PHYS 570 - Introduction to Synchrotron Radiationphys.iit.edu/~segre/phys570/20S/lecture_01.pdfUnderstand the physics behind a variety of experimental techniques Be able to make an

PHYS 570 - Introduction to Synchrotron Radiation

Term: Spring 2020Meetings: Tuesday & Thursday 17:00-18:15Location: 241 Rettaliata Engineering

Instructor: Carlo SegreOffice: 166d/172 Pritzker SciencePhone: 312.567.3498email: [email protected]

Book: Elements of Modern X-Ray Physics, 2nd ed.,J. Als-Nielsen and D. McMorrow (Wiley, 2011)

Web Site: http://csrri.iit.edu/∼segre/phys570/20S

C. Segre (IIT) PHYS 570 - Spring 2020 January 14, 2020 1 / 26

http://csrri.iit.edu/$\sim $segre/phys570/20S

Course objectives

• Understand the means of production of synchrotron x-ray radiation

• Understand the function of various components of a synchrotronbeamline

• Be able to perform calculations in support of a synchrotronexperiment

• Understand the physics behind a variety of experimental techniques

• Be able to make an oral presentation of a synchrotron radiationresearch topic

• Be able to write a General User Proposal in the format used by theAdvanced Photon Source


Course objectives








Course objectives








Course objectives








Course objectives








Course objectives








Course syllabus

• Focus on applications of synchrotron radiation

• Homework assignments

• In-class student presentations on research topics• Choose a research article which features a synchrotron technique

• Timetable will be posted

• Final project - writing a General User Proposal• Start thinking about a suitable project right away

• Synchrotron technique must differ from journal article used in finalpresentation

• Make proposal and get approval before starting


Course syllabus









Course syllabus









Course syllabus









Optional activities

• Visits to Advanced Photon Source• All students who plan to attend will need to request badges from APS

• Go to the APS User Portal and register as a new user:https://beam.aps.anl.gov/pls/apsweb/ufr main pkg.usr start page

• Use MRCAT (Sector 10) as location of experiment

• Use Carlo Segre as local contact

• State that your beamtime will be in the second week of March

• Schedule to be determined but will try to make it during the week ofthe Advanced Photon Source User Meeting (April 20-24, 2020)

• Hands on data analysis training• GSAS for Rietveld refinement of powder diffraction data

https://subversion.xray.aps.anl.gov/trac/pyGSAS

• IFEFFIT suite for x-ray absorption spectroscopy analysishttp://cars9.uchicago.edu/ifeffit


https://beam.aps.anl.gov/pls/apsweb/ufr_main_pkg.usr_start_page


http://cars9.uchicago.edu/ifeffit

Optional activities

• Visits to Advanced Photon Source• All students who plan to attend will need to request badges from APS

• Go to the APS User Portal and register as a new user:https://beam.aps.anl.gov/pls/apsweb/ufr main pkg.usr start page

• Use MRCAT (Sector 10) as location of experiment

• Use Carlo Segre as local contact

• State that your beamtime will be in the second week of March

• Schedule to be determined but will try to make it during the week ofthe Advanced Photon Source User Meeting (April 20-24, 2020)

• Hands on data analysis training• GSAS for Rietveld refinement of powder diffraction data


• IFEFFIT suite for x-ray absorption spectroscopy analysishttp://cars9.uchicago.edu/ifeffit


https://beam.aps.anl.gov/pls/apsweb/ufr_main_pkg.usr_start_page


http://cars9.uchicago.edu/ifeffit

Course grading

33% – Homework assignments

Weekly or bi-weeklyDue at beginning of classMay be turned in via Blackboard

33% – General User Proposal

33% – Final Exam Presentation

Grading scaleA – 80% to 100%B – 65% to 80%C – 50% to 65%E – 0% to 50%


Course grading

33% – Homework assignmentsWeekly or bi-weekly

Due at beginning of classMay be turned in via Blackboard





Course grading

33% – Homework assignmentsWeekly or bi-weeklyDue at beginning of class

May be turned in via Blackboard





Course grading

33% – Homework assignmentsWeekly or bi-weeklyDue at beginning of classMay be turned in via Blackboard





