Intro to Synchrotron Radiation, Bending Magnet Radiationattwood/srms/2007/Lec08.pdf · 13 2.70 Al...
Transcript of Intro to Synchrotron Radiation, Bending Magnet Radiationattwood/srms/2007/Lec08.pdf · 13 2.70 Al...
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
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
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
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
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
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
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 Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
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
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
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
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
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
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
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
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
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
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
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
λu
2γ2
∆λλ
1N
~
θcen –1
γ∗ N
cen =
~
Professor David AttwoodUniv. California, Berkeley Intro Synchrotron Radiation, Bending Magnet Radiation, EE290F, 8 Feb. 2007
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
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
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
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
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?
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?
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
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
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