LONGITUDINAL DYNAMICS
by
Joël Le DuFF
(LALOrsay)
Introductory Level Course on Accelerator Physics
Baden,
1224
September
2004
CASBaden,1224 September2004
1
CASBaden,1224 September2004
2
summary
•
RadioFrequency Acceleration and Synchronism
• Properties of RadioFrequency cavities
• Energy Gain
• Principle of Phase S
tability and Consequences
• Synchronous linear accelerator
• EnergyPhase oscillations
• The capture phenomenon
• The Synchrotron
• RF cavities for Synchrotron
• Dispersion Effects in a Synchrotron
• EnergyPhase Equations in a Synchrotron
• Phase space motions
CASBaden,1224 September2004
3
Bibliography
:
Old Books
M. Stanley Livingston High Energy Accelerators
(Interscience Publishers, 1954)
J.J. Livingood Principles of cyclic Particle Accelerators
(D. Van Nostrand Co Ltd , 1961)
M. Stanley Livingston and J. B. Blewett Particle Accelerators
(Mc Graw Hill Book Company, Inc 1962)
K.G.
Steffen
High Energy optics
(Interscience
Publisher, J.
Wiley
& sons, 1965)
H.
Bruck
Accelera
teurs circu
laires de particules
(PUF, Paris 1966)
M. Stanley
Livingston (editor)
The development
of High Energy Accelerators
(Dover
Publications,
Inc, N. Y. 1966)
A.A. Kolomensky & A.W. Lebedev Theory of cyclic Accelerators
(North Holland Publihers
Company, Amst. 1966)
E. Persico, E. Ferrari, S.E. Segre Principles of Particles Accelerators
(W.A. Benjamin, Inc. 1968)
P.M. Lapostolle
& A.L.
Septier Li
near Acceler
ators
(North Holland Publihers
Company, Amst. 1970)
A.D. Vlasov Theory
of
Linear Accelerators
(Programm
for scientific
translations,
J
erusalem
1968)
CASBaden,1224 September2004
4
Bibliography
:
New Books
M. Conte, W.W. Mac Kay An
Introduction
to the Physic
s of particle Accelerators
(World Scientific, 1991)
P. J. Bryant and K. Johnsen
The Principl
es
of Circular
Accelerators and Storage
Rings
(Cambridge University
Press, 1993)
D. A. Edwards, M. J. Syphers An
Introduction
to the
Physics
of High Energy Accelerators
(J.
Wiley
& sons,
Inc,
1993)
H. Wiedemann
Particle Acceler
ator
Physics
(SpringerVerlag, Berlin, 1993)
M. Reiser
Theory and
Design
of Charged Particles Beams
(J.
Wiley
& sons, 1994)
A. Chao, M. Tigner
Handbook of Accelerato
r Physics and Engineering
(World Scientific
1998)
K. Wille
The Physics of Particle Accelerators: An Introduction
(Oxford University Press, 2000)
E.J.N. Wilson
An introduction to Particle Accelerators
(Oxford University
Press, 200
1)
And CERN Accelerator Schools (CAS) Proceedings
CASBaden,1224 September2004
5
Main Characteristics of an Accelerator
ACCELERATION is the main job of an accelera
tor.
•The accelerator provides kinetic energy
to ch
arged
particles, hence increasing their
momentum.
•In order to do so, it is necessary to have an electric field
, prefera
bly along the
direction of the initial momentum.
E
r
eE
dt
dp
=
BENDING is generated by
a magnetic field
perpendicular
to the plane of the
particle trajectory. The bending radius
ρ
obeys to the relation
:
ρ
B
e
p
=
FOCUSING is a second way of using a magnetic field, in which the
bending
effect is used to bring the particles trajectory closer to the axis, hence
to increase the beam density.
CASBaden,1224 September2004
6
Acceleration & Curvature
θ
o
x, r
z
s
ρ
Within the assumption:
θ
E
E
→
r
z
B
B
→
r
the NewtonLorentz force:
B
v
e
E
e
dt
p
d
r
r
r
r
×
+
=
becomes:
(
)
r
z
r
u
B
ev
u
eE
u
v
m
u
dt
mv
d
r
r
r
r
θ
θ
θ
θ
θ
θ
ρ
−
=
−
2
ρ
θ
θ
θ
z
B
e
p
eE
dt
dp
=
=
leading to:
CASBaden,1224 September2004
7
Energy Gain
In relativistic dynamics, energy and momentum satisfy the relation:
(
)
W
E
E
+
=
0
2
2
2
0
2
c
p
E
E
+
=
Hence:
vdp
dE
=
The rate of energy gain per unit length of acceleration (along z) is then:
z
eE
dt
dp
dz
dp
v
dz
dE
=
=
=
and the kinetic energy gained from the field along the z path is:
eV
dz
E
e
W
dz
eE
dE
dW
z
z
=
∫
=
⇒
=
=
where
V
is just a potential
CASBaden,1224 September2004
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Methods of Acceleration
1_ Electrostatic Field
Energy gain
: W=n.e(V2V1)
limitation :
Vgenerator
=Σ
Vi
Electrostatic accelerator
2_ Radiofrequency Field
Synchronism
:
L=vT/2
Wideroe structure
v=particle velocity
T= RF
period
Methods of Acceleration (2)
3_ Acceleration by induction
From MAXWELL
EQUATIONS
:
The electric field is derived
from a scalar potential
φ
and a vector potential
A
The time variation of the magnetic field
H
generates an electric field
E
A
H
B
t
A
E
r
r
r
r
r
r
r
×
∇
=
=
∂
∂
−
∇
−
=
µ
φ
CASBaden,1224 September2004
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CASBaden,1224 September2004
10
The advantage of Resonant Cavities