Course grading






Course grading






Course grading






Topics to be covered (at a minimum)

• X-rays and their interaction with matter

• Sources of x-rays

• Refraction and reflection from interfaces

• Kinematical diffraction

• Diffraction by perfect crystals

• Small angle scattering

• Photoelectric absorption

• Resonant scattering

• Imaging











• Imaging











• Imaging











• Imaging











• Imaging











• Imaging











• Imaging











• Imaging











• Imaging











• Imaging


Resources for the course

• Orange x-ray data booklet:http://xdb.lbl.gov/xdb-new.pdf

• Center for X-Ray Optics web site:http://cxro.lbl.gov

• Hephaestus from the Demeter suite:http://bruceravel.github.io/demeter

• McMaster data on the Web:http://csrri.iit.edu/periodic-table.html

• X-ray Oriented Programs:https://www.aps.anl.gov/Science/Scientific-Software/XOP


http://xdb.lbl.gov/xdb-new.pdf

http://cxro.lbl.gov

http://bruceravel.github.io/demeter

http://csrri.iit.edu/periodic-table.html

https://www.aps.anl.gov/Science/Scientific-Software/XOP









http://cxro.lbl.gov












http://cxro.lbl.gov












http://cxro.lbl.gov












http://cxro.lbl.gov




Today’s outline - January 14, 2020

• The big picture

• History of x-ray sources

• X-ray interactions with matter

• Thomson scattering

• Atomic form factor

Reading Assignment: Chapter 1.1–1.6; 2.1–2.2



• The big picture








• The big picture








• The big picture








• The big picture








• The big picture








• The big picture







Why synchrotron radiation?

X-rays are the ideal structural probe for interatomic dis-tances with wavelengths of ∼1 A

Laboratory x-ray sources are extremely useful and havegotten much better in the past decades but some exper-iments can only be done at a synchrotron

Synchrotron sources provide superior flux, brilliance, andtunability

Synchrotron sources and particularly FELs produce co-herent beams

The broad range of techniques make synchrotron x-raysources to nearly any science or engineering field






























A bit about my research. . .

0

1

2

aed

cb

Pote

ntia

l (V)

Lithiation Delithiation

SnPx Sn LiySn

0 200 400 6000 200 400 6000

1

2

3

4

f

Capacity (mAh g-1)

R (Å

)

0.00

0.05

0.10

0.15

0.19

0.24

0.29

0.34

0.39

| (R)|(Å-3)

“In situ EXAFS-derived mechanism of highly reversible tin phosphide/graphitecomposite anode for Li-ion batteries,” Y. Ding, Z. Li, E.V. Timofeeva, and C.U. Segre,Adv. Energy Mater. 1702134 (2017).


A bit about my research. . .

0

1

2

aed

cbPo

tent

ial (

V)

Lithiation Delithiation

SnPx Sn LiySn

0 200 400 6000 200 400 6000

1

2

3

4

f

Capacity (mAh g-1)

R (Å

)

0.00

0.05

0.10

0.15

0.19

0.24

0.29

0.34

0.39

| (R)|(Å-3)

“In situ EXAFS-derived mechanism of highly reversible tin phosphide/graphitecomposite anode for Li-ion batteries,” Y. Ding, Z. Li, E.V. Timofeeva, and C.U. Segre,Adv. Energy Mater. 1702134 (2017).


The classical x-ray

The classical plane wave representation of x-rays is:

E(r, t) = εE0ei(k·r−ωt)

where ε is a unit vector in the direction of the electric field, k is thewavevector of the radiation along the propagation direction, and ω is theangular frequency of oscillation of the radiation.

If the energy, E is in keV, the relationship among these quantities is givenby:

~ω = hν = E , λν = c

λ = hc/E= (4.1357× 10−15 eV · s)(2.9979× 108 m/s)/E= (4.1357× 10−18 keV · s)(2.9979× 1018 A/s)/E= 12.398 A · keV/E to give units of A


The classical x-ray





~ω = hν = E , λν = c



The classical x-ray



where ε is a unit vector in the direction of the electric field

, k is thewavevector of the radiation along the propagation direction, and ω is theangular frequency of oscillation of the radiation.


