Transcript ElectronDynamicsCAS16_LRivkinx

Electron dynamics
with
Synchrotron Radiation
Lenny Rivkin
Paul Scherrer Institute (PSI)
and
Swiss Federal Institute of Technology Lausanne (EPFL)
Electron Beam Dynamics, L. Rivkin, Introduction to Accelerator Physics, Budapest
Radiation is emitted into a narrow cone
qe
v~c
v << c
 1
q =   qe
q
v c
Synchrotron radiation power
Power emitted is proportional to:
P  E 2 B 2
4
2

P   c 2  2
3

cC E 4
P 
 2
2 
re
–5
4
m
C =
=
8.858

10
3 mec 2 3
GeV 3

Energy loss per turn:

4
U0 = C  E
 1
=
137
 = 197 Mev  fm
hc

4
4
U0 = hc 
3
Electron Beam Dynamics, L. Rivkin, Introduction to Accelerator Physics, Budapest

dP
Ptot 
=
S
d c c

9 3 
Sx =
x K5 3 x dx
8 x
1

2 2 4
Ptot = hc  2
3


3
c
3
c =
2 
~ 2.1x
1

S x dx = 1
0
3
~ 1.3 x e – x

G1 x = x
0.1
0.01

x
K5 3 x dx
50%

c eV = 665 E 2 GeV B T
0.001
0.001
0.01
0.1
x = c
1
10
Radiation effects in electron storage rings
Average radiated power restored by RF
U 0  10 – 3 of E0
 Electron loses energy each turn to synchrotron radiation VRF > U0
 RF cavities accelerate electrons back to the nominal energy
Radiation damping
 Average rate of energy loss produces DAMPING of electron
oscillations in all three degrees of freedom (if properly arranged!)
Quantum fluctuations
 Statistical fluctuations in energy loss (from quantized emission of
radiation) produce RANDOM EXCITATION of these oscillations
Equilibrium distributions
 The balance between the damping and the excitation of the electron
oscillations determines the equilibrium distribution of particles in the
beam
Radiation damping
Transverse oscillations
Electron Beam Dynamics, L. Rivkin, Introduction to Accelerator Physics, Budapest
Average energy loss and gain per turn
 Every turn electron radiates small
amount of energy
U0
E0
 only the amplitude of the
momentum changes
E1 = E0 – U 0 = E0 1 –
U
U
P1 = P0 – c0 = P0 1 – 0
E0
 Only the longitudinal component
of the momentum is increased in
the RF cavity
 Energy of betatron
oscillation

U0
2
2
A1 = A0 1 –
E0
or

E  A 2
U0
A1  A 0 1 –
2E0
Damping of vertical oscillations
 But this is just the exponential decay law!

A = – U 0
A
2E
t
A A e
0
 The oscillations are exponentially damped
with the damping time (milliseconds!)
2 E T0

U0
the time it would take particle to
‘lose all of its energy’
 In terms of radiation power
2E

and since P  E 4
P
1
 3
E
Adiabatic damping in linear accelerators
In a linear accelerator:
p
 p
x = p decreases  1
E
p
p
p
In a storage ring beam passes many
times through same RF cavity
 Clean loss of energy every turn (no change in x’)
 Every turn is re-accelerated by RF (x’ is reduced)
 Particle energy on average remains constant
Emittance damping in linacs:

W

2

4
W
2
W
4
1
 
or

2
4
const.
Radiation damping
Longitudinal oscillations
Electron Beam Dynamics, L. Rivkin, Introduction to Accelerator Physics, Budapest
Longitudinal motion:
compensating radiation loss U0
 RF cavity provides accelerating field
with frequency
• h – harmonic number
f RF  h  f 0
VRF
U0
 The energy gain:
U RF  eVRF  
 Synchronous particle:
• has design energy
• gains from the RF on the average as much
as it loses per turn U0

Longitudinal motion:
phase stability
VRF
U0

 Particle ahead of synchronous one
• gets too much energy from the RF
• goes on a longer orbit (not enough B)
>> takes longer to go around
• comes back to the RF cavity closer to synchronous part.
 Particle behind the synchronous one
• gets too little energy from the RF
• goes on a shorter orbit (too much B)
• catches-up with the synchronous particle
Longitudinal motion: energy-time oscillations
energy deviation from the design energy,
or the energy of the synchronous particle


longitudinal coordinate measured from the
position of the synchronous electron
Click
Orbitto
Length
edit Master title style
Length element depends on x
dl = 1 + x ds
dl
ds

x
Horizontal displacement has two parts:
x = x + x
 To first order x does not change L
 x – has the same sign around the ring

