1
Large Amplitude Oscillations
For larger phase (or energy) deviations from the reference the
second order differential equation is non

linear:
0
sin
sin
cos
2
s
s
s
(
s
as previously defined)
Multiplying by and integrating gives an invariant of the motion:
I
s
s
s
sin
cos
cos
2
2
2
which for small amplitudes reduces to:
I
s
2
2
2
2
2
(the variable is
and
s
is constant)
Similar equations exist for the second variable :
E
d
/dt
2
Large Amplitude Oscillations (2)
s
s
s
s
s
s
s
s
sin
cos
cos
sin
cos
cos
2
2
2
2
s
s
s
s
m
m
sin
cos
sin
cos
Second value
m
where the separatrix crosses the horizontal axis:
Equation of the separatrix:
When
reaches

s
the force goes
to zero and beyond it becomes non
restoring. Hence

s
is an extreme
amplitude for a stable motion which
in the phase space( ) is shown
as closed trajectories.
,
s
3
Energy Acceptance
From the equation of motion it is seen that reaches an extremum
when , hence corresponding to .
Introducing this value into the equation of the separatrix gives:
0
s
s
s
s
tan
2
2
2
2
2
max
That translates into an acceptance in energy:
This “RF acceptance” depends strongly on
s
and plays an important role
for the electron capture at injection, and the stored beam lifetime.
s
s
s
G
E
h
V
e
E
E
ˆ
2
1
max
s
s
s
s
G
sin
2
cos
2
4
RF Acceptance versus Synchronous Phase
As the synchronous phase
gets closer to 90º the
area of stable motion
(closed trajectories) gets
smaller. These areas are
often called “BUCKET”.
The number of circulating
buckets is equal to “h”.
The phase extension of
the bucket is maximum for
s
=
180º (or 0
°
) which
correspond to no
acceleration . The RF
acceptance increases with
the RF voltage.
5
Potential Energy Function
F
dt
d
2
2
U
F
F
d
F
U
s
s
s
0
0
2
sin
cos
cos
The longitudinal motion is produced by a force that can be derived from
a scalar potential:
The sum of the potential
energy and kinetic energy is
constant and by analogy
represents the total energy
of a non

dissipative system.
6
Ions in Circular Accelerators
A =atomic number
Q =charge state
q = Q e
E
r
= A E
0
m =
m
r
P = m v
E =
r
W = E
–
E
r
P = q B r
E
2
= p
2
c
2
+ E
r
2
2
2
2
r
B
c
q
E
E
r
2
2
0
2
r
B
c
e
A
Q
E
A
W
A
W
Moreover:
r
dr
B
dB
E
E
E
dE
dW
r
2
2
dr/r = 0 synchrotron dB/B = 0 cyclotron
7
From Synchrotron to Linac
In the linac there is no bending magnets, hence there is no
dispersion effects on the orbit and
=0慮搠
㴱=
2
.
s
R
C
2
s
R
C
2
cavity
E
z
s
8
From Synchrotron to Linac (2)
In the linac there is no bending magnets, hence there is no
dispersion effects on the orbit and
=0慮搠
㴱=
2
.
s
R
C
2
s
R
C
2
RF
h
C
RF
cavity
S or z
E
z
9
From Synchrotron to Linac (3)
Moreover one has:
Since in the linac
=0n搠
㴱=
2
, the longitudinal frequency becomes:
p
R
V
e
h
s
s
s
rs
s
2
cos
ˆ
2
2
v
m
p
E
R
V
h
s
s
s
RF
s
0
0
2
ˆ
leading to:
v
m
E
e
s
s
RF
s
3
0
0
2
cos
Since in a linac the independant variable is z rather than t one gets:
v
m
E
e
s
s
s
RF
3
3
0
0
2
cos
2
0
s
10
Adiabatic Damping
Though there are many physical processes that can damp the
longitudinal oscillation amplitudes, one is directly generated by the
acceleration process itself. It will happen in the synchrotron, even
ultra

relativistic, when ramping the energy but not in the ultra

relativistic electron linac which does not show any oscillation.
As a matter of fact, when E
s
varies with time, one needs to be more
careful in combining the two first order energy