~ω = hν = E , λν = c



The classical x-ray



where ε is a unit vector in the direction of the electric field, k is thewavevector of the radiation along the propagation direction

, and ω is theangular frequency of oscillation of the radiation.


~ω = hν = E , λν = c



The classical x-ray





~ω = hν = E , λν = c



The classical x-ray





~ω = hν = E , λν = c



The classical x-ray





~ω = hν = E , λν = c



The classical x-ray





~ω = hν = E , λν = c



Interactions of x-rays with matter

For the purposes of this course, we care most about the interactions ofx-rays with matter.

There are four basic types of such interactions:

1. Elastic scattering

2. Inelastic scattering

3. Absorption

4. Pair production

We will only discuss the first three.







3. Absorption

4. Pair production








3. Absorption

4. Pair production








3. Absorption

4. Pair production








3. Absorption

4. Pair production








3. Absorption

4. Pair production



Elastic scattering

Most of the phenomena we will discuss can be treated classically as elasticscattering of electromagnetic waves (x-rays)

The typical scattering geometry is

k

k

Q

where an incident x-ray of wave number k

scatters elastically from an electron to k′

resulting in a scattering vector Q

or in terms of momentum transfer: ~Q = ~k− ~k′

Start with the scattering from a single electron, then build up to morecomplexity


Elastic scattering



k

k

Q







Elastic scattering



k

k

Q







Elastic scattering



k

k

Q







Elastic scattering



k

k

Q







Elastic scattering



k

k

Q







Elastic scattering



k

k

Q







Thomson scattering

Assumptions:

incident x-ray plane waveelectron is a point chargescattering is elasticscattered intensity ∝ 1/R2

Ein

Erad

The electron is exposed to theincident electric field Ein(t ′) andis accelerated

The acceleration of the electron,ax(t ′), results in the radiation ofa spherical wave with the samefrequency

The observer at R “sees” a scat-tered electric field Erad(R, t) ata later time t = t ′ + R/c

Using this, calculate the elasticscattering cross-section


Thomson scattering

Assumptions:

incident x-ray plane wave

electron is a point chargescattering is elasticscattered intensity ∝ 1/R2

Ein

Erad






Thomson scattering

Assumptions:

incident x-ray plane waveelectron is a point charge

scattering is elasticscattered intensity ∝ 1/R2

Ein

Erad






Thomson scattering

Assumptions:

incident x-ray plane waveelectron is a point chargescattering is elastic

scattered intensity ∝ 1/R2

Ein

Erad






Thomson scattering

Assumptions:


Ein

Erad






Thomson scattering

Assumptions:


Ein

Erad






Thomson scattering

Assumptions:


Ein

Erad

R






Thomson scattering

Assumptions:


Ein

Erad

R






Thomson scattering

Assumptions:


Ein

Erad

R






Thomson scattering

Erad(R, t) = − −e4πε0c2R

ax(t ′) sin Ψ

where t ′ = t − R/c

ax(t ′) = − e

mEx0e

−iωt′

= − e

mEx0e

−iωte iωR/c

ax(t ′) = − e

mEine

iωR/c


Thomson scattering


ax(t ′) sin Ψ where t ′ = t − R/c

ax(t ′) = − e

mEx0e

−iωt′

= − e

mEx0e

−iωte iωR/c

ax(t ′) = − e

mEine

iωR/c


Thomson scattering



ax(t ′) = − e

mEx0e

−iωt′

= − e

mEx0e

−iωte iωR/c

ax(t ′) = − e

mEine

iωR/c


Thomson scattering



ax(t ′) = − e

mEx0e

−iωt′ = − e

mEx0e

−iωte iωR/c

ax(t ′) = − e

mEine

iωR/c


Thomson scattering



ax(t ′) = − e

mEx0e

−iωt′ = − e

mEx0e

−iωte iωR/c

ax(t ′) = − e

mEine

iωR/c


Thomson scattering


−em

EineiωR/c sin Ψ

Erad(R, t)

Ein= − e2

4πε0mc2e iωR/c

Rsin Ψ but k = ω/c


Thomson scattering


−em

EineiωR/c sin Ψ

Erad(R, t)