Length of the off-energy orbitL = dl =


Ds
p E
L = 
ds where  = p =
E
s
x
1 +  ds = L 0 +L
L =  
L
Electron Beam Dynamics, L. Rivkin, Introduction to Accelerator Physics, Budapest
Something funny happens on the way around the ring...
 L
T0 = 0
c
Revolution time changes with energy

T = L – 
T
L


d
dp
1
= 
 Particle goes faster (not much!)
 2 p



while the orbit length increases (more!)

The “slip factor”    since
T =  – 1  dp = dp
p
T
2 p
Ring is above “transition energy”
 >> 12

 1
 2
 tr
isochronous ring:  = 0 or  =  tr
(relativity)
L =  dp
p
L
Not only accelerators work above transition
Dante Aligieri
Divine Comedy
RF Voltage
VRF
U0

V    Vˆ sin h0  s 
here the synchronous phase
U0 

 s  arcsin  
 eVˆ 

Ds
1

ds
L s
Momentum compaction factor
Like the tunes Qx, Qy -  depends on the whole optics

A quick estimate for separated function guide field:

=
1
L 00
mag
D s ds =
1 D L
 = 0 in dipoles
mag
L 0 0
 =  elsewhere
L mag = 2 0
 D
=
R

But

Since dispersion is approximately
 R
D  2    12 typically < 1%
Q
Q
and the orbit change for ~ 1% energy deviation
L = 1    10 – 4
L
Q2
Energy balance
Energy gain from the RF system: U RF = eVRF  = U0 + eVRF  

synchronous particle () will get exactly the energy loss per
turn

dVRF
VRF =
d

we consider only linear oscillations

Each turn electron gets energy from RF and loses energy to
radiation within one revolution time T0
 = U0 + eVRF   – U0 + U  

= 0
d = 1 eV   – U  
dt T0 RF
An electron with an energy deviation will arrive after one turn
at a different time with respect to the synchronous particle
d = – 
E0
dt
Synchrotron oscillations: damped harmonic oscillator
2
d
 + 2 d + W 2 = 0
Combining the two equations
 dt
dt 2
 eV
RF
 where the oscillation frequency W 2 
T0E0
 U
 
typically   <<W
 the damping is slow:
2T0

the solution is then:

 t = 0e –t cos Wt + q 

similarly, we can get for the time delay:

 t = 0e –t cos Wt + q 
Synchrotron (time - energy) oscillations
The ratio of amplitudes at any instant
 =  
WE0
Oscillations are 90 degrees out of phase
q = q + 
2
The motion can be viewed in the phase space of conjugate
variables
, 


, W
E0

ˆ

ˆ



E0
W
V
Vo sin  s
s

U
Stable regime

Separatrix
d (  )
dt

Longitudinal
Phase
Space
Longitudinal motion:
damping of synchrotron oscillations
P  E 2 B 2
During one period of synchrotron oscillation:
 when the particle is in the upper half-plane, it loses more
energy per turn, its energy gradually reduces

U > U0
U < U0

 when the particle is in the lower half-plane, it loses less
energy per turn, but receives U0 on the average, so its
energy deviation gradually reduces
The synchrotron motion is damped
 the phase space trajectory is spiraling towards the origin
Robinson theorem: Damping partition numbers
 Transverse betatron oscillations
are damped with
 Synchrotron oscillations
are damped twice as fast
2 ET0
x z 
U0
ET0
 
U0
 The total amount of damping (Robinson theorem)
depends only on energy and loss per turn

1 + 1 + 1 = 2U0 = U 0 J + J + J
y

x  y   ET0 2ET0 x
the sum of the partition numbers
Jx + J z + J  = 4

PE B
Radiation loss
2
Displaced off the design orbit particle sees fields that
are different from design values
 energy deviation

 different energy:
Pγ  E2
 different magnetic field B
particle moves on a different orbit, defined by the
off-energy or dispersion function Dx
both contribute to linear term in
 betatron oscillations: zero on average
Pγ  
2

PE B
Radiation loss
To first order in 

2
Urad = U0 + U  

electron energy changes slowly, at any instant it is
moving on an orbit defined by Dx

