2 2 0.1%bw.)] mm [Phot./(sec mrad

(1)

School on Pulsed Neutrons - October 2005 - ICTP Trieste

Complementary accelerator generated probes for materials science

Synchrotron Radiation

Prof. Dr. Thomas Brückel

Institute for Scattering Methods

Institute for Solid State Research

(2)

Outline

• Why x-rays ?

• Laboratory x-ray sources

• Synchrotron radiation sources

• The source layout

• Special theory of relativity

• Properties of Synchrotron Radiation

• Insertion devices and free electron lasers

• Summary

(3)

X-Ray and Neutron Penetration

Sword Swallower

(4)

X-Ray and Neutron Penetration

Sword Swallower

(5)

X-Ray and Neutron Penetration

Sword Swallower

Neutron Tomography Disk Drive

(6)

X-Ray and Neutron Penetration

Sword Swallower

Neutron Tomography Disk Drive

Nanostructures:

No adequate lenses → scattering!

(7)

X-Ray and Neutron Cross Sections

0.66

H

^1.76

24

C

^5.55

416

Mn

^1.75

450

Fe

^11.22

522

Ni

^13.30

1408

Pd

^4.39

2986

Ho

^8.06

5631

U

^8.90

x10^-1

1 2

58 60 62

x-ray neutrons

1 6 25 26 28

46 67

92

element

Z

1 barn = 10

^-28

m

²

σ_coh [barn] σ_coh [barn]

(8)

X-Ray Scattering: Structure

(9)

X-Ray Scattering: Structure

(10)

Outline

• Why x-rays ?

• Laboratory x-ray sources

• Synchrotron radiation sources

• The source layout

• Special theory of reflectivity

• Properties of Synchrotron Radiation

• Insertion devices and free electron lasers

• Summary

(11)

Conventional X-Ray Generators

Bombardement of target (anode) material, e.g. Cu, by high energy electrons

Cu Cu Cu Cu Cu Cu Cu Cu Cu Cu

Cu Cu Cu Cu

e

^-

γ

1. Bremsstrahlung

(V: accelerating voltage)

] [

4 . ] 12

min

[

kV Å ≈ V

λ

min max

λ hc

V e E

=

⋅

=

(12)

White Radiation Spectrum

Total conversion of e

^-

energy at λ

_min

unprobable

→ highest intensity at ≈ 1.5 × λ

_min

Total intensity:

I

_white

= A ⋅ I ⋅ Z ⋅ V

ⁿ

constant current atomic number

accelerating voltage

n ≈ 2

(13)

O N

M

L

K

continuum

Characteristic Spectrum

Flourescence:

(14)

O N

M

L

K

continuum

Characteristic Spectrum

Flourescence:

e-

(15)

O N

M

L

K

continuum

Characteristic Spectrum

Flourescence:

e-

K-series α

β γ M

L

K

(16)

O N

M

L

K

continuum

Characteristic Spectrum

Flourescence:

e-

K-series α

β γ M

L

K

(17)

O N

M

L

K

continuum

Characteristic Spectrum

Flourescence:

e-

K-series α

β γ M

L

K

L-series α

β γ

(18)

O N

M

L

K

continuum

Characteristic Spectrum

Flourescence:

e-

K-series α

β γ M

L

K

L-series α

β γ

K α₁ α₂

n = 1, l = 0, j = ½; 1S_1/2

n = 2, l = 1, j = ³/₂; 2P_3/2 (L_III) n = 2, l = 1, j = ½; 2P_1/2 (L_II) n = 2, l = 0, j = ½; 2S_1/2 (L_I)

(19)

O N

M

L

K

continuum

Characteristic Spectrum

Flourescence:

e-

K-series α

β γ M

L

K

L-series α

β γ

K α₁ α₂

n = 1, l = 0, j = ½; 1S_1/2

Nomenclature for x-ray spectra (fluorescence):

with L = 0, 1, 2, … ≡ S, P, D, … K, L, M, N, O ≡ n = 1, 2, 3, 4, 5, … α, β, γ, … ≡ from n + 1, n + 2, n + 3, … shell Selection rule: ∆L = ± 1, ∆J = 0, ± 1