Ein= − e2

4πε0mc2e iωR/c

Rsin Ψ

but k = ω/c


Thomson scattering


−em

EineiωR/c sin Ψ

Erad(R, t)

Ein= − e2

4πε0mc2e iωR/c

Rsin Ψ but k = ω/c


Thomson scattering length

Erad(R, t)

Ein= − e2

4πε0mc2e ikR

Rsin Ψ = −r0

e ikR

Rsin Ψ

r0 =e2

4πε0mc2= 2.82× 10−5A

r0 is called the Thomson scattering length or the “classical” radius of theelectron



Erad(R, t)

Ein= − e2

4πε0mc2e ikR

Rsin Ψ = −r0

e ikR

Rsin Ψ

r0 =e2

4πε0mc2= 2.82× 10−5A




Erad(R, t)

Ein= − e2

4πε0mc2e ikR

Rsin Ψ = −r0

e ikR

Rsin Ψ

r0 =e2

4πε0mc2= 2.82× 10−5A



Scattering cross-section

detector of solid angle ∆Ω located adistance R from electron

incoming beam has cross-section A0

cross section of the scattered beam(into detector) is R2∆Ω

Φ0 ≡I0A0

= c|Ein|2

~ω

Isc ∝ c(R2∆Ω)|Erad |2

~ωIscI0

=|Erad |2

|Ein|2R2∆Ω






Φ0 ≡I0A0

= c|Ein|2

~ω


~ωIscI0

=|Erad |2

|Ein|2R2∆Ω






Φ0 ≡I0A0

= c|Ein|2

~ω


~ωIscI0

=|Erad |2

|Ein|2R2∆Ω






Φ0 ≡I0A0

= c|Ein|2

~ω


~ωIscI0

=|Erad |2

|Ein|2R2∆Ω






Φ0 ≡I0A0

= c|Ein|2

~ω


~ω

IscI0

=|Erad |2

|Ein|2R2∆Ω






Φ0 ≡I0A0

= c|Ein|2

~ω


~ω

IscI0

=|Erad |2

|Ein|2R2∆Ω






Φ0 ≡I0A0

= c|Ein|2

~ω


~ωIscI0

=|Erad |2

|Ein|2R2∆Ω



Differential cross-section is obtained by normalizing

dσ

dΩ

=Isc

Φ0∆Ω=

Isc(I0/A0) ∆Ω

=|Erad |2

|Ein|2R2 = r20 sin2 Ψ

Erad

Ein= −r0

e ikR

R

∣∣ε · ε′∣∣ = −r0e ikR

R

∣∣cos(π2 −Ψ

)∣∣ = −r0e ikR

Rsin Ψ




dσ

dΩ

=Isc

Φ0∆Ω=

Isc(I0/A0) ∆Ω

=|Erad |2


Erad

Ein= −r0

e ikR

R

∣∣ε · ε′∣∣ = −r0e ikR

R

∣∣cos(π2 −Ψ

)∣∣ = −r0e ikR

Rsin Ψ




dσ

dΩ=

IscΦ0∆Ω

=Isc

(I0/A0) ∆Ω=|Erad |2


Erad

Ein= −r0

e ikR

R

∣∣ε · ε′∣∣ = −r0e ikR

R

∣∣cos(π2 −Ψ

)∣∣ = −r0e ikR

Rsin Ψ




dσ

dΩ=

IscΦ0∆Ω

=Isc

(I0/A0) ∆Ω

=|Erad |2


Erad

Ein= −r0

e ikR

R

∣∣ε · ε′∣∣ = −r0e ikR

R

∣∣cos(π2 −Ψ

)∣∣ = −r0e ikR

Rsin Ψ




dσ

dΩ=

IscΦ0∆Ω

=Isc

(I0/A0) ∆Ω=|Erad |2

|Ein|2R2

= r20 sin2 Ψ

Erad

Ein= −r0

e ikR

R

∣∣ε · ε′∣∣ = −r0e ikR

R

∣∣cos(π2 −Ψ

)∣∣ = −r0e ikR

Rsin Ψ




dσ

dΩ=

IscΦ0∆Ω

=Isc

(I0/A0) ∆Ω=|Erad |2

|Ein|2R2

= r20 sin2 Ψ

Erad

Ein= −r0

e ikR

R

∣∣ε · ε′∣∣

= −r0e ikR

R

∣∣cos(π2 −Ψ

)∣∣ = −r0e ikR

Rsin Ψ




dσ

dΩ=

IscΦ0∆Ω

=Isc

(I0/A0) ∆Ω=|Erad |2

|Ein|2R2

= r20 sin2 Ψ

Erad

Ein= −r0

e ikR

R

∣∣ε · ε′∣∣ = −r0e ikR

R

∣∣cos(π2 −Ψ

)∣∣

= −r0e ikR

Rsin Ψ




dσ

dΩ=

IscΦ0∆Ω

=Isc

(I0/A0) ∆Ω=|Erad |2

|Ein|2R2

= r20 sin2 Ψ

Erad

Ein= −r0

e ikR

