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Synchrotron radiation
R. Bartolini
John Adams Institute for Accelerator Science, University of Oxford
and
Diamond Light Source
JUAS 201427-31 January 2014
2/71
Contents
Introduction to synchrotron radiation
properties of synchrotron radiationsynchrotron light sourcesangular distribution of power radiated by accelerated particlesangular and frequency distribution of energy radiatedradiation from undulators and wigglers
Storage ring light sources
electron beam dynamics in storage ringsradiation damping and radiation excitationemittance and brilliancelow emittance latticesdiffraction limited storage rings
R. Bartolini, JUAS, 27-31 January 2014
What is synchrotron radiationElectromagnetic radiation is emitted by charged particles when accelerated
The electromagnetic radiation emitted when the charged particles are accelerated radially (v a) is called synchrotron radiation
It is produced in the synchrotron radiation sources using bending magnets undulators and wigglers
R. Bartolini, JUAS, 27-31 January 2014 3/71
Synchrotron radiation sources properties (I)
Broad Spectrum which covers from microwaves to hard X-rays:
the user can select the wavelength required for experiment;
either with a monochromator or adjusting the emission wavelength of insertion devices
synchrotron light
4/71R. Bartolini, JUAS, 27-31 January 2014
Synchrotron radiation sources properties (II)
High Flux: high intensity photon beam, allows rapid experiments or use of weakly scattering crystals;
High Brilliance (Spectral Brightness): highly collimated photon beam generated by a small divergence and small size source
Polarisation: both linear and circular (with IDs)
Pulsed Time Structure: pulsed length down to
High Stability: submicron source stability in SR
… and it can be computed!
Flux = Photons / ( s BW)
Brilliance = Photons / ( s mm2 mrad2 BW )Partial coherence in SRs
Full T coherence in FELs
10s ps in SRs
10s fs in FELs
Peak Brilliance
1.E+02
1.E+04
1.E+06
1.E+08
1.E+10
1.E+12
1.E+14
1.E+16
1.E+18
1.E+20
Bri
gh
tnes
s (P
ho
ton
s/se
c/m
m2 /m
rad
2 /0.1
%)
Candle
X-ray tube
60W bulb
X-rays from Diamond will be 1012 times
brighter than from an X-ray tube,
105 times brighter than the SRS !
diamond
R. Bartolini, JUAS, 27-31 January 2014 6/71
Layout of a synchrotron radiation source (I)
Electrons are generated and accelerated in a linac, further
accelerated to the required energy in a booster and injected and stored in
the storage ring
The circulating electrons emit an intense beam of synchrotron radiation
which is sent down the beamline
R. Bartolini, JUAS, 27-31 January 2014 8/71
Evolution of synchrotron radiation sources (I)
• First observation:
1947, General Electric, 70 MeV synchrotron
• First user experiments:
1956, Cornell, 320 MeV synchrotron
• 1st generation light sources: machine built for High Energy Physics or other purposes used parasitically for synchrotron radiation
• 2nd generation light sources: purpose built synchrotron light sources, SRS at Daresbury was the first dedicated machine (1981 – 2008)
• 3rd generation light sources: optimised for high brilliance with low emittance and Insertion Devices; ESRF, Diamond, …
R. Bartolini, JUAS, 27-31 January 2014 10/71
Evolution of synchrotron radiation sources (II)
• 4th generation light sources: photoinjectors LINAC based Free Electron Laser sources;
FLASH (DESY) 2007
LCLS (SLAC) 2009
SACLA (Japan) 2011
Elettra (Italy) 2012
and in the near(?) future
• 4th generation light sources storage ring based: diffraction limited storage rings
• …and even a 5th generation with more compact and advanced accelerator technologies e.g. based on laser plasma wakefield accelerators
R. Bartolini, JUAS, 27-31 January 2014 11/71
1992 ESRF, France (EU) 6 GeVALS, US 1.5-1.9 GeV
1993 TLS, Taiwan 1.5 GeV1994 ELETTRA, Italy 2.4 GeV
PLS, Korea 2 GeVMAX II, Sweden 1.5 GeV
1996 APS, US 7 GeVLNLS, Brazil 1.35 GeV