Considering RF acceleration, it is obvious that when particles get high
velocities
the drift spaces get longer and one loses on the efficiency. The
solution consists of using a higher operating frequency.

The power lost by r
adiation, due to circulating currents on the electrodes,
is proportional to the
RF
frequency. The
s
olution consists
of
enclosing
the
system
in a cavity
which resonant frequency matches the RF generator
frequency.
Each such cavity can be independently
powered from the RF generator.

The electromagnetic power is now
constrained in the resonant volume.

Note however that joule losses will
occur in the cavity walls
(unless
made
of superconducting materials)
E
z
r
J
ou
H
r
r
ωRF
CASBaden,1224 September2004
11
The Pill Box Cavity
0
2
2
0
0
2
=
∂
∂
−
∇
tA
A
µ
ε
)
(
H
ou
E
A
=
(
)
kr
J
E
z
0
=
()
kr
J
Z
j
H
1
0
−
=
θ
Ω
=
=
=
=
377
62
,
2
2
0
Z
a
c
k
λ
ω
λ
π
e
t
j
ω
}
E
z
Hθ
From Maxwell’s equations one can derive
the wave equations
:
Solutions for E and H are oscillating modes,
at discrete frequencies, of types TM ou
TE.
For l<2a the most simple mode, TM
010, has
the lowest frequency ,and has only two field
components:
CASBaden,1224 September2004
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The Pill Box Cavity (2)
The design of a pillbox
cavity can
be sophisticated in order to
improve its performances:
A nose cone can be introduced in
order to concentrate the electric
field
around the axis,
Round shaping of the corners
allows a better
distribution of the
magnetic field on the surface and a
reduction of the Joule losses. It
also prevent from multipactoring
effects.
A good cavity is a cavity which
efficiently transforms the RF
power into accelerating voltage.
CASBaden,1224 September2004
13
Transit Time Factor
t
g
V
t
E
E
z
ω
ω
cos
cos
0
=
=
v
t
z
=
Oscillating field at frequency
ω
and which amplitude
is assumed to be constant all along the
gap:
Consider a particle passing through the middle of
the gap at time t=0 :
dz
v
z
g
eV
W
g
g
∫
=
∆
−
2
/
2
/
cos
ω
The total energy gain is:
angle
transit
v
g
ω
θ
=
factor
time
transit
T
eVT
eV
W
=
=
∆
2
/
2
/
sin
θ
θ
( 0 < T < 1 )
CASBaden,1224 September2004
14
Transit Time Factor (2)
Consider the most general case and make use of complex notations:
(
)
dz
e
z
E
e
E
t
j
g
z
e
ω
∫
ℜ
=
∆
0
()
⎥
⎦
⎤
⎢
⎣
⎡
∫
ℜ
=
∆
−
dz
e
z
E
e
e
E
v
z
j
g
z
j
e
p
ω
ψ
0
()
⎥
⎦
⎤
⎢
⎣
⎡
∫
ℜ
=
∆
−
g
v
z
j
z
j
j
e
dz
e
z
E
e
e
e
E
i
p
0
ω
ψ
ψ
()
φ
ω
cos
0
∫
=
∆
g
v
z
j
z
dz
e
z
E
e
E
(
)
()
∫
∫
=
g
z
g
t
j
z
dz
z
E
dz
e
z
E
T
0
0
ω
p
v
z
t
ψ
ω
ω
−
=
i
p
ψ
ψ
φ
−
=
Ψp
is the phase of the particle entering the gap with respect to the RF.
and considering the phase which yields the
maximum energy gain:
Introducing:
CASBaden,1224 September2004
15
Important Parameters of Accelerating Cavities
Shunt Impedance
R
V
Pd
2
=
Relationship between gap
voltage and wall losses.
Quality Factor
Relationship between
stored energy in the
volume and dissipated
power on the walls.
P
W
Q
d
s
ω
=
W
V
Q
R
s
ω
2
=
Filling Time
ω
τ
Q
=
W
Q
dt
W
d
P
s
s
d
ω
=
−
=
Exponential decay of the
stored energy due to losses.
Principle of Phase Stability
Let’s consider a succession of acce
lerating gaps, operating in the 2π mode,
for which the synchronism condition is fulfilled for a phase Φ
s .
For a 2π m
ode,
the electric field
is the same in all
gaps at any given
time.
is the energy gain in one gap for the particle to reach the next
gap with the same RF phase: P1 ,P2, …… are fixed points.
s
V
e
s
eV
Φ
=
sin
ˆ
If an increase in energy is transferred into an increase in velocity, M1 & N1
will move towards P1(stable), while M2
& N2
will go away from P2
(unstable).
CASBaden,1224 September2004
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A Consequence of Phase Stability
Transverse Instability
0
0
<
∂
∂
⇒
>
∂
∂
z
z
E
t
V
Longitudinal phase stability means
:
defocusing
RF force
0
0
0
.
>
∂
∂
⇒
=
∂
∂
+
∂
∂
⇒
=
∇
x
E
z
E
x
E
E
x
z
x
The divergence of the field is
zero according to Maxwell :
External focusing (solenoid,
quadrupole) is then necessary
CASBaden,1224 September2004
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CASBaden,1224 September2004
18
The Traveling Wave Case
(
)
()
0
0
cos
t
t
v
z
v
k
kz
t
E
E
RF
RF
z
−
=
=
−
=
ϕ
ω
ω
velocity
particle
v
velocity
phase
v
=
=
ϕ
The particle travels along with the wave, and
k represents the wave propagation factor.
⎟
⎠
⎞
⎜
⎝
⎛
−
−
=
0
0
cos
φ
ω
ω
ϕ
t
v
v
t
E
E
RF
RF
z
0
0
cos
φ
ϕ
E
E
and
v
v
z
=
=
If synchronism satisfied:
where φ0
is the RF phase seen by the particle.
CASBaden,1224 September2004
19
Multigaps Accelerating Structures:
A
Low Kinetic Energy Linac
(protons,ions)
Mode 2
π
L= vT = βλ
Mode π
L= vT/2
In
«
WIDEROE
»
structure
radiated power
∝ωCV
In order to reduce the
radiated power the gap is
enclosed in a resonant
volume at the operating
frequency.
A common wall
can be suppressed if no
circulating current in it
for the chosen mode.
ALVAREZ
structure
CASBaden,1224 September2004
20
CERN Proton Linac
CASBaden,1224 September2004
21
Multigaps Accelerating Structures:
B
High Energy Electron Linac

