Bending Magnet Critical
Photon Energy and Undulator
Central Radiation Cone
David Attwood
University of California, Berkeley
(http://www.coe.berkeley.edu/AST/srms)
Bending Magnet Critical Photon Energy and Undulator Central Radiation Cone, EE290F, 13 Feb 2007
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
Bending Magnet Critical Photon Energy and Undulator Central Radiation Cone, EE290F, 13 Feb 2007
Ch05_BendMagRadius.ai
Bending Magnet Radiation
(a)
(b)
1
θ= γ
2
A
R sinθ
Radius R
B
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
Bending Magnet Critical Photon Energy and Undulator Central Radiation Cone, EE290F, 13 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
Bending Magnet Critical Photon Energy and Undulator Central Radiation Cone, EE290F, 13 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
Bending Magnet Critical Photon Energy and Undulator Central Radiation Cone, EE290F, 13 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:
•
•
•
Disadvantages: •
Professor David Attwood
Univ. California, Berkeley
covers broad spectral range
least expensive
most accessable
limited coverage of
hard x-rays
• not as bright as undulator
ALS
Ee = 1.9 GeV
Ι = 400 mA
B = 1.27 T
ωc = 3.05 keV
1013
1012
1011
50%
∆θ = 1mrad
∆ω/ω = 0.1%
0.01
Ec
50%
4Ec
0.1
1
10
Photon energy (keV)
Bending Magnet Critical Photon Energy and Undulator Central Radiation Cone, EE290F, 13 Feb 2007
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
Bending Magnet Critical Photon Energy and Undulator Central Radiation Cone, EE290F, 13 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
Bending Magnet Critical Photon Energy and Undulator Central Radiation Cone, EE290F, 13 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
Bending Magnet Critical Photon Energy and Undulator Central Radiation Cone, EE290F, 13 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
Bending Magnet Critical Photon Energy and Undulator Central Radiation Cone, EE290F, 13 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
Bending Magnet Critical Photon Energy and Undulator Central Radiation Cone, EE290F, 13 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
Bending Magnet Critical Photon Energy and Undulator Central Radiation Cone, EE290F, 13 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
Bending Magnet Critical Photon Energy and Undulator Central Radiation Cone, EE290F, 13 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
Bending Magnet Critical Photon Energy and Undulator Central Radiation Cone, EE290F, 13 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.
Bending Magnet Critical Photon Energy and Undulator Central Radiation Cone, EE290F, 13 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
Bending Magnet Critical Photon Energy and Undulator Central Radiation Cone, EE290F, 13 Feb 2007
Exit
slit
1
∆λ

N
λ
θ 1
γ N
Ch05_F13_14VG.ai
Lorentz Space-Time Transformations (Appendix F)
X′
X
(F.1a)
(F.1b)
L′
S′
λ′u
θ′
S
Z′
v
(F.1c)
0
(F.2a)
θ
Z
(F.3)
(F.2a)
(F.2a)
Professor David Attwood
Univ. California, Berkeley
Bending Magnet Critical Photon Energy and Undulator Central Radiation Cone, EE290F, 13 Feb 2007
(F.4)
Ch05_Lorentz.ApxF.ai
Lorentz Transformations:
Frequency, Angles, Length and Time
Doppler frequency shifts
Angular transformations
(F.8a)
(F.9a)
(F.8b)
(F.9b)
Lorentz contraction of length
(F.10b)
(F.12)
(F.10a)
Time dilation
(F.13)
(F.11a)
(F.11b)
Professor David Attwood
Univ. California, Berkeley
Bending Magnet Critical Photon Energy and Undulator Central Radiation Cone, EE290F, 13 Feb 2007
Ch05_LorentzTrans.ai