Intro to Synchrotron Radiation,
Bending Magnet Radiation
Photons
e–
• Many straight
sections containing
periodic magnetic
structures
UV
X-ray
• Tightly controlled
electron beam
(5.80)
(5.82)
(5.85)
2 ns
Ne
70 ps
S
t
N
S
Undulator
radiation
N
e–
N
S
N
S
λu
λ
Professor David Attwood
Univ. California, Berkeley
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
Ch05_F00VG_Feb07.ai
Broadly Tunable Radiation is Needed to Probe
the Primary Resonances of the Elements
Group
IA
1
VIII
1.0079
1
Key
H
Atomic number
1s1
IIA
Hydrogen
3
6.941
1
Li
0.53
4.63
1s2 2s1
Lithium
22.990
1
11
0.97
2.53
Na
4
Be
1.85
12.3
2.23
1s2 2s2
1.74
4.30
3.20
19
0.86
1.33
K
Magnesium
40.08
2
20
1.53
2.30
3.95
[Ar]4s1
[Ar]4s2
Potassium
37
1.53
1.08
85.47
1
Rb
Calcium
87.62
2
38
2.58
1.78
4.30
[Kr]5s1
1.90
0.86
Strontium
132.91
1
Cs
[Xe]6s1
Cesium
87
Sr
[Kr]5s2
Rubidium
55
Ca
(223)
1
Fr
56
3.59
1.58
4.35
137.33
2
Ba
IIIA
21
44.96
3
Sc
2.99
4.01
3.25
[Ar]3d14s2
Scandium
88.91
3
39
Y
4.48
3.03
3.55
[Kr]4d15s2
Yttrium
57
138.91
3
La
[Xe]6s2
Barium
(226)
2
89
Ra
[Rn]7s1
[Rn]7s2
Francium
Radium
Lanthanum
(227)
3
Ac
10.1
2.67
3.76
[Rn]6d17s2
Actinium
IVA
22
Lanthanide series
47.88
4,3
Ti
4.51
5.67
2.89
[Ar]3d24s2
Titanium
91.22
4
40
Zr
6.51
4.30
3.18
[Kr]4d25s2
Zirconium
72
178.49
4
Hf
13.3
4.48
3.13
[Xe]4f145d26s2
Hafnium
(261)
104
Si
5
Gas
Symbol
Rf
23
VIA
50.94
5,4,3,2
V
6.09
7.20
2.62
[Ar]3d34s2
Vanadium
92.91
5,3
41
Nb
8.58
5.56
2.86
[Kr]4d45s1
Niobium
73
180.95
5
Ta
16.7
5.55
2.86
[Xe]4f145d36s2
Tantalum
(262)
105
Db
24
52.00
6,3,2
Cr
7.19
8.33
2.50
[Ar]3d54s1
Chromium
95.94
6,3,2
42
Mo
10.2
6.42
2.73
[Kr]4d55s1
Molybdenum
VIIA
VIIIA
54.94
25 7,6,4,2,3
26
7.47
8.19
Mn
[Ar]3d54s2
Manganese
(98)
7
43
Tc
11.5
7.07
2.70
[Kr]4d55s2
Technetium
55.85
2,3
Fe
7.87
8.49
2.48
[Ar]3d64s2
Iron
27
Ru
Ruthenium
Co
Cobalt
102.91
2,3,4
101.1
44 2,3,4,6,8
45
12.4
7.36
2.65
[Kr]4d75s1
58.93
2,3
8.82
9.01
2.50
[Ar]3d74s2
Rh
12.4
