Syllabus and slides
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Lecture 1: Overview and history of Particle accelerators (EW)
Lecture 2: Beam optics I (transverse) (EW)
Lecture 3: Beam optics II (longitudinal) (EW)
Lecture 4: Liouville's theorem and Emittance (RB)
Lecture 5: Beam Optics and Imperfections (RB)
Lecture 6: Beam Optics in linac (Compression) (RB)
Lecture 7: Synchrotron radiation (RB)
Lecture 8: Beam instabilities (RB)
Lecture 9: Space charge (RB)
Lecture 10: RF (ET)
Lecture 11: Beam diagnostics (ET)
Lecture 12: Accelerator Applications (Particle Physics) (ET)
Visit of Diamond Light Source/ ISIS / (some hospital if possible)
The slides of the lectures are available at
http://www.adams-institute.ac.uk/training/undergraduate
Dr. Riccardo Bartolini (DWB room 622) r.bartolini1@physics.ox.ac.uk
Lecture 10
Synchrotron radiation
properties of synchrotron radiation
synchrotron light sources
Lienard-Wiechert potentials
Angular distribution of power radiated by accelerated particles
Angular and frequency distribution of energy radiated:
Radiation from undulators and wigglers
R. Bartolini, John Adams Institute, 8 May 2013
2/30
What is synchrotron radiation
Electromagnetic 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, John Adams Institute, 8 May 2013
3/30
Synchrotron radiation sources properties
Broad Spectrum which covers
from microwaves to hard X-rays:
the user can select the
wavelength required for
experiment;
synchrotron light
High Flux: high intensity photon beam, allows rapid experiments or use of
weakly scattering crystals;
Flux = Photons / ( s  BW)
High Brilliance (Spectral Brightness): highly collimated photon beam
generated by a small divergence and small size source (partial coherence);
Brilliance = Photons / ( s  mm2  mrad2  BW )
High Stability: submicron source stability
Polarisation: both linear and circular (with IDs)
Pulsed Time Structure: pulsed length down to tens of picoseconds allows the
R. Bartolini,of
John
Adams Institute,
May 2011
4/30
resolution
process
on the18same
time scale
A brief history of storage ring synchrotron
radiation sources
• 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, …
• 4th generation light sources: photoinjectors LINAC based Free Electron
Laser sources; FLASH (DESY), LCLS (SLAC), …
R. Bartolini, John Adams Institute, 8 May 2013
5/30
diamond
1.E+20
1.E+18
1.E+16
2
2
Brightness (Photons/sec/mm /mrad /0.1%)
Peak Brilliance
1.E+14
1.E+12
X-rays from Diamond
will be 1012 times
brighter than from
an X-ray tube,
105 times brighter
than the SRS !
1.E+10
1.E+08
1.E+06
X-ray
tube
60W bulb
Candle
1.E+04
1.E+02
R. Bartolini, John Adams Institute, 8 May 2013
6/30
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, John Adams Institute, 8 May 2013
7/30
Layout of a synchrotron radiation source
Main components of a storage ring
Insertion devices (undulators) to
generate high brilliance radiation
R. Bartolini, John Adams Institute, 8 May 2013
Insertion devices (wiggler) to
reach high photon energies
9/30
Many ways to
use x-rays
photo-emission electronic structure
(electrons)
& imaging
diffraction
crystallography
& imaging
scattering SAXS
& imaging
absorption
from the synchrotron
to the detector
Spectroscopy
EXAFS
XANES
& imaging
fluorescence EXAFS
XRF imaging
R. Bartolini, John Adams Institute, 8 May 2013
10/30
3rd generation storage ring light sources
1992
1993
1994
1996
1997
1998
2000
2004
2006:
2008
2009
2011
ESRF, France (EU)
ALS, US
TLS, Taiwan
ELETTRA, Italy
PLS, Korea
MAX II, Sweden
APS, US
LNLS, Brazil
Spring-8, Japan
BESSY II, Germany
ANKA, Germany
SLS, Switzerland
SPEAR3, US
CLS, Canada
SOLEIL, France
DIAMOND, UK
ASP, Australia3 GeV
MAX III, Sweden
Indus-II, India
SSRF, China
PETRA-III, D
ALBA, E
6 GeV
1.5-1.9 GeV
1.5 GeV
2.4 GeV
2 GeV
1.5 GeV
7 GeV
1.35 GeV
8 GeV
1.9 GeV
2.5 GeV
2.4 GeV
3 GeV
2.9 GeV
2.8 GeV
3 GeV
ESRF
SSRF
700 MeV
2.5 GeV
3.4 GeV
6 GeV
3 GeV
11/30
Diamond Aerial views
June 2003
12/30
Oct 2006
Heuristic approach to synchrotron radiation
Synchrotron radiation is emitted in an arc of circumference with radius ,
Angle of emission of radiation is 1/ (relativistic argument), therefore
T 

c
transit time in the arc of dipole
During this time the electron travels a distance

s  cT 

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
s   1   1   2

  T 
   1 

c c    c  (1   ) 2c 3
The width of the Fourier
transform of the pulse is
 
2c 3

R. Bartolini, John Adams Institute, 8 May 2013
13/30
Lienard-Wiechert Potentials (I)
The equations for vector potential and scalar potential
1  2