R

∣∣ε · ε′∣∣ = −r0e ikR

R

∣∣cos(π2 −Ψ

)∣∣ = −r0e ikR

Rsin Ψ




dσ

dΩ=

IscΦ0∆Ω

=Isc

(I0/A0) ∆Ω=|Erad |2


Erad

Ein= −r0

e ikR

R

∣∣ε · ε′∣∣ = −r0e ikR

R

∣∣cos(π2 −Ψ

)∣∣ = −r0e ikR

Rsin Ψ


Total cross-section

Integrate to obtain the total Thomson scattering cross-section from anelectron.

If displacement is in vertical direction, sin Ψ term is replacedby unity and if the source is unpolarized, it is a combination.

σ =8π

3r20

= 0.665× 10−24 cm2

= 0.665 barn

Polarization factor =

1

sin2 Ψ12

(1 + sin2 Ψ

)


Total cross-section



σ =8π

3r20

= 0.665× 10−24 cm2

= 0.665 barn


1

sin2 Ψ12

(1 + sin2 Ψ

)


Total cross-section



σ =8π

3r20

= 0.665× 10−24 cm2

= 0.665 barn


1

sin2 Ψ12

(1 + sin2 Ψ

)


Total cross-section

Integrate to obtain the total Thomson scattering cross-section from anelectron. If displacement is in vertical direction, sin Ψ term is replacedby unity and if the source is unpolarized, it is a combination.

σ =8π

3r20

= 0.665× 10−24 cm2

= 0.665 barn


1

sin2 Ψ12

(1 + sin2 Ψ

)C. Segre (IIT) PHYS 570 - Spring 2020 January 14, 2020 20 / 26

Atomic scattering

If we have a charge distribution instead of asingle electron, the scattering is more com-plex

A phase shift arises because of scatteringfrom different portions of extended electrondistribution

∆φ(r) = (k− k′) · r = Q · r

r θ

k

k

Q

θwhere the scattering vector, Q is given by

|Q| = 2 |k| sinθ =4π

λsinθ


Atomic scattering



∆φ(r) = (k− k′) · r = Q · r

r θ

k

k

Q


|Q| = 2 |k| sinθ =4π

λsinθ


Atomic scattering



∆φ(r) = (k− k′) · r = Q · r

r θ

k

k

Q


|Q| = 2 |k| sinθ =4π

λsinθ


Atomic scattering



∆φ(r) = (k− k′) · r = Q · r

r θ

k

k

Q

θ

where the scattering vector, Q is given by

|Q| = 2 |k| sinθ =4π

λsinθ


Atomic scattering



∆φ(r) = (k− k′) · r = Q · r

r θ

k

k

Q


|Q| = 2 |k| sinθ =4π

λsinθ


Atomic scattering



∆φ(r) = (k− k′) · r = Q · r

r θ

k

k

Q


|Q| = 2 |k| sinθ =4π

λsinθ


Atomic form factor

The volume element at r contributes −r0ρ(r)d3r with phase factor e iQ·r

for an entire atom, integrate to get the atomic form factor f 0(Q):

−r0f 0(Q) = −r0∫ρ(r)e iQ·rd3r

Electrons which are tightly bound cannot respond like a free electron. Thisresults in a depression of the atomic form factor, called f ′ and a lossy termnear an ionization energy, called f ′′. Together these are the “anomalous”corrections to the atomic form factor.