1997 Spring-8, Japan 8 GeV1998 BESSY II, Germany 1.9 GeV2000 ANKA, Germany 2.5 GeV
SLS, Switzerland 2.4 GeV2004 SPEAR3, US 3 GeV
CLS, Canada 2.9 GeV2006: SOLEIL, France 2.8 GeV
DIAMOND, UK 3 GeV ASP, Australia 3 GeVMAX III, Sweden 700 MeVIndus-II, India 2.5 GeV
2008 SSRF, China 3.4 GeV2009 PETRA-III, Germany 6 GeV2011 ALBA, Spain 3 GeV
3rd generation storage ring light sources
ESRF
SSRF
> 2013 NSLS-II, US 3 GeV
MAX-IV, Sweden 1.5-3 GeVSOLARIS, Poland 1.5 GeVSESAME, Jordan 2.5 GeV
TPS, Taiwan 3 GeV CANDLE, Armenia 3 GeV
PEP-X, USA 4.5 GeVSpring8-II, Japan 6 GeV
3rd generation storage ring light sources
under construction or planned NLSL-II
Max-IV
13/71R. Bartolini, JUAS, 27-31 January 2014
Main components of a storage ring
Dipole magnets to bend the electrons Quadrupole magnets to focus the electrons
Sextupole magnets to focus off-energy electrons (mainly)
RF cavities to replace energy losses due to the emission of synchrotron radiation
Main components of a storage ring
Insertion devices (undulators) to generate high brilliance radiation
Insertion devices (wiggler) to reach high photon energies
16/71R. Bartolini, JUAS, 27-31 January 2014
Photon energy
Brilliance
Flux
Stability
Polarisation
Time structure
Ring energy
Small Emittance
Insertion Devices
High Current; Feedbacks
Vibrations; Orbit Feedbacks; Top-Up
Short bunches; Short pulses
Accelerator physics and technology challengesAccelerator physics and technology challenges
17/71R. Bartolini, JUAS, 27-31 January 2014
The brilliance of the photon beam is determined (mostly) by the electron beam emittance that defines the source size and divergence
Brilliance and low emittanceBrilliance and low emittance
''24 yyxx
fluxbrilliance
2,
2, ephexx
2,
2,'' ' ephexx
2)( xxxx D
2' )'( xxxx D
Brilliance with IDsBrilliance with IDs
Medium energy storage rings with In-vacuum undulators operated at low gaps (e.g. 5-7 mm) can reach 10 keV with a brilliance of 1020 ph/s/0.1%BW/mm2/mrad2
Thanks to the progress with IDs technology storage ring light sources can cover a photon range from few tens of eV to tens 10 keV or more with high brilliance
19/71R. Bartolini, JUAS, 27-31 January 2014
Many ways to use x-rays
scattering SAXS& imaging
diffractioncrystallography& imaging
photo-emission(electrons)
electronic structure& imaging
from the synchrotron
fluorescence EXAFSXRF imaging
absorption
SpectroscopyEXAFSXANES & imaging
to the detector
20/71R. Bartolini, JUAS, 27-31 January 2014
Applications
X-ray fluorescence imaging revealed the hidden text by revealing the iron contained in the ink used by a 10th century scribe. This x-ray image shows the lower left corner of the
page.
Biology
Medicine, Biology, Chemistry, Material Science, Environmental Science and more
A synchrotron X-ray beam at the SSRL facility illuminated an obscured work erased, written
over and even painted over of the ancient mathematical genius Archimedes, born 287 B.C.
in Sicily.
ArcheologyReconstruction of the 3D structure of a nucleosome
with a resolution of 0.2 nm
The collection of precise information on the molecular structure of chromosomes and their
components can improve the knowledge of how the genetic code of DNA is maintained and reproduced
21/71R. Bartolini, JUAS, 27-31 January 2014
Life science examples: DNA and myoglobin
Photograph 51 Franklin-Gosling
DNA (form B)
1952
Franklin and Gosling used a X-ray tube:
Brilliance was 108 (ph/sec/mm2/mrad2/0.1BW)
Exposure times of 1 day were typical (105 sec)
e.g. Diamond provides a brilliance of 1020
100 ns exposure would be sufficient
Nowadays pump probe experiment in life science are performed using 100 ps pulses from storage ring light
sources: e.g. ESRF myoglobin in action
Lienard-Wiechert potentials (II)
Using the properties of the Dirac deltas we can integrate and obtain the Lienard-Wiechert potentials
ret0 R)n1(
e
4
1)t,x(
ret0 R)n1(
e
c4
1)t,x(A
These are the potentials of the em fields generated by the charged particle in motion.
