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Synchrotron Radiation What is it ? Rate of energy loss Longitudinal damping Transverse damping Quantum fluctuations Wigglers
Rende Steerenberg (BE/OP)
13 January 2011
Rende Steerenberg (BE/OP)
13 January 2011
AXEL-2011Introduction to Particle
Accelerators
R. Steerenberg, 13-Jan-2011 AXEL - 2011 2
Acceleration and Electro-Magnetic Radiation
An accelerating charge emits Electro-Magnetic waves.Example:
An antenna is fed by an oscillating current and it emits electro magnetic waves.
In our accelerator we know to types of acceleration:
Longitudinal – RF systemTransverse – Magnetic fields, dipoles, quadrupoles, etc.. am
dtvmd
dtdpF
)(
Momentum change Direction changes but not magnitude
Newton’s lawForce due to magnetic fieldgives change of direction
constantvm
So:
R. Steerenberg, 13-Jan-2011 AXEL - 2011 3
Rate of EM radiation
The rate at which a relativistic lepton radiates EM energy is :
Longitudinal square of energy (E2)
Transverse square of magnetic field (B2)
Force // velocity
Force velocity
PSR E2 B2
In our accelerators: Transverse force > Longitudinal forceTherefore we only consider radiation due to ‘transverse acceleration’ (thus magnetic forces)
R. Steerenberg, 13-Jan-2011 AXEL - 2011 4
Rate of energy loss (1)This EM radiation generates an energy loss of the particle concerned, which can be calculated using:
22
32
03
2FE
cm
rcP
Electron radiusVelocity of lightTotal energy‘Accelerating’ forceLepton rest mass
constant
Our force can be written as: F = evB = ecB
1c
v 22
32
0
32
3
2BE
cm
rceP
ec
E
e
pB
)(
2
4
32
03
2
E
cm
rcP
Thus: but
Which gives us:
R. Steerenberg, 13-Jan-2011 AXEL - 2011 5
Rate of energy loss (2)
We have: 2
4
32
03
2
E
cm
rcP
Finally this gives:
,which gives the energy loss
We are interested in the energy loss per revolution for which we need to integrate the above over 1 turn
c
dsPPdtThus:
c
d
c
ds 2
Bending radius inside the magnets
4
24
320
1
3
4 CEdE
C
cm
ru
Lepton energy
Gets very large if E is large !!!
However:
R. Steerenberg, 13-Jan-2011 AXEL - 2011 6
What about the synchrotron oscillations ?
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 = VsinHowever,
Lower energy particles lose less energy per turnHigher energy particles lose more energy per turn
What will happen…???
4CEu
R. Steerenberg, 13-Jan-2011 AXEL - 2011 7
Synchrotron motion for leptons
All three particles will gain the same energy from the RF systemThe black particle will lose more energy than the red one.
This leads to a reduction in the energy spread, since u varies with E4.
E
t (or )
4CEu
R. Steerenberg, 13-Jan-2011 AXEL - 2011 8
Longitudinal damping in numbers (1)
Remember how we calculated the synchrotron frequency.It was based on the change in energy: Now we have to add an extra term, the energy loss du
becomes
Our equation for the synchrotron oscillation becomes then:
sinVdE
duVdE sin dufVfdt
dErevrev sin
022 22
2
2
duf
E
hVf
E
h
dt
drevrev
Extra term for
energy loss
R. Steerenberg, 13-Jan-2011 AXEL - 2011 9
Longitudinal damping in numbers (2)
This term: dufE
hrev
22
E
dE
dE
dufhrev
2
2 Can be written as:
butrev
rev
f
df
E
dE
1
This now becomes:
dE
dufdfhrevrev
2
dt
drevT
1 dt
d
TdE
du
rev
1
The synchrotron oscillation differential equation becomes now:
021 2
2
2
VfE
h
dt
d
TdE
du
dt
drev
rev
Damped SHM,
as expected
E
dE
dE
du
E
du
R. Steerenberg, 13-Jan-2011 AXEL - 2011 10
Longitudinal damping in numbers (3)
So, we have:
This confirms that the variation of u as a function of E leads to damping of the synchrotron oscillations as we already expected from our reasoning on the 3 particles in the longitudinal phase space.
The damping coefficient revTdE
du 1
021 2
2
2
VfE
h
dt
d
TdE
du
dt
drev
rev
R. Steerenberg, 13-Jan-2011 AXEL - 2011 11
Longitudinal damping time
The damping coefficient is given by: revTdE
du 1
We know that and thus
4CEu
34CE
dE
du
Not totally correct since
E
So approximately: E
u
dE
du 4
For the damping time we have then:
Damping time = u
ETrev
4
1 Energy loss/turn
EnergyRevolution time
4CE
The damping time decreases rapidly (E3) as we increase the beam energy.
4CEu
R. Steerenberg, 13-Jan-2011 AXEL - 2011 12
Damping & Longitudinal emittance
Damping of the energy spread leads to shortening of the bunches and hence a reduction of the longitudinal emittance.
E
E
d
Initial
Later…
R. Steerenberg, 13-Jan-2011 AXEL - 2011 13
Some LHC numbers
Energy loss per turn at:injection at 450 GeV = 1.15 x 10-1 eVCollision at 7 TeV = 6.71 x 103 eV
Power loss per meter in the main dipoles at 7 TeV is 0.2 W/m
Longitudinal damping time at:Injection at 450 GeV = 48489.1 hoursCollision at 7 TeV = 13 hours
R. Steerenberg, 13-Jan-2011 AXEL - 2011 14
What about the betatron oscillations ? (1)
Each photon emission reduces the transverse and longitudinal energy or momentum.Lets have a look in the vertical plane:
particle trajectory
ideal trajectory
particle
Emitted photon (dp)
total momentum (p)
momentum lost dp
R. Steerenberg, 13-Jan-2011 AXEL - 2011 15
What about the 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.
old particle trajectory
ideal trajectory
new 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.