7.27
2.69
[Kr]4d85s1
Rhodium
183.85
186.21
190.2
74 6,5,4,3,2
75 7,6,4,2,–1
76 2,3,4,6,8
77
W
19.3
6.31
2.74
[Xe]4f145d46s2
Tungsten
(266)
106
Sg
Re
21.0
6.80
2.74
[Xe]4f145d56s2
Rhenium
(264)
107
Bh
Os
22.6
7.15
2.68
[Xe]4f145d66s2
Osmium
(277)
108
Hs
192.2
2,3,4,6
Ir
22.6
7.07
2.71
[Xe]4f145d76s2
Iridium
(268)
109
Mt
[Rn]5f146d37s2
[Rn]5f146d47s2
[Rn]5f146d57s2
[Rn]5f146d67s2
[Rn]5f146d77s2
Rutherfordium
Dubnium
Seaborgium
Bohrium
Hassium
Meitnerium
140.12
3,4
Ce
59
140.91
3,4
Pr
60
144.24
3
6.78
2.90
3.64
[Xe]4f36s2
7.00
2.92
3.63
[Xe]4f46s2
90
Praseodymium
231.04
5,4
92
232.05
4
Th
11.7
3.04
3.60
[Rn]6d27s2
Thorium
91
Pa
15.4
4.01
3.21
[Rn]5f26d17s2
Protactinium
61
(145)
3
Nd Pm
6.77
2.91
3.65
[Xe]4f15d16s2
Cerium
Actinide series
VA
Synthetically
prepared
Neodymium
238.03
6,5,4,3
U
19.1
4.82
2.75
[Rn]5f36d17s2
Uranium
[Xe]4f56s2
Promethium
(237)
6,5,4,3
93
20.5
5.20
Np
62
150.36
3,2
Sm
7.54
3.02
3.59
[Xe]4f66s2
Samarium
(244)
6,5,4,3
94
19.8
4.89
63
152.0
3,2
Eu
5.25
2.08
3.97
[Xe]4f76s2
Europium
(243)
6,5,4,3
95
IB
28
58.69
2,3
Ni
8.91
9.14
2.49
[Ar]3d84s2
Nickel
106.4
2,4
46
Pd
12.0
6.79
2.75
[Kr]4d10
Palladium
78
195.08
2,4
Pt
21.4
6.62
2.78
[Xe]4f145d96s1
Platinum
(271)
110
64
157.25
3
Gd
7.87
3.01
3.58
[Xe]4f75d16s2
Gadolinium
(247)
3
96
Pu Am Cm
[Rn]5f46d17s2
[Rn]5f67s2
Neptunium
Plutonium
11.9
2.94
3.61
[Rn]5f77s2
Americium
29
63.55
2,1
Cu
8.93
8.47
2.56
[Ar]3d104s1
Copper
107.87
1
47
Ag
10.5
5.86
2.89
[Kr]4d105s1
Silver
79
197.0
3,1
Au
19.3
5.89
2.88
[Xe]4f145d106s1
Gold
(272)
111
65
158.93
3,4
Tb
8.27
3.13
3.53
[Xe]4f96s2
Terbium
(247)
4,3
97
Bk
IIIB
10.81
3
B
2.47
13.7
1.78
1s2 2s2 2p1
Liquid
Electron
configuration
[Rn]5f146d27s2
58
Professor David Attwood
Univ. California, Berkeley
Solid
(Bold most stable)
Boron
26.98
3
13
References: International Tables for X-ray Crystallography (Reidel, London, 1983) (Ref. 44)
and J.R. De Laeter and K.G. Heumann (Ref. 46, 1991).