J S ¹

L

2 +

(20)

O N

M

L

K

continuum

Characteristic Spectrum

Flourescence:

e-

K-series α

β γ M

L

K

L-series α

β γ

K α₁ α₂

n = 1, l = 0, j = ½; 1S_1/2

Nomenclature for x-ray spectra (fluorescence):

with L = 0, 1, 2, … ≡ S, P, D, … K, L, M, N, O ≡ n = 1, 2, 3, 4, 5, … α, β, γ, … ≡ from n + 1, n + 2, n + 3, … shell Selection rule: ∆L = ± 1, ∆J = 0, ± 1

J S ¹

L

2 +

Intensity of K-line:

I_K = B ⋅ I (V - V_K)^1.5

(21)

Total Spectrum

Example: Cu

max (I

_K

/ I

_white

) for V ≈ 4 ⋅ V

_K

⇒ I

_Kα

≈ 90 ∗ I

_white

(λ

_Kα

) I

_Kα1

≈ 2 ⋅ I

_Kα2

I

_Kα

≈ 5 ⋅ I

_Kβ

Example: E [keV] K_α1 K_α2 K_β1 Cu 8.04778 8.02783 8.90529 Mo 17.47934 17.3743 19.6083

(22)

Sealed-Tube and Rotating-Anode Generators

Operation at ≈ 4 ⋅ V_K

Problem: ≈ 0.1 % of power → x-rays; 99.9 %: heat!

Sealed x-ray tube:

(23)

Sealed-Tube and Rotating-Anode Generators

Operation at ≈ 4 ⋅ V_K

Problem: ≈ 0.1 % of power → x-rays; 99.9 %: heat!

Sealed x-ray tube: Rotating anode:

(24)

Focal spot:

(typical)

Example: 6 kW Rotating Anode for Cu-K_α (≈8 keV)

⇒ V ≈ 32 kV ; I ≈ 0.18 mA 2 ÷ 6 mm

0.3 ÷ 1 mm

"line focus"

Be Be

BeBe

"point focus"

Focus

(25)

Outline

• Why x-rays ?

• Laboratory x-ray sources

• Synchrotron radiation sources

• The source layout

• Special theory of relativity

• Properties of Synchrotron Radiation

• Insertion devices and free electron lasers

• Summary

(26)

X-Ray Sources

10⁶ 10¹⁰ 10¹⁴ 10¹⁸ 10²² 10²⁶

1900 1950 2000

[Phot./(sec mrad2 mm2 0.1%bw.)]

Years

Average Brilliance

x-ray tubes 1 st generation

2 nd generation 3 rd generation

(ESRF)

XFEL

(27)

X-Ray Sources

10⁶ 10¹⁰ 10¹⁴ 10¹⁸ 10²² 10²⁶

1900 1950 2000

[Phot./(sec mrad2 mm2 0.1%bw.)]

Years

Average Brilliance

x-ray tubes 1 st generation

2 nd generation 3 rd generation

(ESRF)

XFEL

Brilliance increases faster than Moor's Law (packing density in information electronics increases exponentially with time):

Moor's

Law

(28)

Synchrotron Radiation Sources World Wide

... their number is increasing even faster ...

(29)

Synchrotron Radiation Sources World Wide

(30)

Synchrotron Radiation Sources World Wide

(31)

Synchrotron Radiation Sources World Wide

(32)

50 m

5 mm properties calculable

small source

size wiggler clean ultra-high

vacuum source

time structure

intense continuous spectrum highly collimated

undulators

.

. polarised

Synchrotron X-Ray Sources

(33)

Examples: 3 ^rd Generation X-Ray Sources

ESRF:

6 GeV

(34)

Examples: 3 ^rd Generation X-Ray Sources

APS:

7 GeV

(35)

Examples: 3 ^rd Generation X-Ray Sources

SPRING8:

8 GeV

(36)

Outline

• Why x-rays ?

