Today’s Outline

We finish introducing the interactions of x-rays with matter and proceedto a discussion of x-ray sources.

Refraction and reflection of x-rays

Magnetic interactions of x-rays

Coherence of x-ray sources

Introduction to x-ray sources

• The x-ray tube

• The synchrotron

• Bending magnet source

C. Segre (IIT) PHYS 570 - Spring 2012 January 17, 2012 1 / 17

• The x-ray tube

• The synchrotron

• The x-ray tube

• The synchrotron

• The x-ray tube

• The synchrotron

• The x-ray tube

• The synchrotron

• The x-ray tube

• The synchrotron

• The x-ray tube

• The synchrotron

• The x-ray tube

• The synchrotron

• The x-ray tube

• The synchrotron

Refraction of X-rays

X-rays can be treated like light when interaction with a medium. However,unlike visible light, the index of refraction of x-rays in matter is very closeto unity:

n = 1− δ + iβ

with δ ∼ 10−5

Snell’s Law

cosα = n cosα′

where α′ < α unlike for visible light

n = 1− δ + iβ

with δ ∼ 10−5

Snell’s Law

cosα = n cosα′

n = 1− δ + iβ

with δ ∼ 10−5

Snell’s Law

cosα = n cosα′

n = 1− δ + iβ

with δ ∼ 10−5

Snell’s Law

cosα = n cosα′

n = 1− δ + iβ

with δ ∼ 10−5

Snell’s Law

cosα = n cosα′

Reflection of X-rays

Because n < 1, at a critical angle αc , we no longer have refraction but

total external reflection

Since α′ = 0 when α = αc

n = cosαc

n ≈ 1− α2c

1− δ + iβ ≈ 1− α2c

δ =α2c

2−→ αc =

√2δ

Because n < 1, at a critical angle αc , we no longer have refraction buttotal external reflection

n = cosαc

n ≈ 1− α2c

1− δ + iβ ≈ 1− α2c

δ =α2c

2−→ αc =

√2δ

n = cosαc

n ≈ 1− α2c

1− δ + iβ ≈ 1− α2c

δ =α2c

2−→ αc =

√2δ

n = cosαc

n ≈ 1− α2c

1− δ + iβ ≈ 1− α2c

δ =α2c

2−→ αc =

√2δ

n = cosαc

n ≈ 1− α2c

1− δ + iβ ≈ 1− α2c

δ =α2c

2−→ αc =

√2δ

n = cosαc

n ≈ 1− α2c

1− δ + iβ ≈ 1− α2c

δ =α2c

2−→ αc =

√2δ

n = cosαc

n ≈ 1− α2c

1− δ + iβ ≈ 1− α2c

δ =α2c

2−→ αc =

√2δ

Uses of Total External Reflection

X-ray mirrors

• harmonic rejection

• focusing & collimation

Evanscent wave experiments

• studies of surfaces

• depth profiling

X-ray mirrors

• depth profiling

X-ray mirrors

• depth profiling

X-ray mirrors

• depth profiling

X-ray mirrors

• depth profiling

X-ray mirrors

• depth profiling

Magnetic Interactions

We have focused on the interaction of x-rays and charged particles.However, electromagnetic radiation also consists of a traveling mag-netic field. In principle, this means it should interact with magneticmaterials as well.

Indeed, x-rays do interact with magnetic materials (and electrons whichhave magnetic moment and spin) but the strength of the interactionis comparatively weak.

Amagnetic

Acharge=

~ωmc2

=5.11× 103 eV

0.511× 106 eV= 0.01

For an x-ray of energy 5.11 keV, interacting with an electron with mass0.511 MeV. Only with the advent of synchrotron radiation sources hasmagnetic x-ray scattering become a practical experimental technique.

Amagnetic

Acharge=

~ωmc2

=5.11× 103 eV

0.511× 106 eV= 0.01

Amagnetic

Acharge=

~ωmc2

=5.11× 103 eV

0.511× 106 eV= 0.01

Amagnetic

Acharge=

~ωmc2

=5.11× 103 eV

0.511× 106 eV= 0.01

For an x-ray of energy 5.11 keV, interacting with an electron with mass0.511 MeV.

Only with the advent of synchrotron radiation sources hasmagnetic x-ray scattering become a practical experimental technique.

