Intro to Synchrotron Radiation, Bending Magnet Radiationattwood/srms/2007/Lec08.pdf · 13 2.70 Al...

27
e Photons X-ray Many straight sections containing periodic magnetic structures Tightly controlled electron beam UV (5.80) (5.82) (5.85) Ch05_F00VG_Feb07.ai Undulator radiation N S N S S N S N e λ u λ t 2 ns 70 ps Ne Intro to Synchrotron Radiation, Bending Magnet Radiation Professor David Attwood Univ. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

Transcript of Intro to Synchrotron Radiation, Bending Magnet Radiationattwood/srms/2007/Lec08.pdf · 13 2.70 Al...

Page 1: Intro to Synchrotron Radiation, Bending Magnet Radiationattwood/srms/2007/Lec08.pdf · 13 2.70 Al 6.02 2.86 26.98 3 [Ne]3s23p1 Aluminum 14 2.33 Si ... Sm 150.36 3,2 [Xe]4f66s2 Samarium

e–

Photons

X-ray• Many straight

sections containingperiodic magneticstructures

• Tightly controlledelectron beam

UV

(5.80)

(5.82)

(5.85)

Ch05_F00VG_Feb07.ai

Undulatorradiation

NS

NS

SN

SN

e–

λu

λ

t

2 ns

70 psNe

Intro to Synchrotron Radiation,Bending Magnet Radiation

Professor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

Page 2: Intro to Synchrotron Radiation, Bending Magnet Radiationattwood/srms/2007/Lec08.pdf · 13 2.70 Al 6.02 2.86 26.98 3 [Ne]3s23p1 Aluminum 14 2.33 Si ... Sm 150.36 3,2 [Xe]4f66s2 Samarium

1H

Hydrogen

1.00791

1s1

3Li

Lithium

0.534.63

6.9411

1s2 2s1

4Be

Beryllium

1.8512.32.23

9.0122

1s2 2s2

2He

Helium

4.0031

1s2

11Na0.97

2.53

22.9901

[Ne]3s1

12Mg1.74

4.303.20

24.312

[Ne]3s2

Sodium

Potassium

13Al2.70

6.022.86

26.983

[Ne]3s23p1

Aluminum

14Si2.33

4.992.35

1.823.54

2.093.92

28.094

[Ne]3s23p2

15P

30.9743,5,4

[Ne]3s23p3

Silicon

14Si2.33

4.992.35

28.094

[Ne]3s23p2

Silicon

Density (g/cm3)

Concentration (1022atoms/cm3)

Nearest neighbor (Å)

Atomic number Atomic weight

References: International Tables for X-ray Crystallography (Reidel, London, 1983) (Ref. 44) and J.R. De Laeter and K.G. Heumann (Ref. 46, 1991).

Key

Symbol

ElectronconfigurationName

Oxidation states(Bold most stable)