• Laboratory x-ray sources

• Synchrotron radiation sources

• The source layout

• Special theory of relativity

• Properties of Synchrotron Radiation

• Insertion devices and free electron lasers

• Summary

(37)

General Source Layout

Example: APS

(38)

LINAC

• Electrostatic acceleration:

(39)

LINAC

- limited to a few MeV by discharges

(40)

LINAC

• Solution: high frequency cavities fed from clystrons

" e- are surf-riding on the electric HF-field (e. g. 500 MHz)"

(41)

LINAC

• Solution: high frequency cavities fed from clystrons

" e- are surf-riding on the electric HF-field (e. g. 500 MHz)"

• LINAC: Disk-loaded structures with defined wavelength

(taken from Wille)

(42)

LINAC

(taken from Wille)

• Single cell cavity in storage ring:

⇒ bunch structure!

(43)

Synchrotron

Idea: Avoid long LINAC by "beam-recycling"

l centripeda 2

Lorentz n F

R m v B

v e

F = × = ⋅ =

ecB R E

B ec

vmc B

e

R mv ⎯ ^v ⎯ → ⎯ ^c =

= ⋅

⇒ ^→

2 2

In practical units:

[T]

B 0.3

[GeV]

R[m] = E

Example: APS: E = 7 GeV; B = 0.6 T → R = 39 m

Synchrotron: R fix → increase B synchronously with E ( • compare: cyclotron)

LINAC

B

(44)

Storage Ring

= "synchrotron with constant beam energy"

Beam orbit = ideal particle track;

vertical field component transverse to orbits:

B_z (x) = a + bx + cx2 + dx3 + ...

Dipole Quadrupole Sextupol Octupole

Dipole: bending of orbit Quadrupole: focusing

Sextupole: correction of chromaticity (∆p/p)

Layout of one Sector at APS

(45)

Magnets

Dipole

Quadrupole

Sextupole

(46)

Beta-Function

Orbit = idealised, stable trajectory particle movement (fluctuation) is a - "transverse" ∆x (Betatron-oscillation) and

- "longitudinal" ∆p → ∆x (Synchrotron-oscillation) oscillation around stable orbit:

x'' (s) - b (s) ⋅ x (s) = 0

Envelope = max. amplitude of all particles described by Betafunction

) ( )

( s s

E = ε ⋅ β

(47)

Emittance

All particle trajectories lie within a phase space ellipse, e. g. along x:

2 / ) ( ' )

( s β s

α = −

) (

) ( ) 1

(

2

s s s

β γ = ⁺ α

Liouville's Theorem: constant area

F = π ⋅ ε

ε = Emittance; characterises particle beam quality third generation sources: small ε !

Significance of beta-function:

Wiggler: small source size desirable → small β

undulator: small divergence desirable → small β' → high β

(48)

Some Parameters:

Electron Gun

LINAC:

200 MeV electron linac e- _→ W-target foil → e+

(Bremsstrahlung pair production) 450 MeV positron linac

Booster-Synchrotron:

368 m circumference

650 MeV → 7 GeV

(49)

Parameters APS Storage Ring

• Vacuum pressure ≈ 10^-9 mbar

(50)

Parameters APS Storage Ring

• Beam life time ≈ 70 h

(gas scattering ≈ 80 h

Touschek ≈ 190 h)

(51)

Parameters APS Storage Ring

Touschek ≈ 190 h)

• Filling time ≈ 1 min

(52)

Parameters APS Storage Ring

Touschek ≈ 190 h)

• Circumference 1104 m

(Diameter ≈ 350 m)

(53)

Parameters APS Storage Ring

Touschek ≈ 190 h)

• Revolution time ≈ 3.7 µs

(speed of light (≈ 3 ⋅ 10⁸ m / s) 271 000 times / s)

=ˆ

(54)

Parameters APS Storage Ring

Touschek ≈ 190 h)

=ˆ

• Nominal energy 7 GeV

(55)

Parameters APS Storage Ring

Touschek ≈ 190 h)