The trajectory itself is determined by external electric and magnetic fields
23/71R. Bartolini, JUAS, 27-31 January 2014
The general solutions for the wave equation driven by a time varying charge and current density read (in the Lorentz gauge)
V
3ret
0
'xd|'xx|
)t,'xx(
4
1)t,x(
where ret means retarted (see next slide)c
|)t(x)t(x|tt ret
ret
V
3ret0 'xd|'xx|
)t,'xx(J
4)t,x(A
Lienard-Wiechert Potentials (III)
c
tRtt
)'('[ ]ret means computed at time t’
Potentials and fields at position x at time t are determined by the characteristic of the electron motion at a time t’ t’ – t is the time it takes for the em radiation to travel the distance R(t’)
i.e. grey is the position of the electron at time t
24/71R. Bartolini, JUAS, 27-31 January 2014
The Lienard-Wiechert fields
The computation has to be done carefully since the potentials depend on t’, not t. The factor dt/dt’ represents the Doppler factor. We get
The electric and magnetic fields are computed from the potentials using
velocity field
retretRn
nn
c
e
Rn
netxE
30
2320 )1(
)(
4)1(4),(
ritEn
c
1)t,x(B
and are called Lienard-Wiechert fields
acceleration fieldR
1
nBE ˆ
2R
1
AB
t
AE
If we consider the acceleration field we have
and the correct dependence as 1/R as for radiation field
25/71R. Bartolini, JUAS, 27-31 January 2014
Power radiated
Power radiated by a particle on a surface is the flux of the Poynting vector
BE1
S0
dntxStS ),())((
Angular distribution of radiated power
22
R)n1)(nS(d
Pd
radiation emitted by the particle
We will analyse two cases:
acceleration orthogonal to the velocity → synchrotron radiation
acceleration parallel to the velocity → bremmstrahlung
26/71R. Bartolini, JUAS, 27-31 January 2014
Assuming and substituting the acceleration field0
2
02
22
0
2
)(4
1
nnc
eER
cd
Pdacc
ret
acc R
nn
c
etxE
)(
4),(
0
Synchrotron radiation: non relativistic motion (I)
27/71R. Bartolini, JUAS, 27-31 January 2014
The angular distribution of the power radiated is given by
Working out the double cross product
We have
Since
22
02
22
sin)4(
c
e
d
Pd
where is the angle between the acceleration and the observation direction, we finally get
Synchrotron radiation: non relativistic motion (II)
28/71R. Bartolini, JUAS, 27-31 January 2014
The angular distribution of power reads
Integrating over the angles gives the total radiated power
2
0
2
c6
eP
Larmor’s formula
Synchrotron radiation: non relativistic motion (III)
29/71R. Bartolini, JUAS, 27-31 January 2014
This integral gives
It shows that radiation is emitted when the particle is accelerated.Using
mc
p
2
320
2
dt
pd
cm6
eP
we have (to be used later for the generalisation to the relativistic case)
In the relativistic case the total radiated power is computed in the same way. Using only the acceleration field (large R)
5
2
02
22
acco
2
)n1(
)n(n
c4
e)n1(ER
c
1
d
Pd
The emission is peaked in the direction of the velocity
The pattern depends on the details of velocity and acceleration but it is dominated by the denominator
Synchrotron radiation: relativistic motion (I)
30/71R. Bartolini, JUAS, 27-31 January 2014
ret
30 R)n1(
)n(n
c4
e~)t,x(E
The angular distribution of the power emitted is (use the retarted time!)
velocity acceleration: synchrotron radiation (I)
Assuming and substituting the acceleration field
22
22
30
2
22
5
2
20
2
22
)cos1(
cossin1
)cos1(
1
4)1(
)(
4
c
e
n
nn
c
e
d
Pd
When the electron velocity approaches the speed of light, the emission pattern is sharply collimated forward
cone aperture
1/
31/71R. Bartolini, JUAS, 27-31 January 2014
32/71R. Bartolini, JUAS, 27-31 January 2014
Courtesy K. Wille
velocity acceleration: synchrotron radiation (II)
Total radiated power via synchrotron radiation
2254
0
4
2
4
0
22
2
320
24
4
2
0
24
2
0
2
66666BE
cm
ece
dt
pd
cm
e
E
E
c
e
c
eP
o
Integrating over the whole solid angle we obtain the total instantaneous power radiated by one electron
• Strong dependence 1/m4 on the rest mass
• proportional to 1/2 ( is the bending radius)
• proportional to B2 (B is the magnetic field of the bending dipole)
The radiation power emitted by an electron beam in a storage ring is very high.