 2


c t 2
0
2
1 2 A
J
 A 2 2  2
c t
c 0
2
with the current and charge densities of a single charged particle, i.e.
 ( x , t )  e (3) ( x  r (t ))
J ( x , t )  ev (t ) (3) ( x  r (t ))
have as solution the Lienard-Wiechert potentials
 ( x ,t ) 

1 
e
40  ( 1    n )R  ret
A( x ,t ) 


e
40 c  ( 1    n )R  ret
1
[ ]ret means computed at time t’
t  t '
R (t ' )
c
R. Bartolini, John Adams Institute, 8 May 2013
14/30
Lineard-Wiechert Potentials (II)
The electric and magnetic fields generated by the moving charge are
computed from the potentials
A
E  V 
B   A
t
and are called Lineard-Wiechert fields

e 
n 
e  n  (n   )   
E (x, t) 



40   2 (1    n ) 3 R 2  ret 40 c  (1    n ) 3 R 
ret
velocity field
acceleration field
B ( x ,t ) 
1

R
1

nE
c

rit

E  B  nˆ
Power radiated by a particle on a surface is the flux of the Poynting vector
S
1
0
E B
  ( S )(t )   S ( x , t )  n d

Angular distribution of radiated power [see Jackson]
d 2P
 ( S  n)(1  n   ) R 2
d
R. Bartolini, John Adams Institute, 8 May 2013
radiation emitted by the particle
15/30
velocity  acceleration: synchrotron radiation
Assuming   
d 2P
e2

d 4 2  0c 2
and substituting the acceleration field [Jackson]

n  (n   )  

2
(1  n   )5
e 
2
2
1

4 2  0c (1   cos  )3

sin 2  cos 2  
1  2
2
  (1   cos  ) 
cone aperture
 1/
When the electron velocity approaches the speed of light the emission
pattern is sharply collimated forward
R. Bartolini, John Adams Institute, 8 May 2013
16/30
Total radiated power via synchrotron radiation
Integrating over the whole solid angle we obtain the total instantaneous power
radiated by one electron (Larmor formula and its relativistic generalisation)
E
e2
dp 2 e 2c  4
e4


4
2 2
P
  





E
B
4
2 3
2
4 5
60c
60c
Eo
60 m c dt
60 
60 m c
e2
2
e2
2
4
2
• 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.
R. Bartolini, John Adams Institute, 8 May 2013
17/30
Energy loss via synchrotron radiation emission
in a storage ring
In the time Tb spent in the bendings
the particle loses the energy U0
2 e 2  4
U 0   Pdt  PTb  P

c
3 0 
i.e. Energy Loss per turn (per electron)
e 2 4
E (GeV ) 4
U 0 (keV ) 
 88.46
3 0 
 (m)
Power radiated by a beam of average
current Ib: this power loss has to be
compensated by the RF system
I b  Trev
e
4
e
E (GeV ) 4 I ( A)
P(kW ) 
I b  88.46
3 0 
 (m)
N tot 
Power radiated by a beam of average
e 4
L(m) I ( A) E (GeV ) 4
P(kW ) 
LI b  14.08
current Ib in a dipole of length L
60  2
 (m) 2
(energy loss per second)
R. Bartolini, John Adams Institute, 8 May 2013
18/30
The radiation integral (I)
The energy received by an observer (per unit solid angle at the source) is


d 2W
d 2P
 
dt  c 0  | RE (t ) |2 dt
d  d

Using the Fourier Transform we move to the frequency space

d 2W
 2c 0  | RE ( ) |2 d
d
0
Angular and frequency distribution of the energy received by an observer
2
d 3W
2 ˆ
 2 0cR E ( )
dd
Neglecting the velocity fields and assuming the observer in the far field:
n constant, R constant
2


3
2
dW
e
n  ( n   )   i ( t  n  r ( t ) / c )
Radiation Integral

e
dt
2
2

dd 40 4 c  (1  n   )

