synchrotron radiation what is it ? rate of energy loss longitudinal damping transverse damping...

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:


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)


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:


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:


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


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


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



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



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


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


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…


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


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


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.


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 <<



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



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


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


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


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


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.


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:


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


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


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 )(


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:


u

ETrev

4

1

4CEu

Transverse:

Vertical damping time:

Horizontal damping time:

urevET21

21

121

u

ET


Questions….,Remarks…?

Synchrotron radiation

Damping

Wigglers

Quantum fluctuations

synchrotron radiation what is it ? rate of energy loss longitudinal damping transverse damping...

Documents

energy loss slide

em energy

energy gain e

lower energy particles

magnets lepton energy

turn higher energy particles

synchrotron radiation

expected slide