Amagnetic

Acharge=

~ωmc2

=5.11× 103 eV

0.511× 106 eV

= 0.01

Amagnetic

Acharge=

~ωmc2

=5.11× 103 eV

0.511× 106 eV= 0.01

Amagnetic

Acharge=

~ωmc2

=5.11× 103 eV

0.511× 106 eV= 0.01

What is coherence?

So far, in our discussion, we have assumed that x-rays are“plane waves”. What does this really mean?

A plane wave has perfect coherence (like a laser).

Real x-rays are not perfect plane waves in two ways:

• they are not perfectly monochromatic

• they do not travel in a perfectly co-linear direction

Because of these imperfections the “coherence length” of anx-ray beam is finite and we can calculate it.

What is coherence?

Longitudinal Coherence

Definition: Distance over which two waves from the same source pointwith slightly different wavelengths will completely dephase.

λ−∆λ

Two waves of slightly different wavelengthsλ and λ−∆λ are emitted from the samepoint in space simultaneously.

After a distance LL, the two waves will beexactly out of phase and after 2LL they willonce again be in phase.

2LL = Nλ2LL = (N + 1)(λ−∆λ)

Nλ = Nλ+ λ− N∆λ−∆λ

0 = λ−N∆λ−∆λ −→ λ = (N + 1)∆λ −→ N ≈ λ

∆λ−→ LL =

2∆λ

λ−∆λ

2LL = Nλ2LL = (N + 1)(λ−∆λ)

0 = λ−N∆λ−∆λ −→ λ = (N + 1)∆λ −→ N ≈ λ

∆λ−→ LL =

2∆λ

λ−∆λ

2LL = Nλ2LL = (N + 1)(λ−∆λ)

0 = λ−N∆λ−∆λ −→ λ = (N + 1)∆λ −→ N ≈ λ

∆λ−→ LL =

2∆λ

λ−∆λ

2LL = Nλ

2LL = (N + 1)(λ−∆λ)

0 = λ−N∆λ−∆λ −→ λ = (N + 1)∆λ −→ N ≈ λ

∆λ−→ LL =

2∆λ

λ−∆λ

2LL = Nλ2LL = (N + 1)(λ−∆λ)

0 = λ−N∆λ−∆λ −→ λ = (N + 1)∆λ −→ N ≈ λ

∆λ−→ LL =

2∆λ

λ−∆λ

2LL = Nλ2LL = (N + 1)(λ−∆λ)

0 = λ−N∆λ−∆λ −→ λ = (N + 1)∆λ −→ N ≈ λ

∆λ−→ LL =

2∆λ

λ−∆λ

2LL = Nλ2LL = (N + 1)(λ−∆λ)

0 = λ−N∆λ−∆λ

−→ λ = (N + 1)∆λ −→ N ≈ λ

∆λ−→ LL =

2∆λ

λ−∆λ

2LL = Nλ2LL = (N + 1)(λ−∆λ)

0 = λ−N∆λ−∆λ −→ λ = (N + 1)∆λ

−→ N ≈ λ

∆λ−→ LL =

2∆λ

λ−∆λ

2LL = Nλ2LL = (N + 1)(λ−∆λ)

0 = λ−N∆λ−∆λ −→ λ = (N + 1)∆λ −→ N ≈ λ

−→ LL =λ2

2∆λ

λ−∆λ

2LL = Nλ2LL = (N + 1)(λ−∆λ)

0 = λ−N∆λ−∆λ −→ λ = (N + 1)∆λ −→ N ≈ λ

∆λ−→ LL =

2∆λ

Transverse Coherence

Definition: The lateral distance along a wavefront over which there is acomplete dephasing between two waves, of the same wavelength, whichoriginate from two separate points in space.