=ˆ

• Dipole bending radius ≈ 39 m

(56)

Parameters APS Storage Ring

Touschek ≈ 190 h)

=ˆ

• Dipole field 0.6 T

(57)

Parameters APS Storage Ring

Touschek ≈ 190 h)

=ˆ

• Critical energy (dipole) 19.5 keV

(58)

Parameters APS Storage Ring

Touschek ≈ 190 h)

=ˆ

• Nominal current (multibunch) 100 mA

(59)

Parameters APS Storage Ring

Touschek ≈ 190 h)

=ˆ

• Bunch length (rms, natural): 5.3 mm ( ⇒ FWHM) ≈ 35 ps

(60)

Parameters APS Storage Ring

Touschek ≈ 190 h)

=ˆ

• Bunch length (rms, natural): 5.3 mm ( ⇒ FWHM) ≈ 35 ps

• Beam size (rms) ≈ 300 µm (H) x 90 µm (V)

(61)

Parameters APS Storage Ring

(62)

Parameters APS Storage Ring

• Beam divergence (rms) ≈ 23 µrad (H) x 9 µrad (V)

(63)

Parameters APS Storage Ring

• Beam emittance: 7.5 nmrad (H) x 0.75 nmrad (V)

(compare: ESRF ≈ 3 nmrad

DORIS III ≈ 415 nmrad)

(64)

Parameters APS Storage Ring

• Max. insertion device length: 5.2 m

(65)

Parameters APS Storage Ring

• Insertion device vacuum chamber aperture: 12 mm

(66)

Parameters APS Storage Ring

• Number of sectors: 40

(67)

Parameters APS Storage Ring

• Max. number of insertion device and BM beamlines: 35

(68)

Parameters APS Storage Ring

• Energy loss per turn:

bending magnet 5.45 MeV

insertion devices 1.25 MeV

total 6.9 MeV

(69)

Parameters APS Storage Ring

total 6.9 MeV

• Source power (@7 GeV, 100 mA): 1.3 MW

(70)

Parameters APS Storage Ring

total 6.9 MeV

• Source power (@7 GeV, 100 mA): 1.3 MW

• Radio frequency 352 MHz

(71)

Highly Relativistic!

Lifetime

(72)

Highly Relativistic!

Lifetime

• Lifetime: ESRF ≈ 50h → s=v·t = 3·10⁸m/s · 50·3600s = 5.4·10¹⁰km Distance Earth – Sun: 1.5·10⁸km

⇒ during the beam life time the electrons cover a distance of about 400 times the distance to the sun without collisions with gas molecules!

(73)

Highly Relativistic!

Lifetime

Mass

(74)

Highly Relativistic!

Lifetime

Mass

• e^- -mass: ESRF 6 GeV → γ = 6·10⁹eV / 511·10³eV ≈ 12000 ⇒ m=γ·m₀

Proton mass: m_p=1836·m_e ; ⇒ m_e(6GeV) ≈ 6.5 m_p(0GeV)

⇒ the moving electron is as heavy as a Li atom!

(75)

In the APS Storage Ring

(76)

Undulator & Front End

(77)

Beamlines

ID 20 @ ESRF MuCAT @ APS

(78)

Outline

• Why x-rays ?

• Laboratory x-ray sources

• Synchrotron radiation sources

• The source layout

• Special theory of relativity

• Properties of Synchrotron Radiation

• Insertion devices and free electron lasers

• Summary

(79)

Excursion: Special Theory of Relativity

Postulates:

1. The same physical laws hold in all reference frames with uniform relative motion

(There is no way to determine velocities on an absolute scale, movements are "relative").