When particles gets ultrarelativistic (v~c) the drift tubes become
very long unless the opera
ting frequency is increased. Late 40’s
the
development of radar led to high power transmitters (klystrons) at very
high frequencies (3 GHz).

Next came the idea of suppressing the drift tubes using traveling
waves. However to get a continuous acceleration the phase velocity of
the wave needs to be adjusted to the particle velocity.
solution
: slow wave guide with irises
iris loaded structure
CASBaden,1224 September2004
22
Iris Loaded Structure for Electron Linac
Photo of a
CGRMeV
structure
4.5 m lo
ng copper structure, equipped
with matched input and output couplers.
Cells are low temperature brazed and a
stainless steel envelope ensures proper
vacuum.
CASBaden,1224 September2004
23
Energyphase Equations

Rate of energy gain for the synchronous particle:
s
s
s
eE
dt
dp
dz
dE
φ
sin
0
=
=

Rate of energy gain for a nonsynchronous particle, expressed in
reduced variables, and
:
s
s
E
E
W
W
w
−
=
−
=
s
φ
φ
ϕ
−
=
()
[]
()
ϕ
ϕ
φ
φ
ϕ
φ
small
eE
eE
dz
dw
s
s
s
.
cos
sin
sin
0
0
≈
−
+
=

Rate of change of the phase with respect to the synchronous one:
()
s
s
RF
s
RF
s
RF
v
v
v
v
v
dz
dt
dz
dt
dz
d
−
−
≅
⎟
⎠
⎞
⎜
⎝
⎛
−
=
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
−
=
2
1
1
ω
ω
ω
ϕ
()
(
)
3
0
2
2
2
s
s
s
s
s
s
v
m
w
c
c
v
v
γ
β
β
β
β
β
≅
−
≅
−
=
−
Since:
CASBaden,1224 September2004
24
Energyphase Oscillations
w
v
m
dz
d
s
s
RF
3
3
0
γ
ω
ϕ
−
=
one gets:
Combining the two first order equations into a second order one:
0
2
2
2
=
Ω
+
ϕ
ϕ
s
dz
d
3
3
0
0
2
cos
s
s
s
RF
s
v
m
eE
γ
φ
ω
=
Ω
with
Stable harmonic oscillations imply:
real
and
s
0
2
>
Ω
hence:
0
cos
>
s
φ
And since acceleration also means:
0
sin
>
s
φ
One finally gets the results:
2
0
π
φ
<
<
s
CASBaden,1224 September2004
25
The Capture Problem