R. Steerenberg, 13-Jan-2011 AXEL - 2011 16
Vertical damping in numbers (1)
The RF system increases the momentum p by dp or energy E by dE
p = longitudinal momentum
pt = transverse momentum
pT= total momentum
p
py
t'
p
dpy
p
dp
p
p
dpp
pynew
tt1'1)'(
dp is small
E
dEy
p
dpydy '''
The change in transverse angle is thus given by:
Tan(α)= αIf α <<
R. Steerenberg, 13-Jan-2011 AXEL - 2011 17
Vertical damping in numbers (2)
A change in the transverse angle alters the betatron oscillation amplitude
dy’
y’
y
ada
sin'..dyda
sin.'.E
dEyda
2
0sin.'.
E
dEyda
Summing over many photon emissions
2
0
2sinE
dEada
sin.a
E
dE
a
da
2
1
R. Steerenberg, 13-Jan-2011 AXEL - 2011 18
Vertical damping in numbers (3)
The change in amplitude/turn is thus:
E
dE
a
da
2
1We found:dE is just the change in
energy per turn u(energy given back by
RF) ada
Which is also: aE
ua
2
aET
u
dt
da
2Thus:
Revolution time
Change in amplitude/second
This shows exponential damping with coefficient: ETu
2
Damping time = u
ET2 (similar to longitudinal case)
4CE
R. Steerenberg, 13-Jan-2011 AXEL - 2011 19
Horizontal damping in numbers
Vertically we found:E
u
a
da
2
1
This is still valid horizontallyHowever, in the horizontal plane, when a particle changes energy (dE) its horizontal position changes too
E
u
E
dE
p
dp
r
drppp
OK since =1
is related to D(s) in the bending magnets
horizontally we get: E
u
a
da
221
Horizontal damping time:
21
12
u
ET
R. Steerenberg, 13-Jan-2011 AXEL - 2011 20
Some intermediate remarks….
Transverse damping for LHC time at:Injection at 450 GeV = 48489.1 hoursCollision at 7 TeV = 26 hours
Longitudinal and transverse emittances all shrink as a function of time.For leptons damping times are typically a few milliseconds up to a few seconds.Advantages:
Reduction in lossesInjection oscillations are damped outAllows easy accumulationInstabilities are damped
Inconvenience:Lepton machines need lots of RF power, therefore LEP was stopped
All damping is due to the energy gain from the RF system an not due to the emission of synchrotron radiation
R. Steerenberg, 13-Jan-2011 AXEL - 2011 21
Is there a limit to this damping ? (1)
Can the bunch shrink to microscopic dimensions ?
No ! , Why not ?
For the horizontal emittance h there is heating term due to the horizontal dispersion.
What would stop dE and v of damping to zero?
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
R. Steerenberg, 13-Jan-2011 AXEL - 2011 22
Is there a limit to this damping ? (2)In the transverse 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 give a finite equilibrium energy spread.
R. Steerenberg, 13-Jan-2011 AXEL - 2011 23
Quantum fluctuations (1)Quantum fluctuation is defined as:
Fluctuation in number of photons emitted in one damping time
Let Ep be the average energy of one emitted photon
Damping time
turnssecondsu
E
u
ET
Energy loss/turn
Revolution time
pE
uNumber of photons emitted/turn =
pp E
E
u
E
E
u
Number of emitted photons in one damping time can then be given by:
R. Steerenberg, 13-Jan-2011 AXEL - 2011 24
Higher energy faster longitudinal damping, but also larger energy spread
Quantum fluctuations (2)
The average photon energy Ep E3
The r.m.s. energy spread E2
pE
ENumber of emitted photons in one damping time = Random
processpE
Er.m.s. deviation =The r.m.s. energy deviation =
pp
p
EEEE
E
Energy of one emitted photon
The damping time E3
R. Steerenberg, 13-Jan-2011 AXEL - 2011 25
Wigglers (1)
The damping time in all planes
If the loss of energy, u, increases, the damping time decreases and the beam size reduces.To be able to control the beam size we add ‘wigglers’
u
ET
N N N N N NS S S S S S
N N N N NS S S S S S N
beam
It is like adding extra dipoles, however the wiggles does not give an overall trajectory change, but increases the photon emission
R. Steerenberg, 13-Jan-2011 AXEL - 2011 26
Wigglers (2)What does the wiggler in the different planes?Vertically:
We do not really need it (no heating term), but the vertical emittance would be reduced
Horizontally:The emittance will reduce.A change in energy gives a change in radial position
We know the dispersion function:
In order to reduce the excitation of horizontal oscillations we should put our wiggler in a dispersion free area (D(s)=0)
E
dEsDdr )(
R. Steerenberg, 13-Jan-2011 AXEL - 2011 27
Wigglers (3)
Longitudinally:The wiggler will increase the number of photons emittedIt will increase the quantum fluctuationsIt will increase the energy spread
Conclusion:
Wigglers increase longitudinal emittance and decrease transverse emittance
SummaryDamping due to addition of longitudinal momentum !Longitudinal:
Energy loss per turn:
Damping time:
R. Steerenberg, 13-Jan-2011 AXEL - 2011 28
u
ETrev
4
1
4CEu
Transverse:
Vertical damping time:
Horizontal damping time:
urevET21
21
121
u
ET
R. Steerenberg, 13-Jan-2011 AXEL - 2011 29
Questions….,Remarks…?
Synchrotron radiation
Damping
Wigglers
Quantum fluctuations