U 
after some algebra one can write
 U
U = 0 2 + D
E0

D  0 only when k  0
dUrad
dE
E0
2
Jx + J z + J  = 4
Damping partition numbers
 Typically we build rings with no vertical dispersion
Jz 1
J x  J  3
 Horizontal and energy partition numbers can be
modified via D :
Jx  1 D
J  2  D
 Use of combined function magnets
 Shift the equilibrium orbit in quads with RF frequency
Equilibrium beam sizes
Electron Beam Dynamics, L. Rivkin, Introduction to Accelerator Physics, Budapest
Radiation effects in electron storage rings
Average radiated power restored by RF
U 0  10 – 3 of E0
 Electron loses energy each turn to synchrotron radiation VRF > U0
 RF cavities accelerate electrons back to the nominal energy
Radiation damping
 Average rate of energy loss produces DAMPING of electron
oscillations in all three degrees of freedom (if properly arranged!)
Quantum fluctuations
 Statistical fluctuations in energy loss (from quantized emission of
radiation) produce RANDOM EXCITATION of these oscillations
Equilibrium distributions
 The balance between the damping and the excitation of the electron
oscillations determines the equilibrium distribution of particles in the
beam
Quantum nature of synchrotron radiation
Damping only
• If damping was the whole story, the beam emittance (size)
would shrink to microscopic dimensions!*
• Lots of problems! (e.g. coherent radiation)
• How small? On the order of electron wavelength

h = C
E =mc 2 = h = hc   e = 1
 mc 
e
C = 2.4 10–12m – Compton wavelength
Diffraction limited electron emittance
Quantum nature of synchrotron radiation
Quantum fluctuations
• Because the radiation is emitted in quanta,
radiation itself takes care of the problem!
• It is sufficient to use quasi-classical picture:
» Emission time is very short
» Emission times are statistically independent
(each emission - only a small change in electron
energy)
Purely stochastic (Poisson) process
Visible quantum effects
I have always been somewhat amazed that a purely quantum
effect can have gross macroscopic effects in large machines;
and, even more,
that Planck’s constant has just the right magnitude needed to
make practical the construction of large electron storage
rings.
A significantly larger or smaller value of

would have posed serious -- perhaps insurmountable -problems for the realization of large rings.
Mathew Sands
Quantum excitation of energy oscillations

3
Photons are emitted with typical energy u ph  htyp = hc 

P
at the rate (photons/second)
N =
u ph
Fluctuations in this rate excite oscillations
During a small interval t electron emits photons
losing energy of
Actually, because of fluctuations, the number is
resulting in spread in energy loss
N = N  t
N  u ph
N  N
 N  u ph
For large time intervals RF compensates the energy loss, providing
damping towards the design energy E0
Steady state: typical deviations from E0
≈ typical fluctuations in energy during a damping time 
Equilibrium energy spread: rough estimate
We then expect the rms energy spread to be
and since

 
E0  u ph
 E
  0
P
and
   N    u ph
P  N  u ph
geometric mean of the electron and photon energies!
Relative energy spread can be written then as:
 
–e
 h

–
 4  10 – 13m


=
e

mec
E0
it is roughly constant for all rings
• typically
  E2
 
~ const ~ 10 – 3
E0
Equilibrium energy spread
More detailed calculations give

0
• for the case of an ‘isomagnetic’ lattice  s = 


E
with
2
Cq E 2
=
J  0

hc = 1.468  10 – 6 m
Cq = 55
32 3 mec 2 3
GeV 2
It is difficult to obtain energy spread < 0.1%
• limit on undulator brightness!
in dipoles
elsewhere
Equilibrium bunch length
Bunch length is related to the energy spread
 Energy deviation and time of arrival
(or position along the bunch)
are conjugate variables (synchrotron oscillations)




 =
Ws E
recall that Ws  VRF

  
=
Ws E
Two ways to obtain short bunches:

 1

RF voltage (power!)