The surface of the vacuum chamber hit by synchrotron radiation must be cooled.
33/71R. Bartolini, JUAS, 27-31 January 2014
velocity || acceleration: bremsstrahlung
Assuming || and substituting the acceleration field
5
2
02
22
5
2
02
2
)cos1(
sin
c4
e
)n1(
)n(n
c4
e
d
dP
2
11151
3
1arccos 2
max
2
320
2
dt
pd
cm6
eP
Integrating over the angles as before gives the total radiated power
34/71R. Bartolini, JUAS, 27-31 January 2014
Back to the general expression for the acceleration field, integrating over the angles gives the total radiated power
The total radiated power can also be computed by relativistic transformation of the 4-acceleration in Larmor’s formula
Relativistic generalization of Larmor’s formula 226
0
2
)()(6
c
eP
2
320
2
dt
pd
cm6
eP
with t
We build the relativistic invariant
anddt
dp
dt
pd
2
2
2
320
2
d
dE
c
1
d
pd
cm6
eP
35/71R. Bartolini, JUAS, 27-31 January 2014
Comparison of radiation from linear and circular trajectories
Comparison of radiation from linear and circular trajectories
2
320
2
dt
pd
cm6
eP
For a linear trajectory
36/71R. Bartolini, JUAS, 27-31 January 2014
The particle energy is
22222 cp)mc(E
Therefore
d
dppc
d
dEE 2 and
d
dpv
d
dE
Inserting in the formula we get
Comparison of radiation from linear and circular trajectories
2
2
2
320
2
d
dE
c
1
d
pd
cm6
eP
Repeating for a circular trajectory
37/71R. Bartolini, JUAS, 27-31 January 2014
The particle energy is now constant
22222 cp)mc(E
Therefore
0d
dE
Inserting in the formula we get the total radiated power in a circular trajectory
22
320
2
dt
pd
cm6
eP
P(v || a) 1/2 P(v a)
This is 2 larger than the linear case
Energy loss via synchrotron radiation emission in a storage ring
i.e. Energy Loss per turn (per electron) )(
)(46.88
3)(
4
0
42
0 m
GeVEekeVU
4
0
2
0 3
2 e
cPPTPdtU b
Power radiated by a beam of average current Ib: this power loss has to be compensated by the RF system
Power radiated by a beam of average current Ib in a dipole of length L
(energy loss per second)
In the time Tb spent in the bendings the particle loses the energy U0
e
TIN revb
tot
)(
)()(46.88
3)(
4
0
4
m
AIGeVEI
ekWP b
2
4
20
4
)(
)()()(08.14
6)(
m
GeVEAImLLI
ekWP b
38/71R. Bartolini, JUAS, 27-31 January 2014
The radiation integral (I)
Angular and frequency distribution of the energy received by an observer
2
)/)((22
0
23
)1(
)(
44
dten
nn
c
e
dd
Wd ctrnti
Neglecting the velocity fields and assuming the observer in the far field: n constant, R constant
22
0
3
)(ˆ2
EcRdd
Wd
Radiation Integral
dttERcdtd
Pd
d
Wd 20
22
|)(|
The energy received by an observer (per unit solid angle at the source) is
0
20
2
|)(|2 dERcd
Wd
Using the Fourier Transform we move to the frequency space
39/71R. Bartolini, JUAS, 27-31 January 2014
The radiation integral (II)
2
)/)((2
0
223
)(44
dtennc
e
dd
Wd ctrnti
determine the particle motion
compute the cross products and the phase factor
integrate each component and take the vector square modulus
)();();( tttr
The radiation integral can be simplified to [see Jackson]
How to solve it?