phase equations in
one second order equation:
0
0
2
2
2
s
s
s
s
s
s
s
s
s
s
s
E
E
E
E
E
E
E
E
dt
d
The damping coefficient is
proportional to the rate of
energy variation and from the
definition of
s
one has:
s
s
s
s
E
E
2
11
Adiabatic Damping (2)
t
t
const
d
t
t
s
sin
ˆ
.
sin
ˆ
0
cos
ˆ
ˆ
2
t
s
s
0
ˆ
ˆ
2
s
s
To integrate the previous equation, with variable coefficients, the
method consists of choosing a solution similar to the one obtained
without the damping term:
4
/
1
2
/
1
ˆ
ˆ
s
s
E
Assuming time derivatives of parameters are small quantities (adiabatic
limit), putting the solution in the equation and neglecting second order
terms one gets:
Integrating:
12
Adiabatic Damping (3)
.
const
d
W
I
W
p
R
h
V
e
t
W
H
s
s
rs
s
2
2
4
1
cos
2
ˆ
)
,
,
(
t
W
W
s
cos
ˆ
t
s
sin
ˆ
So far it was assumed that parameters related to the acceleration
process were constant. Let’s consider now that they vary slowly with
respect to the period of longitudinal oscillation (adiabaticity).
For small amplitude oscillations the hamiltonian reduces to:
with
Under adiabatic conditions the Boltzman

Ehrenfest theorem states
that the action integral remains constant:
(W,
慲攠捡a潮楣慬i癡物扬敳b
W
p
R
h
W
H
dt
d
s
s
rs
2
1
dt
W
p
R
h
dt
dt
d
W
I
s
s
rs
2
2
1
Since:
the action integral becomes:
13
Adiabatic Damping (4)
leads to:
s
W
dt
W
ˆ
2
2
Previous integral over one period:
.
ˆ
2
2
const
W
p
R
h
I
s
s
s
rs
From the quadratic form of the hamiltonian one gets the relation:
ˆ
2
ˆ
rs
s
s
s
h
R
p
W
Finally under adiabatic conditions the long term evolution of the
oscillation amplitudes is shown to be:
E
V
R
E
s
s
s
s
4
/
1
2
4
/
1
cos
ˆ
ˆ
E
E
or
W
s
4
/
1
ˆ
ˆ
iant
in
W
var
ˆ
.
ˆ
14
Synchrotron Radiation Damping
2
2
2
4
0
0
3
2
B
E
c
E
r
P
t
2
0
0
2
0
4
c
m
e
r
Light particles, such as electrons, radiate electro

magnetic energy
when moving on circular orbits; that is typically the case in an electron
synchrotron due to the bending magnetic field.
Damping of longitudinal oscillations comes from the fact that the
radiated power depends on the electron energy and on the magnetic
field felt by the particle on its real trajectory:
Note also that that the radiation leads to an energy lost per turn :
where is the classical electron radius.
m
GeV
keV
turn
turn
E
U
or
E
r
U
4
)
(
4
0
0
5
.
88
3
4
automatically compensated by the RF system (synchronous particle).
15
Synchrotron Radiation Damping (2)
s
E
E
s
s
s
E
E
T
T
dt
dT
2
1
s
s
s
s
s
T
d
dE
dU
U
dT
dT
dV
e
eV
dt
d
0
1
2
2
s
s
s
s
s
E
dT
dV
T
e
dt
d
dE
dU
T
dt
d
)
(
)
(
1
E
U
t
eV
T
T
E
dt
d
s
s
s
s
dE
dU
T
2
1
1
2
s
Consider the energy deviation between a particle and the
reference one:
energy gain in the cavity dispersion effect
Damping term:
Focusing term:
16
Synchrotron Radiation Damping (3)
dt
P
E
U
)
(
s
x
c
ds
dt
1
1
s
x
E
D
x
ds
E
P
D
dE
dP
c
dE
dU
s
s
s
x
s
s
1
The electron radiates on its orbit:
The rate of variation of the energy loss with energy is then
:
17
Synchrotron Radiation Damping (4)
2
2
B
E
P
0
x
B
ds
x
B
B
P
D
cU
E
U
dE
dU
s
s
s
s
x
s
s
s
s
1
2
1
2
Since and considering the existence of a gradient
the previous equation becomes:
which often in the literature is expressed as:
2
2
1
2
2
s
s
s
s
s
x
s
s
s
ds
ds
x
B
B
D
D
with
D
E
U
dE
dU
D
E
U
T
s
s
s
2
2
1
Leading to the damping constant:
18
Dynamics in the Vicinity of Transition Energy
one gets:
2
2
2
1
t
Introducing in the previous expressions:
2
2
4
/
1
cos
ˆ
1
ˆ
t
s
V
2
2
4
/
1
cos
ˆ
1
ˆ
t
s
V
E
2
2
2
/
1
cos
ˆ
t
s
s
V
19
Dynamics in the Vicinity of Transition Energy (2)
In fact close to transition,
adiabatic solution are not
valid since parameters change
too fast. A proper treatment
would show that:
will not go to zero
E will not go to infinity
t
t
t
s
ˆ
E
E
s
ˆ
20
Dynamics in the Vicinity of Transition Energy (3)
0
sin
sin
2
ˆ
s
s
s
s
V
e
dt
d
h
p
R
dt
d
0
2
cos
ˆ
2
2
s
s
s
s
s
s
s
V
e
dt
d
p
dt
d
h
R
dt
d
p
h
R
0
cos
ˆ
2
1
2
0
2
2
C
m
V
he
dt
d
dt
d
dt
d
s
s
s
s
R
C
with
C
m
p
2
2
0
Back to the general second order phase equation:
Deriving the first term and looking for small phase deviations:
Since:
one gets:
21
Dynamics in the Vicinity of Transition Energy (4)
kt
const
k
dt
d
.
.
const
with
t
t
0
1
2
2
t
a
dt
d
t
dt
d
Assuming:
equivalent to:
the resulting phase equation near transition becomes:
with:
k
C
m
V
e
h
a
s
2
0
cos
2
22
Dynamics in the Vicinity of Transition Energy (5)
2
/
3
3
/
2
2
/
3
3
/
2
t
q
N
t
B
t
q
J
t
A
t
a
q
3
2
3
/
2
3
/
2
3
/
2
2
0
q
B
t
The previous equation has Bessel

