Lecture 16
ACCELERATOR PHYSICS
Melbourne
E. J. N. Wilson
Lecture 12 - E. Wilson - 7/13/2017 - Slide 1
Recap of previous lecture
- Electrons 1
 Synchrotron
radiation
 Electrons in circular motion
 Retarded Potential
 Energy loss per turn
 Consequences of Radiation Loss
 Dipole radiation emission pattern
 Tangential observer’s view
 The spectrum
 Rate of emission of quanta
 Virtues of synchrotron radiation
Lecture 12 - E. Wilson - 7/13/2017 - Slide 2
Lecture 12 - Electron beam dynamics II
- contents
 Excitation
of betatron amplitudes
 The effect on emittance
 Summary of radiation integrals
 Energy loss per turn
 Damping of betatron oscillations
 Transverse damping rate
 Equilibrium beam emittance
 Partition numbers
 Vertical plane
 The coupling can only be estimated
 Energy dependence of beam size
 Quantum lifetime
Lecture 12 - E. Wilson - 7/13/2017 - Slide 3
Excitation of betatron amplitudes
 When
a quantum is emitted there is a sudden
change in energy and hence reference orbit
 The effect on the betatron emittance or on the
Courant and Snyder invariant is
instantaneous
 The quantity that remains invariant is the
position of the particle
x  D(s)  E E
Lecture 12 - E. Wilson - 7/13/2017 - Slide 4
The effect on emittance

Reduction in displacement due to dispersion must
match the increase in betatron amplitude
x  D(s)  E E

The effect on the C & S invariant is:
u
 x  D
Es

More exactly
 
 x 2 

u2 D2
 2
Es 
   x 2  2xx' x'2 

Leading to a growth rate
d

dt
N u2
E02
Lecture 12 - E. Wilson - 7/13/2017 - Slide 5
1

H sds

2
Summary of radiation integrals

I1 =
D ds

I2 =
ds
I3 =
2
ds
3
I4 =
D 2k + 1 ds
2

I5 =
H ds
3

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

Lecture 12 - E. Wilson - 7/13/2017 - Slide 6
Damping of betatron oscillations
 Quantum
emission involves a loss of
momentum but does not change the the local
displacement or divergence
 However
at the next RF cavity passage the
cavity tends to only replace the longitudinal
momentum that has been lost
Lecture 12 - E. Wilson - 7/13/2017 - Slide 7
Transverse damping rate
 The
fractional change in divergence is just
  p 
p
p
z 

 z1  
p/ /
p/ /   p
p 

 This
leads to a steady damping of betatron
motion which we can show will be in
equilibrium with the growth due to quantum
excitation.
W   z2  2 zz  z 2
W  2 zz  2 z 
zz   0
z2  W / 2
W
E


W
E
 Thus
the damping rate for betatron motion is
just that for energy (actually half of it)
Lecture 12 - E. Wilson - 7/13/2017 - Slide 8
Equilibrium beam emittance
 Earlier we
 We
had the betatron growth rate
can equate this to the damping rate
 And
obtain the equilibrium emittance
Lecture 12 - E. Wilson - 7/13/2017 - Slide 9
Partition numbers
 The
basic decay time for all three degrees of
freedom is :
P
i  Ji
2 Es
i is x, y or and J is called the
damping partition number
 where
J x  1 

Jz 1
and
J  2  
1
I4
P
D
(
1
/


2
k

)
ds


cU o 
I2
 the
second term applies to combined
function magnets which have negative
damping
 It is a general rule that:
J x  J z  J  4
Lecture 12 - E. Wilson - 7/13/2017 - Slide 10
Vertical plane
 There
is no direct mechanism which can
excite vertical betatron motion following
quantum emission as there is in the
horizontal plane
 Nevertheless random skew quadrupole fields
due for example to small tilts of the main
quadrupoles can couple the two planes.
 It is usual to express this by a constant, k
such that the “natural” equilibrium
emittance one might calculate for the
horizontal plane
 x0

is modified between the two planes
x 
 The
1
 x0
1 k
z 
1
 x0
1 k
coupling can only be estimated
Lecture 12 - E. Wilson - 7/13/2017 - Slide 11
Energy dependence of beam size
 The
strength of the damping increases with
energy and its time constant varies as:
  1/ E3
 The
injection energy must not be too low if a
series of puses, injected from a linac are to be
damped and assimilated in the emittance of
the injected beam.
 Nevertheless the equilibrium energy spread
increases with the square of the energy
2
 2 
55
  
  

 E  64 3 m0c 
 
 1.92  10
 As
does the emittance
Lecture 12 - E. Wilson - 7/13/2017 - Slide 12
13
2

Quantum lifetime

x
2
2 1 xM 
 2   
 
 q   x   e
x M 
Lecture 12 - E. Wilson - 7/13/2017 - Slide 13
Electrons III – Summary
 Excitation
of betatron amplitudes
 The effect on emittance
 Summary of radiation integrals
 Energy loss per turn
 Damping of betatron oscillations
 Transverse damping rate
 Equilibrium beam emittance
 Partition numbers
 Vertical plane
 The coupling can only be estimated
 Energy dependence of beam size
 Quantum lifetime
Lecture 12 - E. Wilson - 7/13/2017 - Slide 14