Calculations are generally quite lengthy: even for simple cases as for the radiation emitted by an electron in a bending magnet they require Airy integrals or the modified Bessel functions (available in MATLAB)
40/71R. Bartolini, JUAS, 27-31 January 2014
Radiation integral for synchrotron radiation
Trajectory of the arc of circumference [see Jackson]
sincossin)( ||
ctctnn
cossin
)( ct
ct
c
trnt
Substituting into the radiation integral and introducing 2/32231
c3
In the limit of small angles we compute
0,sin,cos1)( t
ct
ctr
)(
1)(1
3
2
162
3/122
222
3/2
222
2
20
3
23
KK
cc
e
dd
Wd
41/71R. Bartolini, JUAS, 27-31 January 2014
Critical frequency and critical angle
Using the properties of the modified Bessel function we observe that the radiation intensity is negligible for >> 1
11c3
2/3223
Critical frequency3
2
3
cc
Higher frequencies have smaller critical
angle
Critical angle
3/11
cc
)(
1)(1
3
2
162
3/122
222
3/2
222
2
20
3
23
KK
cc
e
dd
Wd
For frequencies much larger than the critical frequency and angles much larger than the critical angle the synchrotron radiation emission is negligible
42/71R. Bartolini, JUAS, 27-31 January 2014
Frequency distribution of radiated energyIntegrating on all angles we get the frequency distribution of the energy radiated
C
dxxKc
ed
dd
Id
d
dW
C
/
3/50
2
4
3
)(4
3
<< c
3/1
0
2
4
cc
e
d
dW >> c
cec
e
d
dW
c
/
2/1
0
2
42
3
often expressed in terms of the function S() with = /c
dxxKS )(8
39)( 3/5 1)(
0
dS
)(9
2)(
4
3
0
2
/
3/50
2
Sc
edxxK
c
e
d
dW
cc
43/71R. Bartolini, JUAS, 27-31 January 2014
Frequency distribution of radiated energyIt is possible to verify that the integral over the frequencies agrees with the previous expression for the total power radiated [Hubner]
2
4
0
2
0
3/50
2
0
0
6)(
9
211
c
cedxxKd
c
e
Td
d
dW
TT
UP c
bbb
The frequency integral extended up to the critical frequency contains half of the total energy radiated, the peak occurs approximately at 0.3c
50% 50%
3
2
3
ccc
For electrons, the critical energy in practical units reads
][][665.0][
][218.2][ 2
3
TBGeVEm
GeVEkeVc
It is also convenient to define the critical photon energy as
44/71R. Bartolini, JUAS, 27-31 January 2014
Heuristic derivation of critical frequency
Synchrotron radiation is emitted in an arc of circumference with radius , Angle of emission of radiation is 1/ (relativistic argument), therefore
c
T transit time in the arc of dipole
During this time the electron travels a distance
Tcs
The time duration of the radiation pulse seen by the observer is the difference between the time of emission of the photons and the time travelled by the electron in the arc
3
2
c2)1(
1
c1
1
cc
sT
The width of the Fourier transform of the pulse is
3c2
45/71R. Bartolini, JUAS, 27-31 January 2014
Polarisation of synchrotron radiation
In the orbit plane = 0, the polarisation is purely horizontal
Polarisation in the orbit plane
Polarisation orthogonal to the orbit plane
Integrating on all the angles we get a polarization on the plan of the orbit 7 times larger than on the plan perpendicular to the orbit
22
22
2/5220
52
0
32
17
51
)1(
1
64
7
e
ddd
Id
d
Wd
Integrating on all frequencies we get the angular distribution of the energy radiated
)(
1)(1
3
2
162
3/122
222
3/2
222
2
20
3
23
KK
cc
e
dd
Wd
46/71R. Bartolini, JUAS, 27-31 January 2014
Synchrotron radiation emission as a function of beam the energy
Dependence of the frequency distribution of the energy radiated via synchrotron emission on the electron beam energy
No dependence on the energy at longer
wavelengths
3
2
3
ccc
Critical frequency
3
2
3
cc
Critical angle
3/11
cc
Critical energy
47/71R. Bartolini, JUAS, 27-31 January 2014