6.17
2.68
3.73
[Xe]5d16s2
88
Oxidation states
28.09
4
Name
Mg
[Ne]3s2
14
2.33
4.99
2.35
[Ne]3s23p2
Silicon
Beryllium
24.31
2
12
4.003
1
He
Atomic weight
Density (g/cm3)
Concentration (1022atoms/cm3)
Nearest neighbor (Å)
9.012
2
[Ne]3s1
Sodium
39.098
1
2
IIB
30
65.39
2
Zn
7.13
6.57
2.67
[Ar]3d104s2
Zinc
112.41
2
48
Cd
8.65
4.63
2.98
[Kr]4d105s2
Cadmium
80
200.59
2,1
Hg
[Xe]4f145d106s2
Mercury
(283)
Al
2.70
6.02
2.86
[Ne]3s23p1
Aluminum
69.72
3
31
Ga
5.91
5.10
2.44
[Ar]3d104s24p1
Gallium
114.82
3
49
In
7.29
3.82
3.25
[Kr]4d105s25p1
66
162.50
3
Dy
Dysprosium
(251)
3
98
Cf
IVB
12.011
4,2
C
2.27
11.4
1.42
1s2 2s2 2p2
Carbon
28.09
4
14
204.38
3,1
Tl
Si
2.33
4.99
2.35
[Ne]3s23p2
Silicon
72.61
4
32
Ge
5.32
4.42
2.45
[Ar]3d104s24p2
Germanium
118.71
4,2
50
Sn
7.29
3.70
3.02
[Kr]4d105s25p2
Indium
81
Tin
82
Thallium
VB
7
14.007
3,5,4,2
N
1s2 2s2 2p3
Nitrogen
30.974
3,5,4
15
P
1.82
3.54
[Ne]3s23p3
Phosphorus
74.92
3,5
33
As
5.78
4.64
2.51
[Ar]3d104s24p3
Arsenic
121.76
3,5
51
Sb
6.69
3.31
2.90
[Kr]4d105s25p3
Antimony
207.2
4,2
Pb
11.9
11.3
3.50
3.30
3.41
3.50
[Xe]4f145d106s26p1 [Xe]4f145d106s26p2
Lead
(287)
83
208.98
3,5
Bi
9.80
2.82
3.11
[Xe]4f145d106s26p3
Bismuth
8
VIB
16
–2
O
1s2 2s2 2p4
Oxygen
32.066
2,4,6
16
S
2.09
3.92
[Ne]3s23p4
Sulfur
78.96
–2,4,6
34
Se
4.81
3.67
2.32
[Ar]3d104s24p4
Selenium
127.60
–2,4,6
52
Te
6.25
2.95
2.86
[Kr]4d105s25p4
Tellurium
84
(209)
4,2
Po
9
VIIB
Helium
19.00
–1
F
1s2 2s2 2p5
Cl
[Ne]3s23p5
35
Br
[Ar]3d104s24p5
Bromine
126.90
1,5,7
53
I
4.95
2.35
[Kr]4d105s25p5
Iodine
85
Ne
Neon
39.948
1,3,5,7
18
Ar
[Ne]3s23p6
Chlorine
79.904
1,5
3.12
2.35
20.180
1s2 2s2 2p6
Fluorine
35.453
1,3,5,7
17
10
(210)
1,3,5,7
At
Argon
83.80
36
Kr
[Ar]3d104s24p6
Krypton
131.29
54
Xe
[Kr]4d105s25p6
Xenon
86
(222)
Rn
9.27
2.67
3.35
[Xe]4f145d106s26p4 [Xe]4f145d106s26p5 [Xe]4f145d106s26p6
Polonium
Astatine
Radon
114
112
8.53
3.16
3.51
[Xe]4f106s2
6
1s2
67
164.93
3
Ho
8.80
3.21
3.49
[Xe]4f116s2
Holmium
(252)
99
Es
68
167.26
3
Er
9.04
3.26
3.47
[Xe]4f126s2
Erbium
(257)
100
Fm
69
168.93
3,2
Tm
9.33
3.32
3.45
[Xe]4f136s2
Thulium
(258)
101
Md
70
173.04
3,2
Yb
6.97
2.42
3.88
[Xe]4f146s2
Ytterbium
(259)
102
No
71
174.97
3
Lu
9.84
3.39
3.43
[Xe]4f145d16s2
Lutetium
(262)
103
Lr
[Rn]5f76d17s2
[Rn]5f97s2
[Rn]5f107s2
[Rn]5f117s2
[Rn]5f127s2
[Rn]5f137s2
[Rn]5f147s2
[Rn]5f146d17s2
Curium
Berkelium
Californium
Einsteinium
Fermium
Mendelevium
Nobelium
Lawrencium
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
Periodic_Tb_clr_May2007.ai
Synchrotron Radiation from Relativistic Electrons
v <c
~
v << c
λ′
v
λx
λ′
Note: Angle-dependent doppler shift
λ = λ′ (1 – vc cosθ)
λ = λ′ γ (1 – vc cosθ)
γ=
Professor David Attwood
Univ. California, Berkeley
1
v2
1– 2
c
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
Ch05_F09VG_revOct05.ai
Synchrotron Radiation in a Narrow Forward Cone
Frame moving with electron
a′
Θ′
Laboratory frame of reference
sin2Θ′
1
θ~
– γ
2
θ′
(5.1)
(5.2)
Professor David Attwood
Univ. California, Berkeley
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
Ch05_F11VG_Oct05.ai
Relativistic Electrons Radiate
in a Narrow Forward Cone
Dipole radiation
a′
Θ′
sin2Θ′
1
θ γ
2
Frame of reference
moving with electrons
k′
θ′
k′z
k′ = 2π/λ′
Professor David Attwood
