Transcript What is Synchrotron Radiation

Lecture 10
Synchrotron Radiation
Professor Emmanuel Tsesmelis
Directorate Office, CERN
Department of Physics, University of Oxford
ACAS School for Accelerator Physics 2012
Australian Synchrotron, Melbourne
November/December 2012
Contents – Lecture 10
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2
What is Synchrotron Radiation ?
Rate of Energy Loss
Longitudinal Damping
Transverse Damping
Quantum Fluctuations
Wigglers & Undulators
Free Electron Lasers (FELs)
Accelerators for Synchrotron Light
Synchrotron Source of X-rays
Diamond Light Source, Harwell
Science and Innovation Campus, UK
Australian Synchrotron
Melbourne, Australia
Diamond Beamlines
Accelerators for Synchrotron Light

Protein Structures



Proteins are biological
molecules involved in almost
every cellular process.
The protein is produced,
crystallised and illuminated by
X-rays. The interactions
between the X-rays and the
crystal form a pattern that can
be analysed to deduce the
protein structure.
Over 45,000 structures have
been solved by the worldwide
synchrotron community.
Protein
Data
Bank
The trimer of the Lassa
nucleoprotein,
part of the Lassa virus
Acceleration and Electro-Magnetic Radiation


An accelerating charge emits Electro-Magnetic waves.
Example:
An antenna is fed by an oscillating current and it emits electromagnetic waves.

In accelerator there are two types of acceleration:


Longitudinal – RF system
Transverse – Magnetic fields, dipoles, quadrupoles, etc..
Force due to magnetic field
gives change of direction

dp
d
(
m

v
)  m  a
F 
dt
dt
Momentum change
7
Newton’s law
Direction changes but not magnitude
Rate of EM Radiation

The rate at which a relativistic lepton radiates EM
energy is :
Force // velocity


Longitudinal  square of energy (E2)
Force  velocity
Transverse  square of magnetic field (B2)
PSR  E2 B2
In accelerators:
Transverse force > Longitudinal force
Therefore, only consider radiation due to ‘transverse
acceleration’ (thus magnetic forces)
8
Rate of Energy Loss (1)

This EM radiation generates an energy loss of the
particle concerned, which can be calculated using:
constant
 2 rc
P  
 3 m c
2
0

E F

2

3
2
Electron radius
Velocity of light
Total energy
‘Accelerating’ force
Lepton rest mass
Force can be written as: F = evB = ecB
 2 e rc
P  
 3 m c 
2
Thus:
3
2
0
Which gives:
3

E B

2
2
but
 2 rc
P  
 3 m c
2
0
9

3
p E
( B )  
e ec
E


4
2
v
1
c
Rate of Energy Loss (2)

 2 rc
P  
 3 m c
Have:
2
0

3
E


4
2
,which gives the energy loss
Interested in the energy loss per revolution for which need to integrate the
above over one turn.
Thus:
ds
 Pdt   P
c
However:
ds
d
 2 

c
c
Finally, this gives:
Gets very large if
E is large !!!
10
Bending radius
inside the magnets
Lepton
energy
4
r
4 1
u
E  2 d
3
2
3 m0 c 


C

CE 4

What about the Synchrotron Oscillations ?
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The RF system, besides increasing the energy, has to
make up for this energy loss u.
All the particles with the same phase, , w.r.t. RF
waveform will have the same energy gain E = Vsin
However,


Lower energy particles lose less energy per turn.
Higher energy particles lose more energy per turn.
u 
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What will happen…???
CE

4
Synchrotron Motion for Leptons
E
u 
CE
4

t (or )
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12
All three particles gain same energy from the RF system.
The green particle will lose more energy than red one.
This leads to a reduction in the energy spread,
since u varies with E4.
Longitudinal Damping (1)
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Remember how synchrotron frequency was calculated.
Based on the change in energy.
Now need to add an extra term, the energy loss du
dE  V sin 
dE  V sin   du
dE
 frevV sin   frev du
dt

Equation for synchrotron oscillation becomes:
d 2  2 h
2 h
2
2



f

V


f


rev
rev du  0
2
E
dt
 E

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Extra term for
energy loss
Longitudinal Damping (2)

2 h
2
f rev du
E
This term:
du du dE

E dE E
2 h f
Can be written as:
This now becomes:
d
dt
2
rev
du dE
dE E
but
du
 2 h df f
dE
rev
d   2 h

f
dt  E
2
14
2
rev
rev
rev
du 1 d

dE T dt
rev
1
T
rev
rev
The synchrotron oscillation differential equation
becomes:
2
dE
1 df