Brilliance with IDs (medium energy light sources)Brilliance with IDs (medium energy light sources)
Medium energy storage rings with in-vacuum undulators operated at low gaps (e.g. 5-7 mm) can reach 10 keV with a brilliance of 1020 ph/s/0.1%BW/mm2/mrad2
Brilliance dependence
with current
with energy
with emittance
48/71R. Bartolini, JUAS, 27-31 January 2014
Radiation from undulators and wigglers
Radiation emitted by undulators and wigglers
Types of undulators and wigglers
49/71R. Bartolini, JUAS, 27-31 January 2014
Undulators and wigglersPeriodic array of magnetic poles providing a sinusoidal magnetic field on axis:
),0),sin(,0( 0 zkBB u
ztK
ctxtK
tr uu
zuu ˆ)2cos(
16ˆsin
2)(
2
2
mc
eBK u
20
Undulator parameter
Constructive interference of radiation emitted at different poles
Solution of equation of motions:
nd uu cos
21
2
11
2
2
Kz
22
2
2 21
2
K
nu
n
50/71R. Bartolini, JUAS, 27-31 January 2014
From the lecture on synchrotron radiation
Continuous spectrum characterized by c = critical energy
c(keV) = 0.665 B(T)E2(GeV)
eg: for B = 1.4T E = 3GeV c = 8.4 keV
(bending magnet fields are usually lower ~ 1 – 1.5T)
Quasi-monochromatic spectrum with peaks at lower energy than a wiggler
undulator - coherent interference K < 1Max. angle of trajectory < 1/
wiggler - incoherent superposition K > 1Max. angle of trajectory > 1/
bending magnet - a “sweeping searchlight”
2
2 21
2 2u u
n
K
n n
2
2
[ ]( ) 9.496
[ ] 12
n
u
nE GeVeV
Km
51/71R. Bartolini, JUAS, 27-31 January 2014
Radiation integral for a linear undulator (I)The angular and frequency distribution of the energy emitted by a wiggler is computed again with the radiation integral:
2
)/ˆ(2
0
223
)ˆ(ˆ44
dtennc
e
dd
Wd crnti
Using the periodicity of the trajectory we can split the radiation integral into a sum over Nu terms
2)1(2
22/
2/
)/ˆ(2
0
223
...1)ˆ(ˆ44
u
u
u
Niiic
c
crnti eeedtennc
e
dd
Wd
)(
2
res
where
22
2
2 21
2)(
Ku
res)(
2)(
resres
c
52/71R. Bartolini, JUAS, 27-31 January 2014
Radiation integral for a linear undulator (II)
),,()(4 0
2223
KFNL
c
Ne
dd
Wdn
res
22/
2/
)/ˆ(0
0
)ˆ(ˆ),,(
c
c
crntin dtennKF
= - n res()
))(/(sin
))(/(sin
)( 22
2
res
res
res N
NNL
The sum on generates a series of sharp peaks in the frequency spectrum harmonics of the fundamental wavelength
The radiation integral in an undulator or a wiggler can be written as
The integral over one undulator period generates a modulation term Fn which depends on the angles of observations and K
53/71R. Bartolini, JUAS, 27-31 January 2014
Radiation integral for a linear undulator (II)
e.g. on axis ( = 0, = 0): )0,0,()0(4 0
2223
KFNLc
Ne
dd
Wdn
res
2
2
1
2
12
22
)()()2/1(
)0,0,(
ZJZJ
K
KnKF nnn )2/1(4 2
2
K
nKZ
Only odd harmonic are radiated on-axis;
as K increases the higher harmonic becomes stronger
Off-axis radiation contains many harmonics
54/71R. Bartolini, JUAS, 27-31 January 2014
Angular patterns of the radiation emitted on harmonics
Angular spectral flux as a function of frequency for a linear undulator; linear polarisation solid, vertical polarisation dashed (K = 2)
21
2
2
21
Ku
Fundamental wavelength emitted by the undulator
21
2
2nd harmonic, not emitted on-axis !
31
2
3rd harmonic, emitted on-axis !
55/71R. Bartolini, JUAS, 27-31 January 2014
Synchrotron radiation emission from a bending magnet
Dependence of the frequency distribution of the energy radiated via synchrotron emission on the electron beam energy
No dependence on the energy at longer wavelengths
3
2
3
ccc
Critical frequency
3
2
3
cc
Critical angle
3/11
cc
Critical energy
56/71R. Bartolini, JUAS, 27-31 January 2014
Undulators and wigglers (large K)
For large K the wiggler spectrum becomes similar to the bending magnet spectrum, 2Nu times larger.