If we assume that the two wavesoriginate from points with a smallangular separation ∆θ, Thetransverse coherence length is givenby:

2LT= tan ∆θ ≈ ∆θ

R= tan ∆θ ≈ ∆θ

LT =λR

2LT= tan ∆θ

≈ ∆θ

R= tan ∆θ

≈ ∆θ

LT =λR

Coherence Lengths at the APS

For a typical 3rd generation undulator source, such as at the AdvancedPhoton Source the vertical source size is D = 100µm and we are typicallyR = 50m away with our experiment. If we assume a typical wavelength ofλ = 1A, and a monochromator resolution of ∆λ/λ = 10−5 we have for thevertical direction:

LL =λ2

2∆λ

2· λ

∆λ=

1× 10−10

2 · 10−5= 5µm

LT =λR

=(1× 10−10) · 50

2 · (100× 10−6)= 25µm

LL =λ2

2∆λ

2· λ

∆λ=

1× 10−10

2 · 10−5= 5µm

LT =λR

=(1× 10−10) · 50

2 · (100× 10−6)= 25µm

LL =λ2

2∆λ=λ

2· λ

=1× 10−10

2 · 10−5= 5µm

LT =λR

=(1× 10−10) · 50

2 · (100× 10−6)= 25µm

LL =λ2

2∆λ=λ

2· λ

∆λ=

1× 10−10

2 · 10−5

= 5µm

LT =λR

=(1× 10−10) · 50

2 · (100× 10−6)= 25µm

LL =λ2

2∆λ=λ

2· λ

∆λ=

1× 10−10

2 · 10−5= 5µm

LT =λR

=(1× 10−10) · 50

2 · (100× 10−6)= 25µm

LL =λ2

2∆λ=λ

2· λ

∆λ=

1× 10−10

2 · 10−5= 5µm

LT =λR

=(1× 10−10) · 50

2 · (100× 10−6)= 25µm

LL =λ2

2∆λ=λ

2· λ

∆λ=

1× 10−10

2 · 10−5= 5µm

LT =λR

(1× 10−10) · 50

2 · (100× 10−6)

= 25µm

LL =λ2

2∆λ=λ

2· λ

∆λ=

1× 10−10

2 · 10−5= 5µm

LT =λR

(1× 10−10) · 50

2 · (100× 10−6)= 25µm

X-Ray Tube Schematics

Fixed anode tube

Rotating anode tube

• low power

• low maintenance

• high power

• high maintenance

X-Ray Tube Schematics

Fixed anode tube Rotating anode tube

• low power

• low maintenance

• high power

• high maintenance

X-Ray Tube Spectrum

• Minimum wavelength(maximum energy) setby acceleratingpotential

• Bremßtrahlungradiation providessmooth background(charged particledeceleration)

• Highest intensity at the characteristic fluorescence emission energy ofthe anode material

• Unpolarized, incoherent x-rays emitted in all directions from anodesurface, must be collimated with slits

X-Ray Tube Spectrum

Synchrotron Sources

Bending magnet

• Wide horizontal beam

• Broad spectrum to highenergies

Undulator

• Highly collimated beam

• Highly peaked spectrumwith harmonics

Synchrotron Sources

Bending magnet

Undulator

Synchrotron Sources

Bending magnet

Undulator

Synchrotron Sources

Bending magnet

Undulator

Synchrotron Sources

Bending magnet

Undulator

Synchrotron Sources

Bending magnet

Undulator

Bending Magnet Spectra

0 20000 40000 60000 80000 1e+050

Lower energy sources, such as NSLS have lower peak energy and higherintensity at the peak.Higher energy sources, such as APS have higher energy spectrum and areonly off by a factor of 2 intensity at low energy.

0 20000 40000 60000 80000 1e+050

Lower energy sources, such as NSLS have lower peak energy and higherintensity at the peak.

Higher energy sources, such as APS have higher energy spectrum and areonly off by a factor of 2 intensity at low energy.

0 20000 40000 60000 80000 1e+050

Lower energy sources, such as NSLS have lower peak energy and higherintensity at the peak.Higher energy sources, such as APS have higher energy spectrum and areonly off by a factor of 2 intensity at low energy.

100 1000 10000 1e+051e+05

Logarithmic scale shows clearly how much more energetic and intense thebending magnet sources at the 6 GeV and 7 GeV sources are.

100 1000 10000 1e+051e+05

Logarithmic scale shows clearly how much more energetic and intense thebending magnet sources at the 6 GeV and 7 GeV sources are.

