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

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


C. Segre (IIT) PHYS 570 - Fall 2016 November 07, 2016 1 / 19



• Photoemission








• Photoemission








• Photoemission








• Photoemission








• Photoemission








• Photoemission






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


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


In situ battery box

Pouch cell clamped against front window in helium environment


In situ battery box

Suitable for both transmission and fluorescence measurements



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



-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



-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)





-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)





-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



-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.



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



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





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





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).


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






Ekin =~2q2v2m

= ~ω − φ− EB

EB = EF − Ei






Ekin =~2q2v2m

= ~ω − φ− EB

EB = EF − Ei






Ekin =~2q2v2m

= ~ω − φ− EB

EB = EF − Ei


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

















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.





f (~Q, ω) = f 0(~Q) + f ′(ω) + if ′′(ω)







f (~Q, ω) = f 0(~Q) + f ′(ω) + if ′′(ω)







f (~Q, ω) = f 0(~Q) + f ′(ω) + if ′′(ω)







f (~Q, ω) = f 0(~Q) + f ′(ω) + if ′′(ω)







f (~Q, ω) = f 0(~Q) + f ′(ω) + if ′′(ω)







f (~Q, ω) = f 0(~Q) + f ′(ω) + if ′′(ω)




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




−iωt .



x + Γx + ω2s x = −

(eE0

m

)e−iωt

x = x0e−iωt



(−ω2 − iωΓ + ω2s )x0e

−iωt = −(eE0

m

)e−iωt

x0 = −(eE0

m

)1

(ω2s − ω2 − iωΓ)





−iωt .



x + Γx + ω2s x = −

(eE0

m

)e−iωt

x = x0e−iωt



(−ω2 − iωΓ + ω2s )x0e

−iωt = −(eE0

m

)e−iωt

x0 = −(eE0

m

)1

(ω2s − ω2 − iωΓ)





−iωt .



x + Γx + ω2s x = −

(eE0

m

)e−iωt

x = x0e−iωt



(−ω2 − iωΓ + ω2s )x0e

−iωt = −(eE0

m

)e−iωt

x0 = −(eE0

m

)1

(ω2s − ω2 − iωΓ)





−iωt .



x + Γx + ω2s x = −

(eE0

m

)e−iωt

x = x0e−iωt



(−ω2 − iωΓ + ω2s )x0e

−iωt = −(eE0

m

)e−iωt

x0 = −(eE0

m

)1

(ω2s − ω2 − iωΓ)





−iωt .



x + Γx + ω2s x = −

(eE0

m

)e−iωt

x = x0e−iωt



(−ω2 − iωΓ + ω2s )x0e

−iωt = −(eE0

m

)e−iωt

x0 = −(eE0

m

)1

(ω2s − ω2 − iωΓ)





−iωt .



x + Γx + ω2s x = −

(eE0

m

)e−iωt

x = x0e−iωt



(−ω2 − iωΓ + ω2s )x0e

−iωt = −(eE0

m

)e−iωt

x0 = −(eE0

m

)1

(ω2s − ω2 − iωΓ)





−iωt .



x + Γx + ω2s x = −

(eE0

m

)e−iωt

x = x0e−iωt



(−ω2 − iωΓ + ω2s )x0e

−iωt = −(eE0

m

)e−iωt

x0 = −(eE0

m

)1

(ω2s − ω2 − iωΓ)





−iωt .



x + Γx + ω2s x = −

(eE0

m

)e−iωt

x = x0e−iωt



(−ω2 − iωΓ + ω2s )x0e

−iωt = −(eE0

m

)e−iωt

x0 = −(eE0

m

)1

(ω2s − ω2 − iωΓ)





−iωt .



x + Γx + ω2s x = −

(eE0

m

)e−iωt

x = x0e−iωt



(−ω2 − iωΓ + ω2s )x0e

−iωt = −(eE0

m

)e−iωt

x0 = −(eE0

m

)1

(ω2s − ω2 − iωΓ)



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)


Radiated field


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

)


ω2

(ω2 − ω2s + iωΓ)


Radiated field


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

)


ω2

(ω2 − ω2s + iωΓ)


Radiated field


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

)


ω2

(ω2 − ω2s + iωΓ)


Radiated field


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

)


ω2

(ω2 − ω2s + iωΓ)


Radiated field


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

)


ω2

(ω2 − ω2s + iωΓ)


Radiated field


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)


Radiated field


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

)


ω2

(ω2 − ω2s + iωΓ)


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




and since Γ� ωs


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


(ω2 − ω2s )2 + (ωΓ)2




and since Γ� ωs


fs =ω2 + (−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


(ω2 − ω2s )2 + (ωΓ)2




and since Γ� ωs


fs =ω2 + (−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


(ω2 − ω2s )2 + (ωΓ)2




and since Γ� ωs


fs =ω2 + (−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


(ω2 − ω2s )2 + (ωΓ)2




and since Γ� ωs

the second term being thedispersion correction

whosereal and imaginary compo-nents can be extracted

fs =ω2 + (−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


(ω2 − ω2s )2 + (ωΓ)2




and since Γ� ωs



fs =ω2 + (−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


(ω2 − ω2s )2 + (ωΓ)2




and since Γ� ωs



fs =ω2 + (−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


(ω2 − ω2s )2 + (ωΓ)2




and since Γ� ωs


fs =ω2 + (−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


(ω2 − ω2s )2 + (ωΓ)2




and since Γ� ωs


fs =ω2 + (−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


(ω2 − ω2s )2 + (ωΓ)2




and since Γ� ωs


fs =ω2 + (−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


(ω2 − ω2s )2 + (ωΓ)2




and since Γ� ωs


fs =ω2 + (−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


(ω2 − ω2s )2 + (ωΓ)2




and since Γ� ωs


fs =ω2 + (−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


(ω2 − ω2s )2 + (ωΓ)2


Single oscillator dispersion terms

These dispersion terms giveresonant corrections to thescattering factor

f ′s =ω2s (ω2 + ω2

s )

(ω2 − ω2s )2 + (ωΓ)2


(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




f ′s =ω2s (ω2 + ω2

s )

(ω2 − ω2s )2 + (ωΓ)2


(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




f ′s =ω2s (ω2 + ω2

s )

(ω2 − ω2s )2 + (ωΓ)2


(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


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


Total cross-section





σT =

(8π

3

)(ω

ωs

)4

r20


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


Total cross-section





σT =

(8π

3

)(ω

ωs

)4

r20


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


Total cross-section





σT =

(8π

3

)(ω

ωs

)4

r20


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


Total cross-section





σT =

(8π

3

)(ω

ωs

)4

r20


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


Total cross-section





σT =

(8π

3

)(ω

ωs

)4

r20


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


Total cross-section





σT =

(8π

3

)(ω

ωs

)4

r20


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


Total cross-section





σT =

(8π

3

)(ω

ωs

)4

r20


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


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


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


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


Documents

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