Fixed B0, to reach the bending magnet critical wavelength we need:
Radiated intensity emitted vs K
K 1 2 10 20
n 1 5 383 3015
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Undulator tuning curve (with K)
Brightness of a 5 m undulator 42 mm period with maximum K = 2.42 (ESRF) Varying K one varies the wavelength emitted at various harmonics (not all wavelengths of this graph are emitted at a single time)
high K
low K
mc
eBK u
20
Undulator parameter
K decreases by opening the gapof the undulator (reducing B)
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Spectral brightness of undulators of wiggler
Comparison of undulators for a 1.5 GeV ring for three harmonics (solid dashed and dotted) compared with a wiggler and a bending magnet (ALS)
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Diamond undulators and wigglerSpectral brightness for undulators and wigglers in state-of-the-art 3rd generation light sources
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Summary of radiation characteristics of undulators or wiggler
Undulators have weaker field or shorter periods (K< 1)
Produce narrow band radiation and harmonics
Intensity is proportional to Nu2
Wigglers have higher magnetic field (K >1)
Produce a broadband radiation
Intensity is proportional to Nu
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Type of undulators and wigglers
Electromagnetic undulators: the field is generated by current carrying coils; they may have iron poles;
Permanent magnet undulators: the field is generated by permanent magnets Samarium Cobalt (SmCo; 1T) and Neodymium Iron Boron (NdFeB; 1.4T); they may have iron poles (hybrid undulators);
APPLE-II: permanent magnets arrays which can slide allowing the polarisation of the magnetic field to be changed from linear to circular
In-vacuum: permanent magnets arrays which are located in-vacuum and whose gap can be closed to very small values (< 5 mm gap!)
Superconducting wigglers: the field is generated by superconducting coils and can reach very high peak fields (several T, 3.5 T at Diamond)
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Electromagnetic undulators (I)
Period 64 mm
14 periods
Min gap 19 mm
Photon energy < 40 eV (1 keV with EM undulators)
HU64 at SOLEIL:
variable polarisation electromagnetic undulator
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Electromagnetic undulators (II)
Depending on the way the coil power supplies are powered it can generate linear H, linear V or circular polarisations
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Permanent magnet undulators
Halback configuration
hybrid configuration with steel poles
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In-vacuum undulatorsU27 at Diamond
27 mm, 73 periods 7 mm gap, B = 0.79 T; K = 2
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Apple-II type undulators (I)
HU64 at Diamond; 33 period of 64 mm; B = 0.96 T; gap 15 mm; Kmax = 5.3
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Apple-II type undulators (II)
Four independent arrays of permanent magnets
Diagonally opposite arrays move longitudinal, all arrays move vertically
Sliding the arrays of magnetic pole it is possible to control the polarisation of the radiation emitted
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Superconducting Wigglers
Superconducting wigglers are used when a high magnetic field is required
3 - 10 T
They need a cryogenic system to keep the coil superconductive
Nb3Sn and NbTi wires
SCMPW60 at Diamond
3.5 T coils cooled at 4 K
24 period of 64 mm
gap 10 mm
Undulator K = 21
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Summary and bibliographyAccelerated charged particles emit electromagnetic radiation
Synchrotron radiation is stronger for light particles and is emitted by bending magnets in a narrow cone within a critical frequency
Undulators and wigglers enhance the synchrotron radiation emission
J. D. Jackson, Classical Electrodynamics, John Wiley & sons.
E. Wilson, An Introduction to Particle Accelerators, OUP, (2001)
M. Sands, SLAC-121, (1970)
R. P. Walker, CAS CERN 94-01 and CAS CERN 98-04
K. Hubner, CAS CERN 90-03
J. Schwinger, Phys. Rev. 75, pg. 1912, (1949)
B. M. Kincaid, Jour. Appl. Phys., 48, pp. 2684, (1977).
R.P. Walker: CAS 98-04, pg. 129
A. Ropert: CAS 98-04, pg. 91
P. Elleaume in Undulators, Wigglers and their applications, pg. 69-107
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