E
 f
du 1 d

V  
0
dE T dt

rev
Damped SHM,
as expected
Longitudinal Damping (3)

So,
d   2 h

f
dt  E
2
2
The damping coefficient is
2
rev
du 1 d

V  
0
dE T dt

rev

du 1
dE T
rev
This confirms that the variation of u as a function of E leads to damping
of the synchrotron oscillations.
15
Longitudinal Damping Time

du 1
The damping coefficient is given by:
dE T
CE
du
4CE
Know that
and
thus
u 


dE

CE
du
4u
Not totally
So, approximately:
u 

correct since 

dE
E
E

rev
4
3
4
Giving for the damping time:
Damping time =
1
ET

 4u
rev
Energy
Revolution time

CE

4
Energy loss/turn
The damping time decreases rapidly (E3) as beam energy increases.
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Damping & Longitudinal Emittance

Damping of the energy spread leads to shortening of
the bunches and hence a reduction of the longitudinal
emittance.
E
Initial
Later…
E

d
17
Betatron Oscillations (1)
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
Each photon emission reduces the transverse and longitudinal
energy or momentum.
In the vertical plane:
Emitted photon (dp)
total momentum (p)
momentum lost dp
ideal trajectory
particle
particle trajectory
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Betatron Oscillations (2)
The RF system must make up for the loss in longitudinal energy dE or
momentum dp.
However, the cavity only supplies energy parallel to ideal trajectory.
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
ideal trajectory
new particle trajectory
old particle trajectory
Each passage in the cavity increases only the longitudinal energy.
This leads to a direct reduction of the amplitude of the betatron
oscillation.
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Vertical Damping (1)

The RF system increases the momentum p by dp or
energy E by dE
pt = transverse momentum
tan(α)= α
if α << 1
pT= total momentum
pt
y' 
p
p = longitudinal momentum
dp is small
pt
pt  dp 
 dp 
new( y ' ) 
 1    y ' 1  
p  dp p 
p
p

The change in transverse angle is thus given by:
dp
dE
dy'   y '
  y'
p
E
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Vertical Damping (2)

A change in the transverse angle alters the betatron
oscillation amplitude
y’

a.sin 
da   .dy'.sin 
dE
da    . y ' .sin 
E
dE
da    . y ' .sin 
E
dE
da  a
 sin 
E
da
1 dE

a
2 E
2
y
dy’
 0
2
 0
a
da
Summing over many
photon emissions
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2
Vertical Damping (3)
Have found:

da
1 dE

a
2 E
dE is just the change in
energy per turn u
(energy given back by RF)
da  a
The change in amplitude/turn is thus:
u
a
Which is also: a  
2E
da
u
Thus:

a
dt
2 ET
Change in amplitude/second
Revolution time
This shows exponential damping with coefficient:
Damping time =
22
2 ET
u
 u
(similar to longitudinal case)

CE

4
2ET
Horizontal Damping

Vertically found:
da
1u

a
2E
This is still valid horizontally.
However, in the horizontal plane, when a particle changes energy (dE) its
horizontal position changes too.
OK since =1
dr
dp
dE
u
p
p
p
r
p
E
E
Horizontally get:
Horizontal damping time:
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 is related to D(s) in the
bending magnets
da
u
 1  2 
a
2E
2 ET  1 


u  1  2 
Ok provided
 small
Some intermediate remarks….
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Longitudinal & transverse emittances all shrink as a function of time.
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Damping times of few milliseconds up to few seconds for leptons.
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Advantages:
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Inconvenience:
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Reduction in losses.
Injection oscillations are damped out.
Allows easy accumulation.
Instabilities are damped.
Lepton machines need lots of RF power !!!
All damping is due to the energy gain from the RF system and not due to the
emission of synchrotron radiation.
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Is There a Limit to this Damping ? (1)
Can the bunch shrink to microscopic dimensions ?
•
•
For the horizontal emittance h there is heating term due to the horizontal
dispersion.
•
What would stop dE and v of damping to zero?
•
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No ! , Why not ?
For v there is no heating term. So v can get very small. Coupling with
motion in the horizontal plane finally limits the vertical beam size.
Is There a Limit to this Damping ? (2)

In the vertical plane the damping seems to be limited.