Review of Special Relativity

1− β2

E = γmc2

√1− 1

γ2−→ β ≈ 1− 1

use binomial expansion since 1/γ2 << 1

Let’s calculate these quantitiesfor an electron at NSLS andAPS

me = 0.511 MeV/c2

NSLS: E = 1.5 GeV

γ =1.5× 109

0.511× 106= 2935

APS: E = 7.0 GeV

γ =7.0× 109

0.511× 106= 13700

1− β2

E = γmc2

√1− 1

γ2−→ β ≈ 1− 1

me = 0.511 MeV/c2

NSLS: E = 1.5 GeV

γ =1.5× 109

0.511× 106= 2935

APS: E = 7.0 GeV

γ =7.0× 109

0.511× 106= 13700

1− β2

E = γmc2

√1− 1

γ2−→ β ≈ 1− 1

me = 0.511 MeV/c2

NSLS: E = 1.5 GeV

γ =1.5× 109

0.511× 106= 2935

APS: E = 7.0 GeV

γ =7.0× 109

0.511× 106= 13700

1− β2

E = γmc2

√1− 1

γ2−→ β ≈ 1− 1

me = 0.511 MeV/c2

NSLS: E = 1.5 GeV

γ =1.5× 109

0.511× 106= 2935

APS: E = 7.0 GeV

γ =7.0× 109

0.511× 106= 13700

1− β2

E = γmc2

√1− 1

−→ β ≈ 1− 1

me = 0.511 MeV/c2

NSLS: E = 1.5 GeV

γ =1.5× 109

0.511× 106= 2935

APS: E = 7.0 GeV

γ =7.0× 109

0.511× 106= 13700

1− β2

E = γmc2

√1− 1

γ2−→ β ≈ 1− 1

me = 0.511 MeV/c2

NSLS: E = 1.5 GeV

γ =1.5× 109

0.511× 106= 2935

APS: E = 7.0 GeV

γ =7.0× 109

0.511× 106= 13700

1− β2

E = γmc2

√1− 1

γ2−→ β ≈ 1− 1

me = 0.511 MeV/c2

NSLS: E = 1.5 GeV

γ =1.5× 109

0.511× 106= 2935

APS: E = 7.0 GeV

γ =7.0× 109

0.511× 106= 13700

1− β2

E = γmc2

√1− 1

γ2−→ β ≈ 1− 1

me = 0.511 MeV/c2

NSLS: E = 1.5 GeV

γ =1.5× 109

0.511× 106= 2935

APS: E = 7.0 GeV

γ =7.0× 109

0.511× 106= 13700

1− β2

E = γmc2

√1− 1

γ2−→ β ≈ 1− 1

me = 0.511 MeV/c2

NSLS: E = 1.5 GeV

γ =1.5× 109

0.511× 106= 2935

APS: E = 7.0 GeV

γ =7.0× 109

0.511× 106= 13700

1− β2

E = γmc2

√1− 1

γ2−→ β ≈ 1− 1

me = 0.511 MeV/c2

NSLS: E = 1.5 GeV

γ =1.5× 109

0.511× 106= 2935

APS: E = 7.0 GeV

γ =7.0× 109

0.511× 106= 13700

“Headlight” Effect

In electron rest frame:

emission is symmetric about theaxis of the acceleration vector

In lab frame:

emission is pushed into the direc-tion of motion of the electron

“Headlight” Effect

In electron rest frame:

emission is symmetric about theaxis of the acceleration vector

In lab frame:

emission is pushed into the direc-tion of motion of the electron

Relativistic Emission

the electron is in constant trans-verse acceleration due to theLorentz force from the magneticfield of the bending magnet

~F = e~v × ~B = me~a

the aperture angle of the radiationcone is 1/γ

the angular frequency of the elec-tron in the ring is ωo ≈ 106 andthe cutoff energy for emission is

Emax ≈ γ3ωo

for the APS, with γ ≈ 104 we have

Emax ≈ (104)3 · 106 = 1018

~F = e~v × ~B = me~a

Emax ≈ γ3ωo

Emax ≈ (104)3 · 106 = 1018

~F = e~v × ~B = me~a

the angular frequency of the elec-tron in the ring is ωo ≈ 106

andthe cutoff energy for emission is

Emax ≈ γ3ωo

Emax ≈ (104)3 · 106 = 1018

~F = e~v × ~B = me~a

Emax ≈ γ3ωo

Emax ≈ (104)3 · 106 = 1018

~F = e~v × ~B = me~a

Emax ≈ γ3ωo

Emax ≈ (104)3 · 106 = 1018

Today’s Outline - January 17, 2012

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