Momentum compaction factor in the limit of  = 0
isochronous ring: particle position along the bunch is
frozen


VRF
  

Excitation of betatron oscillations
x  x  x
x  D 

E
x  x  x
x  x  x  0
x   D 

E
Courant Snyder invariant x   D 

E
  
   x2  2x x  x2  D 2  2DD  D2   
E
2
Excitation of betatron oscillations
Electron emitting a photon
• at a place with non-zero dispersion
• starts a betatron oscillation around a
new reference orbit
x  D 

E
Horizontal oscillations: equilibrium
Emission of photons is a random process
 Again we have random walk, now in x. How far particle
will wander away is limited by the radiation damping
 The balance is achieved on the time scale of the damping
time x = 2 
 x  N  x  D 


E
 2D

E
Typical horizontal beam size ~ 1 mm
Quantum effect visible to the naked eye!

Vertical size - determined by coupling
Beam emittance

Area
= 
Betatron oscillations
• Particles in the beam execute betatron oscillations with
x’
different amplitudes.
Transverse beam distribution
• Gaussian (electrons)
• “Typical” particle: 1 -  ellipse
(in a place where  = ’ = 0)

 x2
Emittance 


Units
of  m  rad

x =  
 x =  / 
x’
x
x

 =  x   x
 x
=
x
Equilibrium horizontal emittance
Detailed calculations for isomagnetic lattice

H mag

CqE
x0 
=


Jx

2
x
2

where
H = D 2 + 2DD + D 2
= 1 D 2 + D + D

and
H
mag
2
is average value in the bending magnets
2-D Gaussian distribution
x
Electron rings emittance definition
 1 -  ellipse
x’

n x dx =
1 e –x 2 / 2 2dx
2 
x
x
Area = x


Probability to be inside 1- ellipse
Probability to be inside n- ellipse
P1 = 1 – e – 1 2 = 0.39
Pn = 1 –
n2 2
–
e
FODO cell lattice
FODO lattice
emittance
100

2
D
H ~  ~ R3
Q
Emittance

10

CqE 2 R 1
x0 
 3
Jx
Q
 2
E
3

q FFODO 
Jx
1
0
20
40 60 80 100 120 140 160 180
Phase advance per cell [degrees]
Ionization cooling
E
p
p||
absorber
acceleration



 =
 02
2
+  MS
 0 >>  MS
similar to radiation
damping, but there is
multiple scattering
in the absorber that
blows up the
emittance
to minimize the
blow up due to
multiple
scattering in the
absorber we can
focus the beam
Minimum emittance lattices


 x0
Cq E 2 3

 θ  Flatt
Jx
Fmin =
1
12 15
Quantum limit on emittance

Electron in a storage ring’s dipole fields is accelerated,
interacts with vacuum fluctuations: «accelerated
thermometers show increased temperature»

synchrotron radiation opening angle is ~ 1/-> a lower
limit on equilibrium vertical emittance

independent of energy
G(s) =curvature, Cq = 0.384 pm
isomagnetic lattice

in case of SLS: 0.2 pm
 y  0.09 pm 
y

Mag
Vertical emittance record
Beam size 3.6  0.6 m
Emittance 0.9  0.4 pm
SLS beam cross section compared to a human hair:
80 m
4 m
Summary of radiation integrals
Momentum compaction factor
 I1
=
2R
Energy loss per turn

U0 = 1 C E 4  I 2
2
re
–5
4
m
C =
=
8.858

10
3 mec 2 3
GeV 3



I1 =
D ds

I2 =
ds
I3 =
2
ds
3
I4 =
D 2k + 1 ds
2

I5 =
H ds
3


Summary of radiation integrals (2)
I4
D =
I2
Damping parameter
Damping times, partition numbers

J  = 2 + D , Jx = 1 – D , J y = 1

I1 =
D ds

I2 =
ds
I3 =
2
ds
3
 2ET0
 0
i=
0 =
1 ds
Ji
I4 = D
2k
+
U0

2
Equilibrium energy spread
H ds
I
=

2
5
3
 2 Cq E I 3
=

E
J
I2

55
hc = 1.468  10 – 6 m
C
=
q
Equilibrium emittance
32 3 mec 2 3
GeV 2
 2 C E2 I
x 0 = x = q  5

H = D 2 + 2DD + D 2
J
I
x
2
Damping wigglers
Increase the radiation loss per turn U0 with WIGGLERS
 reduce damping time
E

P  Pwig
 emittance control
wigglers at high dispersion:
blow-up emittance
e.g. storage ring colliders for high energy physics
wigglers at zero dispersion:
decrease emittance
e.g. damping rings for linear colliders
e.g. synchrotron light sources (PETRAIII, 1 nm.rad)
END