Univ. California, Berkeley
k′x
Laboratory frame of reference
k
Lorentz
transformation
kx = k′x
θ
kz = 2γk′z(Relativistic Doppler shift)
k
k′
tanθ′
1
x
θ k  x =

2γk′z
2γ
2γ
z
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
Ch05_F11modif_VG.ai
Some Useful Formulas for Synchrotron Radiation
γ=
1
2
1 – v2
c
=
1
1 – β2
v
; β= c
Ee = γmc2, p = γmv
Ee
γ=
= 1957 Ee(GeV)
mc2
ω  λ = 1239.842 eV  nm
Bending Magnet:
Undulator:
,
;
where
Professor David Attwood
Univ. California, Berkeley
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
Ch05_UsefulFormulas.ai
Three Forms of Synchrotron Radiation
λ
1
γ
Bending magnet
radiation
F
e–
ω
>>
e–
1
γ
Wiggler radiation
F
λ
ω
1
γ N
e–
Undulator radiation
F
λ
ω
Professor David Attwood
Univ. California, Berkeley
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
Ch05_F01_03VG.ai
Modern Synchrotron Radiation Facility
Older Synchrotron
Radiation Facility
Modern Synchrotron
Radiation Facility
Circular
electron
motion
e–
Photons
e–
• Many straight
a
Continuous e–
trajectory
bending
Photons
sections containing
periodic magnetic
structures
UV
X-ray
• Tightly controlled
electron beam
“X-ray
light bulb”
“Bending
magnet
radiation”
ω
“Undulator
and wiggler
radiation”
• Partially
coherent
P
• Tunable
ω
Professor David Attwood
Univ. California, Berkeley
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
Ch05_F04VG.ai
The ALS with San Francisco
in the Background
1.9 GeV, γ = 3720, 197 m circumference
Professor David Attwood
Univ. California, Berkeley
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
ALS_SF_Feb07.ai
France’s ESRF is Well Situated
6 GeV; γ = 11, 800; 884m circumference
Professor David Attwood
Univ. California, Berkeley
Bounded by two rivers
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
ESRF.ai
SPring-8 in Hyogo Prefecture, Japan
8 GeV; γ = 15,700; 1.44 km circumference
Professor David Attwood
Univ. California, Berkeley
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
SPring8.ai
A Single Storage Ring Serves
Many Scientific User Groups
Spectromicroscopy
of Surfaces
Environmental
Spectromicroscopy
and Biomicroscopy
Surface and
Materials Science
7.0
8.0
6.0
Protein
Crystallography
U5
5.0
4.0
U10
9.0
U10
U5
EW
20
Booster
Synchrotron
U3.65
10.0
U8
io
n
2.0
U10
In
je
ct
EPU
RF
Professor David Attwood
Univ. California, Berkeley
Photoemission Spectroscopy
of Strongly Correlated Systems
Linac
W16
Magnetic Materials
Polarization Studies
Chemical Dynamics
12.0
Atomic and
Molecular Physics
11.0
Elliptical Wiggler for Materials
Science and Biology
EUV Interferometry
and Coherent Optics
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
Ch05_F05VG.ai
Bending Magnet Radius
The Lorentz force for a relativistic electron
in a constant magnetic field is
V
where p = γmv. In a fixed magnetic field
the rate of change of electron energy is
R
F
B
thus with Ee = γmc2
v = βc
∴ γ = constant
and the force equation becomes
dp
=
dt
β→1
–v2
a=
R
∴
Professor David Attwood
Univ. California, Berkeley
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
Ch05_BendMagRadius.ai
Bending Magnet Radiation
(a)
A
(b)
1
θ= γ
2
B
R sinθ
Radius R
B′
1
θ= γ
2
The cone half angle
θ sets the limits of
arc-length from which
radiation can be
observed.