2. The vacuum speed of light has the same isotropic value c in all uniformly moving reference frames = inertial reference frames (Michelson-experiment)

(80)

Relativistic Kinematics

Galilei-transformation:

t ' = t

t v r r ' = −

x

z y

x'

z' y'

v

(81)

Relativistic Kinematics

t ' = t

t v r r ' = −

x

z y

x'

z' y'

v

contradicts postulate 2:

r & ' = r & − v ⇒ c ' = c − v

(82)

Relativistic Kinematics

t ' = t

t v r r ' = −

x

z y

x'

z' y'

v

r & ' = r & − v ⇒ c ' = c − v ' ( ) ) (

' ' '

c z t

t

ct z

z

y y

x x

γ β

β γ

+

⋅

=

+

⋅

=

Lorentz-transformation:

with

c

= v β

and

γ = ( 1 − β

²

)

⁻¹^/²

(83)

Relativistic Kinematics

t ' = t

t v r r ' = −

x

z y

x'

z' y'

v

r & ' = r & − v ⇒ c ' = c − v ' ( ) ) (

' ' '

c z t

t

ct z

z

y y

x x

γ β

β γ

+

⋅

=

+

⋅

=

Lorentz-transformation:

with

c

= v β

and

γ = ( 1 − β

²

)

⁻¹^/²

Important consequences:

Length contraction:

" a moving ruler appears shorter than a stationary one"

γ /

' z

z = ∆

∆

Time dilatation:

∆ t ' = γ ⋅ ∆ t

" a moving clock runs slower than a stationary clock"

with

c

= v β

and

γ = ( 1 − β

²

)

⁻

(84)

Relativistic Dynamics

Newtons form can be kept for the spatial components, if a speed dependent mass m is

introduced (m_o = rest mass): ₂

1 / β

γ = −

= m

_o

m

_o

m

and

dt

p F d

v m

p = ⋅ ; =

⇒ kinetic energy of a relativistic particle:

2 4 2

2 2

2

m c ; E p c m c

mc

E = = γ

_o

= +

_o

⇒ in addition to pure energy due to movement there is a constant rest energy m_oc²:

2 2

0

2 1 m v c

m

E ⎯

_β

⎯ → ⎯

_→ _o

+

_o

(85)

Relativistic Electrodynamics

This formulation of classical electrodynamics is not invariant against Lorentz-transformation!

Example:

stationary frame: static, spatially varying B-field

⇒ moving frame: in addition E-field (law of induction)

Form-invariant formulation of Maxwell equations via the relativistic field tensor , which's components are the components of E and B.

F

_νµ

special case: Lorentz-transformation along z:

) (

'

₁ ₂

1

E B

E = γ − β ) (

'

₂ ₁

2

E B

E = γ − β

3 3

' E E =

) (

'

₁ ₂

1

B E

B = γ − β ) (

'

₂ ₁

2

B E

B = γ − β

3 3

' B

B =

(86)

Outline

• Why x-rays ?

• Laboratory x-ray sources

•Synchrotron radiation sources

• The source layout

• Special theory of relativity

• Properties of Synchrotron Radiation

• Insertion devices and free electron lasers

• Summary

(87)

The Pointingvector

E B S = µ ×

0

1

determines the flow of energy through a unit area per unit time at the position of the observer

Total power emitted by particle of charge e and mass m₀: Result from classical Electrodynamics (see e.g. Jackson):

2 2 3

2

6 ⎟⎟

⎠

⎜⎜ ⎞

⎝

= ⎛

dt p d c

m P e

o o

s

πε

Galilei: emitted power independent of uniform motion; only accelerated movement

"shakes off" the field!

Azimutal angular distribution: Hertz Dipole (radio-antennas):

⎟⎟ Ψ

⎠

⎜⎜ ⎞

⎝

= ⎛ Ω

2 2

2 3 2

2

16 dt sin

p d c

m e d

dP

o o s

ε π

Emitted Power

(88)

Radiation of Accelerated Relativistic Charged Particles

Classical formula not relativistic invariant: change of reference frame changes dt!