Phosphorus

16S

32.0662,4,6

[Ne]3s23p4

Sulfur

17Cl35.4531,3,5,7

[Ne]3s23p5

Chlorine

18Ar39.9481,3,5,7

[Ne]3s23p6

ArgonMagnesium

19K0.86

1.33

39.0981

[Ar]4s1

20Ca1.53

2.303.95

40.082

[Ar]4s2

Calcium

21Sc2.99

4.013.25

44.963

[Ar]3d14s2

Scandium

22Ti4.51

5.672.89

47.884,3

[Ar]3d24s2

Titanium

23V6.09

7.202.62

50.945,4,3,2

[Ar]3d34s2

Vanadium

24Cr7.19

8.332.50

52.006,3,2

[Ar]3d54s1

Chromium

25Mn7.47

8.19

54.947,6,4,2,3

[Ar]3d54s2

Manganese

26Fe7.87

8.492.48

55.852,3

[Ar]3d64s2

Iron

27Co8.82

9.012.50

58.932,3

[Ar]3d74s2

Cobalt

28Ni8.91

9.142.49

58.692,3

[Ar]3d84s2

Nickel

29Cu8.93

8.472.56

63.552,1

[Ar]3d104s1

Copper

30Zn7.13

6.572.67

65.392

[Ar]3d104s2

Zinc

31Ga5.91

5.102.44

69.723

[Ar]3d104s24p1

Gallium

32Ge5.32

4.422.45

72.614

[Ar]3d104s24p2

Germanium

33As5.78

4.642.51

74.923,5

[Ar]3d104s24p3

Arsenic

34Se4.81

3.672.32

78.96–2,4,6

[Ar]3d104s24p4

Selenium

35Br3.12

2.35

79.9041,5

[Ar]3d104s24p5

Bromine

36Kr

83.80

[Ar]3d104s24p6

Krypton

Rubidium

37Rb1.53

1.08

85.471

[Kr]5s1

38Sr2.58

1.784.30

87.622

[Kr]5s2

Strontium

39Y4.48

3.033.55

88.913

[Kr]4d15s2

Yttrium

40Zr6.51

4.303.18

91.224

[Kr]4d25s2

Zirconium

41Nb8.58

5.562.86

92.915,3

[Kr]4d45s1

Niobium

42Mo Tc10.2

6.422.73

95.946,3,2

[Kr]4d55s1

Molybdenum

4311.5 7.072.70

(98)7

[Kr]4d55s2

Technetium

44Ru12.4

7.362.65

101.12,3,4,6,8

[Kr]4d75s1

Ruthenium

45Rh12.4

7.272.69

102.912,3,4

[Kr]4d85s1

Rhodium

46Pd12.0

6.792.75

106.42,4

[Kr]4d10

Palladium

47Ag10.5

5.862.89

107.871

[Kr]4d105s1

Silver

48Cd8.65

4.632.98

112.412

[Kr]4d105s2

Cadmium

49In7.29

3.823.25

114.823

[Kr]4d105s25p1

Indium

50Sn7.29

3.703.02

118.714,2

[Kr]4d105s25p2

Tin

51Sb6.69

3.312.90

121.763,5

[Kr]4d105s25p3

Antimony

52Te6.25

2.952.86

127.60–2,4,6

[Kr]4d105s25p4

Tellurium

53I4.95

2.35

126.901,5,7

[Kr]4d105s25p5

Iodine

54Xe131.29

[Kr]4d105s25p6

Xenon

Cesium

55Cs1.90

0.86

132.911

[Xe]6s1

56Ba3.59

1.584.35

137.332

[Xe]6s2

Barium

57La6.17

2.683.73

138.913

[Xe]5d16s2

Lanthanum

72Hf13.3

4.483.13

178.494

[Xe]4f145d26s2

Hafnium

73Ta16.7

5.552.86

180.955

[Xe]4f145d36s2

Tantalum

74W19.3

6.312.74

183.856,5,4,3,2

[Xe]4f145d46s2

Tungsten

Francium

87Fr

(223)1

[Rn]7s1

88Ra

(226)2

[Rn]7s2

Radium

Cerium

58Lanthanide series

Actinide series

Ce6.772.913.65

140.123,4

[Xe]4f15d16s2

59Pr6.78

2.903.64

140.913,4

[Xe]4f36s2

Praseodymium

60Nd Pm7.00

2.923.63

7.543.023.59

144.243

[Xe]4f46s2

Neodymium

61 (145)3

[Xe]4f56s2

Promethium

62Sm

150.363,2

[Xe]4f66s2

Samarium

5.252.083.97

63Eu

152.03,2

[Xe]4f76s2

Europium

7.873.013.58

64Gd

157.253

[Xe]4f75d16s2

Gadolinium

8.273.133.53

65Tb158.93

3,4

[Xe]4f96s2

Terbium

8.533.163.51

66Dy

162.503

[Xe]4f106s2

Dysprosium

8.803.213.49

67Ho

164.933

[Xe]4f116s2

Holmium

9.043.263.47

68Er167.26

3

[Xe]4f126s2

Erbium

9.333.323.45

69Tm

168.933,2

[Xe]4f136s2

Thulium

6.972.423.88

70Yb

173.043,2

[Xe]4f146s2

Ytterbium

9.843.393.43

71Lu174.97

3

[Xe]4f145d16s2

Lutetium

Thorium

90Th11.7

3.043.60

232.054

[Rn]6d27s2

91Pa15.4

4.013.21

231.045,4

[Rn]5f26d17s2

Protactinium

92U Np Pu Am Cm Bk Cf Es Fm Md No Lr19.1

4.822.75

19.84.89

238.036,5,4,3

[Rn]5f36d17s2

Uranium

9320.55.20

(237)6,5,4,3

[Rn]5f46d17s2

Neptunium

94 (244)6,5,4,3

[Rn]5f67s2

Plutonium

11.92.943.61

95 (243)6,5,4,3

[Rn]5f77s2

Americium

96 (247)3

[Rn]5f76d17s2

Curium

97 (247)4,3

[Rn]5f97s2

Berkelium

98 (251)3

[Rn]5f107s2

Californium

99 (252)

[Rn]5f117s2

Einsteinium

100 (257)