3000 4000 5000 6000 7000Energy (eV)

f

f

the total atomic scattering factor is

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


Atomic form factor





3000 4000 5000 6000 7000Energy (eV)

f

f


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


Atomic form factor





3000 4000 5000 6000 7000Energy (eV)

f

f


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


Atomic form factor





3000 4000 5000 6000 7000Energy (eV)

f

f


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


Atomic form factor





3000 4000 5000 6000 7000Energy (eV)

f

f


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


Atomic form factor





3000 4000 5000 6000 7000Energy (eV)

f

f


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


Atomic form factor





3000 4000 5000 6000 7000Energy (eV)

f

f


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


Atomic form factor

0 0.1 0.2 0.3 0.4 0.5 0.6

sin(θ)/λ [Å-1

]

0

5

10

15

20

f [el

ectr

ons]

The atomic form factor hasan angular dependence

Q =4π

λsin θ

lighter atoms have a broaderform factor

forward scattering countselectrons, f (0) = Z

O, Cl,

Cl−, Ar


Atomic form factor

0 0.1 0.2 0.3 0.4 0.5 0.6

sin(θ)/λ [Å-1

]

0

5

10

15

20

f [el

ectr

ons]


Q =4π

λsin θ



O, Cl,

Cl−, Ar


Atomic form factor

0 0.1 0.2 0.3 0.4 0.5 0.6

sin(θ)/λ [Å-1

]

0

5

10

15

20

f [el

ectr

ons]


Q =4π

λsin θ



O, Cl,

Cl−, Ar


Atomic form factor

0 0.1 0.2 0.3 0.4 0.5 0.6

sin(θ)/λ [Å-1

]

0

5

10

15

20

f [el

ectr

ons]


Q =4π

λsin θ



O, Cl,

Cl−, Ar


Atomic form factor

0 0.1 0.2 0.3 0.4 0.5 0.6

sin(θ)/λ [Å-1

]

0

5

10

15

20

f [el

ectr

ons]


Q =4π

λsin θ



O, Cl,

Cl−, Ar


Atomic form factor

0 0.1 0.2 0.3 0.4 0.5 0.6

sin(θ)/λ [Å-1

]

0

5

10

15

20

f [el

ectr

ons]


Q =4π

λsin θ



O, Cl, Cl−,

Ar


Atomic form factor

0 0.1 0.2 0.3 0.4 0.5 0.6

sin(θ)/λ [Å-1

]

0

5

10

15

20

f [el

ectr

ons]


Q =4π

λsin θ



O, Cl, Cl−, Ar


Scattering from atoms: all effects

Scattering from an atom is built up from component quantities:

Thomson scattering from a single electron

atomic form factor

anomalous scattering terms

polarization factor

−r0 = − e2

4πε0mc2

f 0(Q) =

∫ρ(r)e iQ·rd3r

f ′(~ω) + if ′′(~ω)

P =

1

sin2 Ψ12(1 + sin2 Ψ)

− r0f (Q, ~ω) sin2 Ψ = −r0[f 0(Q)

+ f ′(~ω) + if ′′(~ω)

]sin2 Ψ





atomic form factor


polarization factor

−r0 = − e2

4πε0mc2

f 0(Q) =

∫ρ(r)e iQ·rd3r

f ′(~ω) + if ′′(~ω)

P =

1


− r0

f (Q, ~ω) sin2 Ψ

= −r0

[f 0(Q)

+ f ′(~ω) + if ′′(~ω)

]sin2 Ψ





atomic form factor


polarization factor

−r0 = − e2

4πε0mc2

f 0(Q) =

∫ρ(r)e iQ·rd3r

f ′(~ω) + if ′′(~ω)

P =

1


− r0f (Q, ~ω)

sin2 Ψ

= −r0[f 0(Q)

+ f ′(~ω) + if ′′(~ω)