Neumann type solutions symetric with
respect to t:
where
A
and
B
are initial conditions and:
Expanding Bessel and Neumann function for small t leads to:
Since energy deviation is proportional to the time derivative of phase
deviation, deriving J and N and expanding one gets:
k
h
q
AR
t
E
s
s
3
/
5
2
0
3
/
2
2
2
3
/
1
showing that energy deviation does not go to infinity.
23
Stationary Bucket
This is the case sin
s
=0 (no acceleration) which means
s
=0
or
. The
equation of the separatrix for
s
=
(above transition) becomes:
2
2
2
cos
2
s
s
2
sin
2
2
2
2
2
s
Replacing the phase derivative by the canonical variable W:
rs
s
s
rs
h
R
p
E
W
2
2
and introducing the expression
for
s
leads to the following
equation for the separatrix:
2
sin
2
ˆ
2
h
E
V
e
c
C
W
s
with C=2
R
s
W
0
2
bk
24
Stationary Bucket (2)
Setting
=
楮瑨攠灲p癩潵猠敱畡楯i杩g敳e瑨攠桥楧桴潦⁴攠扵捫整b
The area of the bucket is:
h
E
V
e
c
C
A
s
bk
2
ˆ
16
2
0
2
d
W
A
bk
Since:
2
0
4
2
sin
d
one gets:
h
E
V
e
c
C
W
s
bk
2
ˆ
2
8
A
W
bk
bk
25
Bunch Matching into a Stationary Bucket
A particle trajectory inside the separatrix is described by the equation:
W
0
2
bk
W
b
m
2

m
I
s
s
s
sin
cos
cos
2
2
2
s
=
I
s
cos
2
2
2
m
s
s
cos
cos
2
2
2
2
cos
cos
2
m
s
2
cos
2
cos
8
2
2
m
bk
A
W
The points where the trajectory
crosses the axis are symmetric with
respect to
s
=
26
Bunch Matching into a Stationary Bucket (2)
Setting
in the previous formula
allows to calculate the bunch height:
2
cos
8
m
bk
b
A
W
2
cos
m
bk
b
W
W
or:
2
cos
m
s
RF
s
b
E
E
E
E
This formula shows that for a given bunch energy spread the proper
matching of a shorter bunch will require a bigger RF acceptance, hence a
higher voltage ( short bunch means
m
close to
).
27
Effect of a Mismatch
Starting with an injected bunch with short lenght and large energy spread,
after a quarter of synchrotron period the bunch rotation shows a longer
bunch with a smaller energy spread.
W
W
2
2
16
m
bk
A
W
1
16
2
2
m
m
bk
A
W
2
16
m
bk
b
A
A
For small oscillation amplitudes the equation of the ellipse reduces to:
Ellipse area is called longitudinal emittance
28
Capture of a Debunched Beam with Adiabatic Turn

On
29
Capture of a Debunched Beam with Fast Turn

On
30
Debunching
f
R
v
p
p
f
f
h
R
s
rev
2
p
p
h
f
v
s
t
rev
debunch
/
2
1
.
Switching off the RF the stationary bucket vanishes and the
bunch will debunch:
W
Φ
Φ
W
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