[Rn]5f127s2

Fermium

101 (258)

[Rn]5f137s2

Mendelevium

102 (259)

[Rn]5f147s2

Nobelium

103 (262)

[Rn]5f146d17s2

Lawrencium

89Ac Sg Bh Hs MtRf Db10.1

2.673.76

(227)3

[Rn]6d17s2

Actinium

104 (261)

[Rn]5f146d27s2

Rutherfordium

105 (262)

[Rn]5f146d37s2

Dubnium

106 (266)

[Rn]5f146d47s2 [Rn]5f146d57s2 [Rn]5f146d67s2 [Rn]5f146d77s2

Seaborgium

107 (264)

Bohrium

108 (277)

Hassium

109 (268)

Meitnerium

75Re21.0

6.802.74

186.217,6,4,2,–1

[Xe]4f145d56s2

Rhenium

76Os22.6

7.152.68

190.22,3,4,6,8

[Xe]4f145d66s2

Osmium

77Ir22.6

7.072.71

192.22,3,4,6

[Xe]4f145d76s2

Iridium

78Pt21.4

6.622.78

195.082,4

[Xe]4f145d96s1

Platinum

79Au19.3

5.892.88

197.03,1

[Xe]4f145d106s1

Gold

80Hg

200.592,1

[Xe]4f145d106s2

Mercury

81Tl11.9

3.503.41

204.383,1

[Xe]4f145d106s26p1

Thallium

82Pb11.3

3.303.50

207.24,2

[Xe]4f145d106s26p2

Lead

83Bi9.80

2.823.11

208.983,5

[Xe]4f145d106s26p3

Bismuth

84Po9.27

2.673.35

(209)

114 (287)

4,2

[Xe]4f145d106s26p4

Polonium

85At

(210)1,3,5,7

[Xe]4f145d106s26p5

Astatine

86Rn

(222)

110 (271) 111 (272) 112 (283)

[Xe]4f145d106s26p6

Radon

5B

Boron

2.4713.71.78

10.813

1s2 2s2 2p1

6C

Carbon

2.2711.41.42

12.0114,2

1s2 2s2 2p2

7N

Nitrogen

14.0073,5,4,2

1s2 2s2 2p3

8O

Oxygen

16–2

1s2 2s2 2p4

9F

Fluorine

19.00–1

1s2 2s2 2p5

10Ne

Neon

20.180

1s2 2s2 2p6

GroupIA

IIA

IIIA IVA VA VIA VIIA VIIIA IB IIB

IIIB IVB VB VIB VIIB

VIII

Solid

Gas

Liquid

Synthetically prepared

Periodic_Tb_clr_May2007.ai

Broadly Tunable Radiation is Needed to Probethe Primary Resonances of the Elements

Professor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

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Ch05_F09VG_revOct05.ai

Synchrotron Radiation from Relativistic Electrons

λ′

λ′ λxv

Note: Angle-dependent doppler shift

v << c v c~<

λ = λ′ (1 – cosθ) λ = λ′ γ (1 – cosθ)

γ =

vc

v2

c2

1

1 –

vc

Professor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

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Ch05_F11VG_Oct05.ai

Synchrotron Radiation in a Narrow Forward Cone

a′ sin2Θ′

θ′

Θ′θ – 1

2γ~

(5.1)

(5.2)

Frame moving with electron Laboratory frame of reference

Professor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

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Ch05_F11modif_VG.ai

Relativistic Electrons Radiatein a Narrow Forward Cone

a′ sin2Θ′

k′ k

k′ = 2π/λ′

Lorentztransformationk′

k′

Θ′θ

θθ′

12γ

kxkz

k′2γk′

tanθ′2γ

12γθ

Dipole radiation

Frame of referencemoving with electrons

Laboratory frame of reference

x

z

kx = k′

kz = 2γk′ (Relativistic Doppler shift)z

xz

=

x

Professor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

Page 6: Intro to Synchrotron Radiation, Bending Magnet Radiationattwood/srms/2007/Lec08.pdf · 13 2.70 Al 6.02 2.86 26.98 3 [Ne]3s23p1 Aluminum 14 2.33 Si ... Sm 150.36 3,2 [Xe]4f66s2 Samarium

Ch05_UsefulFormulas.ai

Some Useful Formulas for Synchrotron Radiation

ω λ = 1239.842 eV nm

Ee = γmc2, p = γmv

γ = = 1957 Ee(GeV)

γ = = ; β =

where

Undulator: ;

