 +
(e2ρ
ε0m
)ω2s − ω2
(ω2s − ω2)2 + (ωΓ)2
+ iωΓ
(ω2s − ω2)2 + (ωΓ)2
-0.5
0
0.5
1
1.5
2
2.5
3
0.1 1 10
Re
[n2]
ω/ωs
0
0.5
1
1.5
2
2.5
3
0.1 1 10
Im[n
2]
ω/ωs
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Refractive index
n2 = 1 +
(e2ρ
ε0m
)1
(ω2s − ω2 − iωΓ)
= 1 +
(e2ρ
ε0m
)ω2s − ω2
(ω2s − ω2)2 + (ωΓ)2
+ iωΓ
(ω2s − ω2)2 + (ωΓ)2
-0.5
0
0.5
1
1.5
2
2.5
3
0.1 1 10
Re
[n2]
ω/ωs
0
0.5
1
1.5
2
2.5
3
0.1 1 10
Im[n
2]
ω/ωs
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Refractive index
n2 = 1 +
(e2ρ
ε0m
)1
(ω2s − ω2 − iωΓ)
= 1 +
(e2ρ
ε0m
)ω2s − ω2
(ω2s − ω2)2 + (ωΓ)2
+ iωΓ
(ω2s − ω2)2 + (ωΓ)2
-0.5
0
0.5
1
1.5
2
2.5
3
0.1 1 10
Re
[n2]
ω/ωs
0
0.5
1
1.5
2
2.5
3
0.1 1 10
Im[n
2]
ω/ωs
C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 19 / 19