e – Photons X-ray • Many straight sections containing periodic magnetic structures • Tightly controlled electron beam UV (5.80) (5.82) (5.85) Ch05_F00VG_Feb09.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 SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
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e–
Photons
X-ray
• Many straightsections containingperiodic magneticstructures
• Tightly controlledelectron beam
UV
(5.80)
(5.82)
(5.85)
Ch05_F00VG_Feb09.ai
Undulatorradiation
N
SN
S
S
NS
N
e–
λu
λ
t
2 ns
70 psNe
Intro to Synchrotron Radiation,Bending Magnet Radiation
Professor David AttwoodUniv. California, Berkeley SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
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).
Broadly Tunable Radiation is Needed to Probethe Primary Resonances of the Elements
Professor David AttwoodUniv. California, Berkeley SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
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 SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
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 SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
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
x
z=
x
Professor David AttwoodUniv. California, Berkeley SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
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 SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
e–
λ
1γ
F
ω
e–
λ
1γ N
F
ω
e–
1γ
λ
>>
ω
F
Ch05_F01_03VG.ai
Bending magnetradiation
Wiggler radiation
Undulator radiation
Three Forms of Synchrotron Radiation
Professor David AttwoodUniv. California, Berkeley SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
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
• TunableP
Older SynchrotronRadiation Facility
Modern SynchrotronRadiation Facility
“X-raylight bulb”
“Bendingmagnet
radiation”
ω
Professor David AttwoodUniv. California, Berkeley SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
The ALS with San Franciscoin the Background
ALS_SF_Feb09.ai
1.9 GeV, γ = 3720, 197 m circumference
Professor David AttwoodUniv. California, Berkeley SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
France’s ESRF is Well Situated
ESRF.ai
6 GeV; γ = 11, 800; 884m circumference
Bounded by two riversProfessor David AttwoodUniv. California, Berkeley SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
SPring-8 in Hyogo Prefecture, Japan
SPring8.ai
8 GeV; γ = 15,700; 1.44 km circumference
Professor David AttwoodUniv. California, Berkeley SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
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
Inje
ctio
n
Linac
BoosterSynchrotron
EW20
6.0
7.08.0
9.0
11.0
10.0
12.0
ProteinCrystallography
2.0
5.0
4.0
W16
U3.65
Professor David AttwoodUniv. California, Berkeley SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
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
=dp
dt
∴
R FB
V
v = βc
β → 1
a =–v2
R
Professor David AttwoodUniv. California, Berkeley SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
Ch05_BendMagRad_Feb09.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 =m
2eΒγ2
γmc
eΒand R but (1 – β)
Professor David AttwoodUniv. California, Berkeley SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
Ch05_BendMagRad2_Feb07.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 SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
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) = 1.454
G1(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 SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
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 accessable
Disadvantages: • limited coverage of hard x-rays • not as bright as undulator
4Ec
50%
ALS
Professor David AttwoodUniv. California, Berkeley SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
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 –
2θ
λu
λu2γ2
∆λλ
1N
~
θcen –1
γ∗ N
cen =
~
Professor David AttwoodUniv. California, Berkeley SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
An Undulator Up Close
Undulator_Close.ai
ALS U5 undulator, beamline 7.0, N = 89, λu = 50 mmProfessor David AttwoodUniv. California, Berkeley SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
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 SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
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
Δλλ
θcen1
γ N
θcen 40 µrad
λʹ = λu
γ
Bandwidth:
λʹ N
λ = λʹγ(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
~–
~–
~–
~–Δλʹ
typically
Professor David AttwoodUniv. California, Berkeley SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
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 SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
Ch05_Eq11VG.ai
(5.11)
and the observed wavelength is
Give examples.
By definition γ = ; γ2 =
thus
1
1 – β2
1(1 – β)(1 + β)
12(1 – β)
Professor David AttwoodUniv. California, Berkeley SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
Physically, where does theλ = λu/2γ2 come from?
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 SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
What about the off-axis θ 0 radiation?
The Undulator’s “Central Radiation Cone”
Ch05_Eq13_15VG_Feb09.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
λu
2γ2
1N
λ0 =λu
2γ2
cen
Professor David AttwoodUniv. California, Berkeley SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
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 ax
is
Professor David AttwoodUniv. California, Berkeley SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
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 SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
Ch05_Lorentz.ApxF.ai
Lorentz Space-Time Transformations (Appendix F)
S′
S
X
X′
Z
Z′v
L′
λ′uθ′
θ0
(F.1a)
(F.1b)
(F.1c)
(F.2a)
(F.2a)
(F.2a)
(F.3)
(F.4)
Professor David AttwoodUniv. California, Berkeley SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9
Ch05_LorentzTrans.ai
Lorentz Transformations: Frequency, Angles, Length and Time
Doppler frequency shifts
Lorentz contraction of length
Time dilation
Angular transformations
(F.8a)
(F.8b)
(F.12)
(F.13)
(F.9a)
(F.9b)
(F.10b)
(F.10a)
(F.11a)
(F.11b)
Professor David AttwoodUniv. California, Berkeley SXR & EUV Radiation, Spring 2009 / EE213 & AST210 / Intro to Synchrotron Radiation, Bendng Magnet Radiation / Lec 9