→ relativistic form-invariant generalisation:

2 2

1 ; 1 1

c m dt E

d dt

o

− =

=

→ γ β

τ γ

2 3

2

1

1 ; ;

, ,

, E m m E mc

mv c mv

mv p

p

_µ

⎟ =

_o

=

⎠

⎜ ⎞

⎝

= ⎛

→ γ

⇒

( ) ^⎥ ^⎥ _⎦

⎤

⎢ ⎢

⎣

⎡ ⎟

⎠

⎜ ⎞

⎝

− ⎛

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

2 2

2 2 2

2

1 1

6 πε τ d τ

dE c

d p d c

m c

P e

o o s

• same prefactor as classical formula

• However, for relativistic case emission depends on direction of acceleration and on direction of movement!

(89)

limiting cases:

- linear acceleration: E increases with p

→ partial compensation of terms

⇒ radiation losses are not relevant for LINACS - circular acceleration:

→ in the rest frame of the particle, the emission is identical to the classical case Observation in laboratory system of moving particle →

• increase of mass of inertia

• time dilatation For circular movement:

= 0 τ d dE

R E c

m E R

E R

mc c

R p v dt

pd dt

dp d

dp

o

⋅

=

≈

=

γ

2

γ

α γ γ

τ γ

c v ≈

Radiation of Accelerated Relativistic Charged Particles

(90)

Emitted power for circular movement

⇒

( )

²

4 2 4

2

1 6 R

E c

m c

P e

o o

s

⁼ πε

For electrons (positrons), the energy loss per turn amounts in practical units to:

∫ ⁼

=

∆ [ ]

] 5 [

. 88 ]

[

4

m R

GeV dt E

P keV

E

_s

Radiation of Accelerated Relativistic Charged Particles

(91)

⇒

( )

²

4 2 4

2

1 6 R

E c

m c

P e

o o

s

⁼ πε

∫ ⁼

=

∆ [ ]

] 5 [

. 88 ]

[

4

m R

GeV dt E

P keV

E

_s

Radiation of Accelerated Relativistic Charged Particles

Note:

(92)

⇒

( )

²

4 2 4

2

1 6 R

E c

m c

P e

o o

s

⁼ πε

∫ ⁼

=

∆ [ ]

] 5 [

. 88 ]

[

4

m R

GeV dt E

P keV

E

_s

Radiation of Accelerated Relativistic Charged Particles

Note:

• in relativistic case: strong energy dependence of emitted power (~ E⁴)

(93)

⇒

( )

²

4 2 4

2

1 6 R

E c

m c

P e

o o

s

⁼ πε

∫ ⁼

=

∆ [ ]

] 5 [

. 88 ]

[

4

m R

GeV dt E

P keV

E

_s

Radiation of Accelerated Relativistic Charged Particles

Note:

• only e^- and e⁺ are effective for production of SR (compare: COSY):

(94)

⇒

( )

²

4 2 4

2

1 6 R

E c

m c

P e

o o

s

⁼ πε

∫ ⁼

=

∆ [ ]

] 5 [

. 88 ]

[

4

m R

GeV dt E

P keV

E

_s

Radiation of Accelerated Relativistic Charged Particles

Note:

• only e^- and e⁺ are effective for production of SR (compare: COSY):

4 13 4

2 2 ,

, 1 10

511 .

0 938 ⎟⎟ ≈ ⋅

⎠

⎜⎜ ⎞

⎝

= ⎛

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

= ⎛

MeV MeV c

m c m P

P

e p p

s

e

s

(95)

Width of Angular Distribution

• consider a photon emitted in the restframe perpendicular to movement and to acceleration:

in restframe:

→

E

_s

= h ω

_s

= h ck

_s

= c ⋅ p

_s µ,s

⁽ ⁰ ^,

s

^, ⁰ ^, ¹ E

s

⁾

p c p =

Lorentztransformation into laboratory frame:

⎟ ⎠

⎜ ⎞

⎝

= ⎛

⎟ ⎠

⎜ ⎞

⎝

= ⎛

c p E

p c p

E c

p E

p

_µ_,_l

0 ,

_s

, γβ

^s

, γ

^s

:

_x_,_l

,

_y_,_l

,

_z_,_l

,

^l

(96)

⇒ consequences of optical Doppler effect:

• photon has additional momentum along direction of movement in laboratory frame; larger by factor γβ compared to perpendicular component

⇒ angle of emittance in laboratory frame:

E c m p

p

_o

s s

2 1

tan Θ = ⎯ ⎯ → ⎯

_→

Θ ≈ 1 = γ

γβ

^β

⇒ opening angle of 1/γ

e. g. E = 4.5 GeV ⇒ Θ ≈ 0.1 mrad

≈ 0.007°

≈ 23 "

→ in 10 m distance 1.1 mm width!