With θ  1/2γ , sinθ  θ
Ι
2∆τ
Radiation
pulse
Time
With v = βc
but (1 – β)
∴ 2∆τ =
and
R
γmc
eΒ
m
2eΒγ2
Professor David Attwood
Univ. California, Berkeley
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
Ch05_BendMagRad_April04.ai
Bending Magnet Radiation (continued)
From Heisenberg’s Uncertainty Principle for rms pulse duration and photon energy
thus
∆Ε ≥

2∆τ
∆Ε ≥

m/2eΒγ2
(5.4b)
Thus the single-sided rms photon energy width (uncertainty) is
(5.4c)
A more detailed description of bending magnet radius finds the critical photon energy
(5.7a)
In practical units the critical photon energy is
(5.7b)
Professor David Attwood
Univ. California, Berkeley
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
Ch05_BendMagRad2_April04.ai
Bending Magnet Radiation
10
G1(1) = 0.6514
H2(1) = 1.454
ψ
θ
G1(y) and H2(y)
e–
1
0.1
G1(y)
H2(y)
50%
50%
0.01
0.001
0.001
0.01
0.1
y = E/Ec
1
4 10
(5.7a)
(5.6)
(5.7b)
(5.5)
Professor David Attwood
Univ. California, Berkeley
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
(5.8)
Ch05_F07_T2.ai
Bending Magnet Radiation Covers a Broad
Region of the Spectrum, Including the
Primary Absorption Edges of Most Elements
e–
ψ
θ
Photon flux (ph/sec)
1014
(5.7a)
(5.7b)
(5.8)
Advantages:
ALS
1013
1012
1011
50%
∆θ = 1mrad
∆ω/ω = 0.1%
0.01
•
•
•
Disadvantages: •
Professor David Attwood
Univ. California, Berkeley
Ee = 1.9 GeV
Ι = 400 mA
B = 1.27 T
ωc = 3.05 keV
covers broad spectral range
least expensive
most accessable
limited coverage of
hard x-rays
• not as bright as undulator
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
Ec
50%
4Ec
0.1
1
10
Photon energy (keV)
100
Ch05_F07_revJune05.ai
Narrow Cone Undulator Radiation,
Generated by Relativistic Electrons
Traversing a Periodic Magnet Structure
Magnetic undulator
(N periods)
λu
λ
2θ
Relativistic
electron beam,
Ee = γmc2
λu
~
λ–
2γ2
1
θcen ~
–
γ∗ N
∆λ  = 1
 λ cen N
Professor David Attwood
Univ. California, Berkeley
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
Ch05_F08VG.ai
An Undulator Up Close
Professor David Attwood
Univ. California, Berkeley
ALS U5 undulator, beamline 7.0, N = 89, λu = 50 mm
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
Undulator_Close.ai
Installing an Undulator at Berkeley’s
Advanced Light Source
Professor David Attwood
Univ. California, Berkeley
ALS Beamline 9.0 (May 1994), N = 55, λu = 80 mm
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
Undulator_Install.ai
Undulator Radiation
Laboratory Frame
of Reference
Frame of
Moving e–
λu
N
S
N
S
S
N
S
N
e–
Frame of
Observer
Following
Monochromator
1
θ ~
– 2γ
sin2Θ
θcen
e–
e– radiates at the
Lorentz contracted
wavelength:
E = γmc2
1
γ=
1–
v2
c2
N = # periods
λ
λ′ = γu
Bandwidth:
λ′
∆λ′
~
– N
Doppler shortened
wavelength on axis:
λ = λ′γ(1 – βcosθ)
λ=
λu
2γ2
(1 + γ2θ2)
1
For ∆λ ~
–
λ
N
1
θcen ~
– γ
N
typically
θcen ~
– 40 rad
Accounting for transverse
motion due to the periodic
magnetic field:
λ=
λu
2γ
K2 γ 2 2
(1
+
+ θ )
2
2
where K = eB0λu /2πmc
Professor David Attwood
Univ. California, Berkeley
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
Ch05_LG186.ai
Physically, where does the
λ = λu/2γ 2 come from?