R. Bartolini, John Adams Institute, 8 May 2013

19/30
The radiation integral (II)
The radiation integral can be simplified to [see Jackson]
d 3W
e 2 2

dd 40 4 2 c

2
i ( t  n  r ( t ) / c )
n

(
n


)
e
dt


How to solve it?
 determine the particle motion
r (t );  (t );  (t )
 compute the cross products and the phase factor
 integrate each component and take the vector square modulus
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)
R. Bartolini, John Adams Institute, 8 May 2013
20/30
Radiation integral for synchrotron radiation
Trajectory of the arc of circumference [see Jackson]
 

c 
c
r ( t )    1  cos t , sin
t , 0 
 

 

In the limit of small angles we compute


 ct 
 ct 
    cos
 sin  
n  (n   )     || sin 
  
  


   ct 

 n  r (t ) 
 cos  
 t 
   t  sin 
c 

 c   

Substituting into the radiation integral and introducing
2


2 2 3/ 2

1


 
3
3c

d 3W
e 2  2 
 2 2
2 2 2
2
2



1



K
(

)

K
(

)
1/ 3
 2/3

2 2
d d 16 3 0 c  3c 2 
1






R. Bartolini, John Adams Institute, 8 May 2013

21/30
Critical frequency and critical angle
 2
d3W
e2


d d 16 3 0 c  3c 2
2


2 2


2 2


2
2
2
 1    K 2 / 3 ( ) 
K
(

)

1/ 3

1   2 2



Using the properties of the modified Bessel function we observe that the
radiation intensity is negligible for  >> 1


2 2 3/ 2

1


   1
3c 3
Higher frequencies
have smaller critical
angle
3c 3

2
Critical frequency
c 
Critical angle
1  
c   c 

1/ 3
For frequencies much larger than the critical frequency and angles much
larger than the critical angle the synchrotron radiation emission is negligible
R. Bartolini, John Adams Institute, 8 May 2013
22/30
Frequency distribution of radiated energy
It is possible to verify that the integral over the frequencies agrees with
the previous expression for the total power radiated [Hubner]



U 0 1 dW
1 2e 2
e 2c  4
P
 
d 
c   d  K 5 / 3 ( x)dx 
Tb Tb 0 d
Tb 9 0c 0
6 0c  2

The frequency integral extended up to the critical frequency contains
half of the total energy radiated, the peak occurs approximately at 0.3c
It is also convenient to define the critical
photon energy as
 c  c 
3 c 3

2 
For electrons, the critical energy in
practical units reads
50% 50%
E [GeV ]3
 c [keV ]  2.218
 0.665  E[GeV ]2  B [T ]
[m]
R. Bartolini, John Adams Institute, 8 May 2013
23/30
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
Critical frequency
3c 3
c 

2
Critical angle
No dependence on
the energy at longer
wavelengths
1  
c   c 

Critical energy
 c  c 
E [GeV ]3
 c [keV ]  2.218
 0.665  E[GeV ]2  B [T ]
[m]
R. Bartolini, John Adams Institute, 8 May 2013
1/ 3
3 c 3

2 
for electrons! How does this
change for protons?
24/30
Brilliance with IDs (medium energy light sources)
Brilliance dependence
with current
with energy
with emittance
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
R. Bartolini, John Adams Institute, 8 May 2013
25/30
Polarisation of synchrotron radiation
2

d 3W
e 2  2 
 2 2
2 2 2
2
2



1



K
(

)

K
(

)
1/ 3
 2/3

2 2
d d 16 3 0 c  3c 2 
1







Polarisation in
the orbit plane
Polarisation orthogonal
to the orbit plane
In the orbit plane  = 0, the polarisation is purely horizontal
Integrating on all frequencies we get the angular distribution of the energy
radiated

d 2W
d 3I
7e 2 5
1

d 
d 0 d d
640  (1   2 2 )5 / 2
 5  2 2 
1 
2 2
 7 1   
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
R. Bartolini, John Adams Institute, 8 May 2013
26/30
Undulators and wigglers
Periodic array of magnetic
poles providing a sinusoidal
magnetic field on axis:
B  (0, B0 sin( ku z ), 0,)
Insertion devices (undulators) to generate high brilliance radiation
Constructive interference of radiation
emitted by different electrons at
different poles
R. Bartolini, John Adams Institute, 8 May 2013
27/30
Synchrotron radiation from undulators and wigglers
Continuous spectrum characterized
by c = critical energy
bending magnet - a “sweeping searchlight”
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)
wiggler - incoherent superposition K > 1
Quasi-monochromatic spectrum with
peaks at lower energy than a wiggler
undulator - coherent interference K < 1
u  K 2  u
n 
1
 2
2 
2n 
2  n
nE[GeV ]2
 n (eV )  9.496
 K2 
u [m] 1 

2


R. Bartolini, John Adams Institute, 8 May 2013
28/30
Summary
Accelerated 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
Synchrotron radiation has unique characteristics and many applications
R. Bartolini, John Adams Institute, 8 May 2013
29/30
Bibliography
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).
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