(but: convolution with e^--beam divergence!)

Width of Angular Distribution

(97)

⇒ frequency shift by factor γ to higher frequencies:

propagation of light wave with c in both reference frames (in contrast to acoustic Doppler-effect), but frequency shift!

Width of Angular Distribution

(98)

Doppler Effect:

Rest frame:

emittance for one frequency

Laboratory frame:

Power flow in for-

ward direction (E = hν);

highly collimated

Optical Doppler effect due to time dilatation in addition to “classical” Doppler effect:

frequency change by factor γ!

(99)

light flash due to sharp collimation

Time Structure

γ : θ 1

=

3 3

4 sin 2

2 cγ R

c θ R

cβ Rθ

t t

∆t _e _γ

≈

−

=

−

=

⇒ typical frequency:

3 0 3

2 3 2

2 3 π ω γ

R πc

∆t

ω

_typ

= π = γ =

_⇒ line spectrum from fundamental ω₀ up to

≈ ω_typ. Smeared due to Betatron oscillations

(100)

for radiation from a Dipole magnet (time-averaged) Characterization of a source of radiation:

Flux, Brightness and Brilliance

Spectral Flux F(E)

Brilliance F(E, Ψ)

Brightness F(E, Ψ, x, z)

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

⋅ ∆

⋅ E

mrad E s

Photons

% 1 . 0

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

⋅ ∆

⋅ E

mrad E s

Photons

% 1 .

2 0

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

⋅ ∆

⋅

⋅ E

mm E mrad

s

Photons

% 1 .

2 0

2

∫

=

. .

) , ( )

(

distr vert

d E

F E

F ψ ψ

D↔E

(101)

The flux at the sample position is in general determined by the Brightness, which is conserved in an optical system according to Liouville's theorem.

Liouville's theorem

(102)

see e. g. Jackson:

Universal curves, identical for all storage rings, if normalised:

Ordinate: on e^- -energy and e^--current Abszissa: on "critical wavelength"

Spectral Distribution

3

4 π γ λ

_c

= ^R

] [

] 6 [

. 5 ]

[

₃ ₃

GeV E

m Å R

c

=

λ

; in practical units:

with E [keV] =

] [

4 . 12 λ Å

] [ / ] [

218 .

2 ]

[ keV E

³

GeV

³

R m

E

_c

=

(103)

for a bending magnet:

Spectrum: Universal Curves

FLUX BRILLIANCE

critical wavelength: equal emitted power on both sides of spectrum:

∫

∫ ⁼

^∞

c s c

s

d P d

P

λ

( λ ) λ ( λ ) λ

0

(104)

for a bending magnet:

Spectrum: Universal Curves

FLUX BRILLIANCE

critical wavelength: equal emitted power on both sides of spectrum:

∫

∫ ⁼

^∞

c s c

s

d P d

P

λ

( λ ) λ ( λ ) λ

0

(105)

λ_c determines, whether bending magnet radiation from a storage ring is usable for x-ray experiments or not:

hard x-ray:

APS: R = 39 m E = 7 GeV → E_c = 19.5 keV

soft x-ray:

BESSY II: R = 4.4 m E = 1.7 GeV → E_c = 2.5 keV

Critical Wavelength

(106)

Transformation of Hertz' Dipole radiation into laboratory system:

Polarization

"relative compression of vertical components by factor 1/γ"

(107)

Parallel component: Gaussian function with width σ_p:

Universal functions for the vertical intensity distribution for both polarisation states:

Degree of polarisation for the DORIS ring at 3.5 GeV:

Polarization

(108)

Outline

• Why x-rays ?