Bending Magnet: ,

Eemc2

1

1 –

vcv2

c2

1

1 – β2

Professor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

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e–

λ

F

ω

e–

λ

1γ N

F

ω

e–

λ

>>

ω

F

Ch05_F01_03VG.ai

Bending magnetradiation

Wiggler radiation

Undulator radiation

Three Forms of Synchrotron Radiation

Professor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

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Ch05_F04VG.ai

Modern Synchrotron Radiation Facility

Continuous e–trajectorybending

Circularelectronmotion

Photons

e– e–

Photons

X-ray• Many straight sections containing periodic magnetic structures

• Tightly controlled electron beam

UV

a

“Undulatorand wigglerradiation”

ω

• Partially coherent• Tunable

P

Older SynchrotronRadiation Facility

Modern SynchrotronRadiation Facility

“X-raylight bulb”

“Bendingmagnet

radiation”

ω

Professor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

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The ALS with San Franciscoin the Background

ALS_SF_Feb07.ai

1.9 GeV, γ = 3720, 197 m circumferenceProfessor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

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France’s ESRF is Well Situated

ESRF.ai

6 GeV; γ = 11, 800; 884m circumference

Bounded by two riversProfessor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

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SPring-8 in Hyogo Prefecture, Japan

SPring8.ai

8 GeV; γ = 15,700; 1.44 km circumference

Professor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

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Ch05_F05VG.ai

A Single Storage Ring ServesMany Scientific User Groups

Surface and Materials Science

Spectromicroscopy of Surfaces

Atomic andMolecular Physics

EUV Interferometryand Coherent Optics

Chemical Dynamics

Magnetic MaterialsPolarization Studies

EnvironmentalSpectromicroscopyand Biomicroscopy

Elliptical Wiggler for Materials Science and Biology

Photoemission Spectroscopy of Strongly Correlated Systems

U5

EPU

U10

U5

U10

U10

U8

RF

Injec

tion

Linac

BoosterSynchrotron

EW20

6.07.0

8.0

9.0

11.0

10.0

12.0

ProteinCrystallography

2.0

5.0

4.0

W16

U3.65

Professor David AttwoodUniv. California, Berkeley

Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

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Ch05_BendMagRadius.ai

Bending Magnet Radius

The Lorentz force for a relativistic electronin a constant magnetic field is

where p = γmv. In a fixed magnetic fieldthe rate of change of electron energy is

∴ γ = constant

thus with Ee = γmc2

and the force equation becomes

=dpdt

R FB

V

v = βc

β → 1

a =–v2

R

Professor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

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Ch05_BendMagRad_April04.ai

Bending Magnet Radiation

Radiationpulse

Time

2∆τ

Ι

B′BA

Radius R

R sinθ

θ =

θ = 12γ

The cone half angleθ sets the limits ofarc-length from whichradiation can beobserved.

(a) (b)12γ

With θ 1/2γ , sinθ θ

With v = βc

∴ 2∆τ = m2eΒγ2

γmceΒ

and R but (1 – β)

Professor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

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Ch05_BendMagRad2_April04.ai

Bending Magnet Radiation (continued)

From Heisenberg’s Uncertainty Principle for rms pulse duration and photon energy

thus

Thus the single-sided rms photon energy width (uncertainty) is

A more detailed description of bending magnet radius finds the critical photon energy

In practical units the critical photon energy is

(5.4b)

(5.4c)

(5.7a)

(5.7b)

∆Ε ≥

2∆τ

∆Ε ≥

m/2eΒγ2

Professor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

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Ch05_F07_T2.ai

Bending Magnet Radiation

10

1

0.1

0.01

0.0010.001 0.01 0.1 1 4 10

y = E/Ec

G1(

y) a

nd H

2(y)

H2(1) = 1.454G1(1) = 0.6514

G1(y)

H2(y)

50% 50%

(5.7a)

(5.7b)

(5.6)

(5.8)(5.5)

ψ

θ

e–

Professor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

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Ch05_F07_revJune05.ai

Bending Magnet Radiation Covers a BroadRegion of the Spectrum, Including thePrimary Absorption Edges of Most Elements

1013

1014

1012

1011

0.01 0.1 1 10 100Photon energy (keV)

Pho

ton

flux

(ph/

sec)

Ec

50%

Ee = 1.9 GeVΙ = 400 mAB = 1.27 Tωc = 3.05 keV

(5.7a)

(5.7b)

(5.8)

ψθ

e–

∆θ = 1mrad∆ω/ω = 0.1%

Advantages: • covers broad spectral range • least expensive • most accessableDisadvantages: • limited coverage of hard x-rays • not as bright as undulator