]

sin2 Ψ





atomic form factor


polarization factor

−r0 = − e2

4πε0mc2

f 0(Q) =

∫ρ(r)e iQ·rd3r

f ′(~ω) + if ′′(~ω)

P =

1


− r0f (Q, ~ω)

sin2 Ψ

= −r0[f 0(Q) + f ′(~ω) + if ′′(~ω)

]

sin2 Ψ





atomic form factor


polarization factor

−r0 = − e2

4πε0mc2

f 0(Q) =

∫ρ(r)e iQ·rd3r

f ′(~ω) + if ′′(~ω)

P =

1


− r0f (Q, ~ω) sin2 Ψ = −r0[f 0(Q) + f ′(~ω) + if ′′(~ω)

]sin2 Ψ


Scattering from molecules

Recall for a single atom we have a form factor

f (Q) = f 0(Q) + f ′(~ω) + if ′′(~ω)

extending to a molecule . . .

Fmolecule(Q) =∑j

fj(Q)e iQ·rj

Fmolecule(Q) = f1(Q)e iQ·r1 + f2(Q)e iQ·r2 + f3(Q)e iQ·r3




f (Q) = f 0(Q) + f ′(~ω) + if ′′(~ω)


Fmolecule(Q) =∑j

fj(Q)e iQ·rj





f (Q) = f 0(Q) + f ′(~ω) + if ′′(~ω)


Fmolecule(Q) =∑j

fj(Q)e iQ·rj





f (Q) = f 0(Q) + f ′(~ω) + if ′′(~ω)


Fmolecule(Q) =∑j

fj(Q)e iQ·rj





f (Q) = f 0(Q) + f ′(~ω) + if ′′(~ω)


Fmolecule(Q) =∑j

fj(Q)e iQ·rj





f (Q) = f 0(Q) + f ′(~ω) + if ′′(~ω)


Fmolecule(Q) =∑j

fj(Q)e iQ·rj

Fmolecule(Q) =

f1(Q)e iQ·r1 + f2(Q)e iQ·r2 + f3(Q)e iQ·r3




f (Q) = f 0(Q) + f ′(~ω) + if ′′(~ω)


Fmolecule(Q) =∑j

fj(Q)e iQ·rj

Fmolecule(Q) = f1(Q)e iQ·r1 +

f2(Q)e iQ·r2 + f3(Q)e iQ·r3




f (Q) = f 0(Q) + f ′(~ω) + if ′′(~ω)


Fmolecule(Q) =∑j

fj(Q)e iQ·rj

Fmolecule(Q) = f1(Q)e iQ·r1 + f2(Q)e iQ·r2 +

f3(Q)e iQ·r3




f (Q) = f 0(Q) + f ′(~ω) + if ′′(~ω)


Fmolecule(Q) =∑j

fj(Q)e iQ·rj



Scattering from a crystal

and similarly, to a crystal lattice ...

. . . which is simply a periodic array of molecules

F crystal(Q) = FmoleculeF lattice

F crystal(Q) =∑j

fj(Q)e iQ·rj∑n

e iQ·Rn

The lattice term,∑

e iQ·Rn , is a sum over a large number so it is alwayssmall unless Q · Rn = 2πm where Rn = n1a1 + n2a2 + n3a3 is a real spacelattice vector and m is an integer.






F crystal(Q) =∑j

fj(Q)e iQ·rj∑n

e iQ·Rn








F crystal(Q) =∑j

fj(Q)e iQ·rj∑n

e iQ·Rn








F crystal(Q) =∑j

fj(Q)e iQ·rj∑n

e iQ·Rn








F crystal(Q) =∑j

fj(Q)e iQ·rj∑n

e iQ·Rn


e iQ·Rn , is a sum over a large number

so it is alwayssmall unless Q · Rn = 2πm where Rn = n1a1 + n2a2 + n3a3 is a real spacelattice vector and m is an integer.






F crystal(Q) =∑j

fj(Q)e iQ·rj∑n

e iQ·Rn




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PHYS 570 - Introduction to Synchrotron Radiationphys.iit.edu/~segre/phys570/20S/lecture_01.pdfUnderstand the physics behind a variety of experimental techniques Be able to make an