• Laboratory x-ray sources

• Synchrotron radiation sources

• The source layout

• Special theory of relativity

• Properties of Synchrotron Radiation

• Insertion devices and free electron lasers

• Summary

(109)

Magnet arrangement in straight section with

Insertion Devices

∫ ^Bds ⁼ ⁰

^→ orbit not disturbed 1. Wave length shifter:

] [ ] [

665 .

0 ]

[ keV E

²

GeV

²

B T

E

_c

= ⋅

Example BESSY II: E_c = 1.7 keV soft x-ray range with 7 T Wiggler: E_c = 10 keV hard x-rays

(110)

Array of (permanent) magnets of alternating polarity in straight section:

Wiggler & Undulators

Electromagnetic Hybrid

Permanentmagnets

(111)

Properties of radiation determined by

"K"- or

Undulatorparameter

Wiggler & Undulator

] [ ]

[ 934

.

0

₀

cm B

₀

T

K = α ⋅ γ ≈ ⋅ λ ⋅

max. angle of deviation from ideal orbit

1/θ natural opening angle

undulator- period (N-S-N)

field amplitude

(α determined from equations of movement in external field)

(112)

Wiggler

K >> 1 → α >> 1/θ (typically K = 10)

→ incoherent superposition of radiation from 2 N dipole magnets

→ spectrum & polarization ≈ dipole (B₀) I_Wiggler ≈ 2 N ⋅ I_Dipole

→ horizontal opening angle 2 α = 2 K / γ

(113)

Undulator

K ≈ 1 → same magnet structure as wiggler,

smaller field strength by e. g. larger gap opening

→ coherent superposition of radiation from all poles for a certain wavelength

¾ Intensity

¾ spectral width

¾ angular width

(diffraction limited):

N

~ 1 λ λ

∆

0

/ 1 λ λ

λ

σ = = ⋅

L N

N 2

I ∝

(114)

Interference Condition:

in moving frame: period of magnet-structure is Lorentz contracted: λ₀' = λ₀/γ

→ harmonic oszillations with frequency

' ' 2

λ π

0

ω = ^c

in laboratory frame: frequency shift due to optical Doppler-effect:

ω ' γ ω = ⋅

2 0

/ γ λ λ =

⇒

(115)

Undulator: Interference Condition

Angular dependence:

⎟

⎠

⎜ ⎞

⎝

⎛ + +

= ⁰ ₂ 1 ² 2 ² ²

2 γ θ

γ

λ λ ^K

→ • monochromatic radiation in foreward direction (pinhole)

• tunable via gap size → B₀ → K

(note: larger gap → smaller field → shorter wavelength or higher energy!)

• spectral „tail“ to longer wavelength for finite slit

• stronger field → longitudinal oscillations (e^- performs movement „ " in reference frame moving on orbit) → higher harmonics

• on axis: only odd harmonics (1, 3, 5, ...) off axis: also even harmonics (2, 4, ...)

8

(116)

Undulator Spectrum

(117)

BM-, Wiggler-, Undulator- Spectra

(118)

XFEL

schematics: SASE-Principle:

→interaction e-m-field ↔ e^--bunches

⇒ microbunching ⇒ coherent radiation from e^- within microbunch ~ !!!

(compare undulator: ~N_pole² ; only radiation between poles is coherent, not between electrons)

2

n

el

(119)

TESLA XFEL

(120)

Realisation

(121)

Realisation

(122)

Realisation

(123)

Realisation

(124)

Realisation

(125)

Outline

• Why x-rays ?

• Laboratory x-ray sources

• Synchrotron radiation sources

• The source layout

• Special theory of relativity

• Properties of Synchrotron Radiation

• Insertion devices and free electron lasers

• Summary

(126)

50 m

5 mm properties calculable

small source

size wiggler clean ultra-high

vacuum source

time structure

intense continuous spectrum highly collimated

undulators

.

. polarised

Synchrotron X-Ray Sources