4Ec

50%

ALS

Professor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

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Ch05_F08VG.ai

Narrow Cone Undulator Radiation,Generated by Relativistic ElectronsTraversing a Periodic Magnet Structure

Magnetic undulator(N periods)

Relativisticelectron beam,Ee = γmc2

λ

λ –

λu

λu

2γ2

∆λλ

1N

~

θcen –1

γ∗ N

cen =

~

Professor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

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An Undulator Up Close

Undulator_Close.ai

ALS U5 undulator, beamline 7.0, N = 89, λu = 50 mmProfessor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

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Installing an Undulator at Berkeley’sAdvanced Light Source

Undulator_Install.ai

ALS Beamline 9.0 (May 1994), N = 55, λu = 80 mmProfessor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

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Undulator Radiation

e–

N

S S

N

N N

S S

λu

E = γmc2

γ =1

1 – v2

c2

N = # periods

e–sin2Θ θ ~– 1

2γ θcen

e– radiates at theLorentz contractedwavelength:

Doppler shortenedwavelength on axis:

Laboratory Frameof Reference

Frame ofMoving e–

Frame ofObserver

FollowingMonochromator

For 1N

∆λλ

θcen1

γ N

θcen 40 rad

λ′ = λuγ

Bandwidth:

λ′ N

λ = λ′γ(1 – βcosθ)

λ = (1 + γ2θ2)

Accounting for transversemotion due to the periodicmagnetic field:

λu

2γ2

λu

2γ 2λ = (1 + + γ 2θ2)K2

2

where K = eB0λu /2πmc

Ch05_LG186.ai

~–

~–

~–

~–∆λ′

typically

Professor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

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Physically, where does theλ = λu/2γ2 come from?

Ch05_Eq09_10VG.ai

(5.10)

(5.9)

The electron “sees” a Lorentz contracted period

and emits radiation in its frame of reference at frequency

On-axis (θ = 0) the observed frequency is

Observed in the laboratory frame of reference, this radiationis Doppler shifted to a frequency

Professor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

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Ch05_Eq11VG.ai

(5.11)

and the observed wavelength is

Give examples.

By definition γ = ; γ2 =

thus

11 – β2

1(1 – β)(1 + β)

12(1 – β)

Professor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

Physically, where does theλ = λu/2γ2 come from?

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Ch05_Eq10_12VG.ai

(5.10)

(5.12)

For θ ≠ 0, take cos θ = 1 – + . . . , then

exhibiting a reduced Doppler shift off-axis, i.e., longer wavelengths.This is a simplified version of the “Undulator Equation”.

The observed wavelength is then

θ2

2

= =c/λu

1 – β (1 – θ2/2 + . . . )c/λu

1 – β + βθ2/2 – . . .c/(1 – β)λu

1 + βθ2/2(1 – β). . .

Professor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

What about the off-axis θ 0 radiation?

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The Undulator’s “Central Radiation Cone”

Ch05_Eq13_15VG_Jan06.ai

(5.14)

(5.13)

(5.15)

With electrons executing N oscillations as they traverse the periodic magnet structure, and thus radiating a wavetrain of N cycles, it is of interest to know what angular cone contains radiation of relative spectral bandwidth

Write the undulator equation twice, once for on-axis radiation (θ = 0) and once for wavelength-shifted radiation off-axis at angle θ:

divide and simplify to

Combining the two equations (5.13 and 5.14)

This is the half-angle of the “central radiation cone”, defined as containing radiation of ∆λ/λ = 1/N.

λ0 + ∆λ = (1 + γ2θ2)

defines θcen : γ2θ2 , which gives

λu2γ2

1N

λ0 =λu

2γ2

cen

Professor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

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Ch05_F12VG.ai

The Undulator Radiation Spectrumin Two Frames of Reference

Frequency, ω′

Execution of N electron oscillationsproduces a transform-limitedspectral bandwidth, ∆ω′/ω′ = 1/N.

The Doppler frequency shift has astrong angle dependence, leadingto lower photon energies off-axis.

Frequency, ω

dP′dΩ′

dPdΩ

ω′∆ω′

~ N

Off-axis

Nea

r axi

s

Professor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007

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Ch05_F13_14VG.ai

The Narrow (1/N) Spectral Bandwidth of UndulatorRadiation Can be Recovered in Two Ways

ω ω

dPdΩ

dPdΩ ∆λ

λ

θ

Pinholeaperture

Gratingmonochromator

Exitslit

θ

With a pinhole aperture

With a monochromator

1N

1 γ N

∆λλ

2θ 1γ

∆λλ 1

Professor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007