The electron “sees” a Lorentz contracted period
(5.9)
and emits radiation in its frame of reference at frequency
Observed in the laboratory frame of reference, this radiation
is Doppler shifted to a frequency
(5.10)
On-axis (θ = 0) the observed frequency is
Professor David Attwood
Univ. California, Berkeley
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
Ch05_Eq09_10VG.ai
Physically, where does the
λ = λu/2γ 2 come from?
By definition γ =
1
1 – β2
; γ2 =
1
1

(1 – β)(1 + β)
2(1 – β)
thus
and the observed wavelength is
(5.11)
Give examples.
Professor David Attwood
Univ. California, Berkeley
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
Ch05_Eq11VG.ai
What about the off-axis θ  0 radiation?
θ2
For θ ≠ 0, take cos θ = 1 –
+ . . . , then
2
(5.10)
c/λu
c/λu
c/(1 – β)λu
=
=
1 – β (1 – θ2/2 + . . . )
1 – β + βθ2/2 – . . .
1 + βθ2/2(1 – β). . .
The observed wavelength is then
(5.12)
exhibiting a reduced Doppler shift off-axis, i.e., longer wavelengths.
This is a simplified version of the “Undulator Equation”.
Professor David Attwood
Univ. California, Berkeley
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
Ch05_Eq10_12VG.ai
The Undulator’s “Central Radiation Cone”
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
(5.14)
Write the undulator equation twice, once for on-axis radiation (θ = 0) and once for wavelengthshifted radiation off-axis at angle θ:
λ0 + ∆λ =
λ0 =
λu
2γ2
(1 + γ2θ2)
λu
2γ2
(5.13)
divide and simplify to
Combining the two equations (5.13 and 5.14)
defines θcen :
γ2θ2cen 
1
N
, which gives
(5.15)
This is the half-angle of the “central radiation cone”, defined as containing radiation of ∆λ/λ = 1/N.
Professor David Attwood
Univ. California, Berkeley
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
Ch05_Eq13_15VG_Jan06.ai
The Undulator Radiation Spectrum
in Two Frames of Reference
ω′ ~ N
–
∆ω′
is
ax
f
f
O
Frequency, ω′
Execution of N electron oscillations
produces a transform-limited
spectral bandwidth, ∆ω′/ω′ = 1/N.
Professor David Attwood
Univ. California, Berkeley
Near axis
dP
dΩ
dP′
dΩ′
Frequency, ω
The Doppler frequency shift has a
strong angle dependence, leading
to lower photon energies off-axis.
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
Ch05_F12VG.ai
The Narrow (1/N) Spectral Bandwidth of Undulator
Radiation Can be Recovered in Two Ways
θ
With a pinhole aperture
Pinhole
aperture
dP
dΩ
dP
dΩ
∆λ
λ
ω
ω
Grating
monochromator
With a monochromator
2θ 
1
γ
∆λ
1
λ
Professor David Attwood
Univ. California, Berkeley
Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
Exit
slit
1
∆λ

N
λ
θ 1
γ N
Ch05_F13_14VG.ai