What about the longitudinal plane ?
Whenever a photon is emitted, the particle energy changes.
This leads to small changes in the synchrotron oscillations.
This is a random process.
Adding many such random changes (quantum fluctuations)
causes the amplitude of the synchrotron oscillation to grow.
When growth rate = damping rate then damping stops,
which gives a finite equilibrium energy spread.
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Quantum Fluctuations (1)

Quantum fluctuations are defined as:

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Fluctuation in number of photons emitted in one damping time.
Let Ep be the average energy of one emitted photon.
Revolution time
Damping time 
ET
E
seconds  turns
u
u
Energy loss/turn
Number of photons emitted/turn =
u
E
p
Number of emitted photons in one damping time given by:
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u E E

Ep u Ep
Quantum Fluctuations (2)
Number of emitted photons in one damping time =
r.m.s. deviation =
E
E
E
Ep
Random
process
p
The r.m.s. energy deviation =
E
E  EE
E
p
Energy of one
emitted photon
p
p

The average photon energy Ep  E3
The r.m.s. energy spread  E2
The damping time  E3
Higher energy  faster longitudinal damping,
but also larger energy spread
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Depends on
the lattice
parameters
Wigglers and Undulators

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Intense & tightly
collimated synchrotron
radiation can be
produced by special
magnets consisting of
periodic series of short
bending magnets wigglers and undulators.
Field of W/U magnet is
periodic along the beam
axis, z, with a period
length u.

Along beam axis periodically
varying field has peak value
B0 = field at centre of pole-tip.
 g = gap height.
for given u, g increases and the
field at the beam falls rapidly.


Field along beam axis becomes
W/U Magnet Designs

Various designs used to construct W/U magnets:

Classical electromagnet with conventional iron plates.

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Periodic field by arranging alternate permanent magnets.


u  25 cm. with B  2 T; ohmic heating
SmCo5; B  0.8-1.0 T
Hybrid magnets – combination of iron poles excited by permanent
magnets – field strengths of > 2 T reached.
W/U Magnet Arrangement

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W/U magnets installed in specially-reserved
straight sections of storage ring – insertions.
Insertions much have zero dispersion.

Switching on W/U magnets (dipoles) in non-zero
dispersion region increases beam emittance.
W/U magnet must not
cause bending of beam orbit.
Difference between Wiggler & Undulator

Difference lies in bending strength, quantified
with parameter

Conclusion:


Undulators – bending is weak; radiation emitted in
parallel with very small opening angle.
Wigglers – considerably stronger bending; emit
fan of radiation.
Undulator Radiation
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Electrons passing through undulator
perform transverse oscillations.
 Resulting from periodic B-field.
Coherence condition for undulator
radiation of wavelength w:
Wavelength of radiation determined by:
 Undulator period u
 Beam energy 
 Undulator parameter K
 Emission angle wrt to s-axis 0

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Finite wave-train of
coherent radiation
from undulator.
Electrons continuously
emit radiation with
frequency w
Undulator Radiation
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Higher order oscillations also present.
 Intensity decreases with increasing order.

Undulator radiation overlays spontaneous synchrotron radiation.
 For sufficiently large Nu, undulator radiation can be several orders of magnitude
more intense than spontaneous synchrotron radiation.
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Wigglers – strong bending causes radiation to be emitted in broad fan,
preventingcoherent superposition of radiation produced in individual period.
Free Electron Laser (FEL)
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Classical laser v FEL

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Laser – wave travels
through energy reservoir
inducing stimulated
emission; upon exiting
medium laser has same
frequency but amplified.
FEL – energy reservoir is
high-energy electron beam;
exchange of energy through
EM interactions with freelymoving beam electrons in
B-field.

Comparison of classical laser
(upper diagramme) with FEL.
Free Electron Laser (FEL)

Phase condition between laser field and
electron beam oscillating in an undulator.
Result is overall
gain in energy by
laser field.
4th Generation Light Source – Free Electron Laser
European XFEL
European XFEL
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4th Generation light source under construction at
DESY, Germany
Parameters
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Spatially-coherent short photon pulses  80 fs
Peak brilliance 1032-1034 photons/s/mm2/mrad2/0.1%BW
Photon energy range from 0.26 to 29.2 keV at electron
beam energies 10.5 GeV, 14 GeV,17.5 GeV
4th Generation Light
Source –
X-ray FELLCLS at SLAC