Previous results show that at ultrarelativistic energies (γ
>> 1) the
longitudinal motion is frozen. Since this is rapidly the case for electrons, all
traveling wave structures can be made identical (phase velocity=c).

Hence the question is: can we capture low kinetic electrons energies (γ
< 1),
as they come out from a gun, using an iris loaded structure matched to c ?
(
)
t
E
E
z
φ
sin
0
=
e
β0 < 1
vϕ=c
gun
structure
The electron entering the structure, with velocity v < c, is not
synchronous
with the wave. The path difference, after a time dt, between the wa
ve and
the particle is:
(
)
dt
v
c
dz
−
=
Since:
c
v
k
factor
n
propagatio
with
kz
t
RF
RF
RF
ω
ω
ω
φ
ϕ
=
=
−
=
φ
π
λ
φ
ω
d
d
c
dz
g
RF
2
=
=
()
β
λ
π
φ
−
=
1
2
c
dt
d
g
one gets:
and
CASBaden,1224 September2004
26
The Capture Problem (2)
()
()
()
φ
β
β
βγ
sin
1
0
2
1
2
0
0
eE
dt
d
c
m
dt
d
c
m
mv
dt
d
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−
=
=
From NewtonLorentz:
Introducing a suitable variable:
α
φ
α
2
0
0
sin
sin
c
m
eE
dt
d
−
=
the equation becomes:
2
1
1
2
cos
cos
2
1
0
0
0
2
0
0
≤
⎟
⎠
⎞
⎜
⎝
⎛
+
−
=
−
β
β
λ
π
φ
φ
E
e
c
m
g
α
α
α
λ
π
φ
φ
d
eE
c
m
d
g
2
0
2
0
sin
cos
1
2
sin
−
=
−
dt
d
d
d
dt
d
α
α
φ
φ
=
Using
Integrating from t0
to t
(from β=β0
to β=1)
2
1
0
0
2
0
0
1
1
⎟
⎠
⎞
⎜
⎝
⎛
+
−
≥
β
β
λ
π
g
e
c
m
E
Capture condition
CASBaden,1224 September2004
27
Improved Capture With Prebuncher
A long bunch coming
from the gun enters an
RF cavity; the
reference particle is
the one which has no
velocity change. The
others get accelerated
or decelerated. After
a distance L bunch
gets shorter while
energies are sprea
d:
bunching effect. This
short bunch can now
be captured more
efficiently by a TW
structure (vϕ=c).
CASBaden,1224 September2004
28
Improved Capture With Prebuncher (2)
The bunching effect is a space modulation that results from a velocity
modulation and is similar to the phase stability phenomenon. Let’s look at
particles in the vicinity of the reference one and use a classical approach.
Energy gain as a function of cavity crossing time:
φ
φ
0
0
0
0
2
0
sin
2
1
eV
eV
v
v
m
v
m
W
≈
=
∆
=
⎟
⎠
⎞
⎜
⎝
⎛
∆
=
∆
0
0
0
v
m
eV
v
φ
=
∆
Perfect linear bunching will occur after a time delay τ, corresponding to
a distance L, when the path difference is compensated between a
particle and the reference one:
RF
v
t
v
z
v
ω
φ
τ
0
0
.
=
=
∆
=
∆
(assuming the reference particle
enters the cavity at time t=0)
Since L = vτ
one gets:
RF
eV
W
v
L
ω
0
0
2
=
CASBaden,1224 September2004
29
The Synchrotron
The synchrotron is a synchronous accelerator since there is a synchronous RF
phase for which the energy gain fits the increase of the magnetic field at each
turn. That implies the following operating conditions:
B
e
P
B
cte
R
cte
h
cte
V
e
r
RF
s
⇒
=
=
=
=
=
Φ
=
Φ
Φ
ρ
ρ
ω
ω
sin
^
Energy gain per turn
Synchronous particle
RF synchronism
Constant orbit
Variable magnetic field
If
v =
c,
ωr
hence
ωRF
remain constant (ultrarelativistic e
)
The Synchrotron (2)
Energy ramping is simply obtained by varying the B field:
v
B
R
e
r
T
B
e
turn
p
B
e
dt
dp
eB
p
′
=
′
=
∆
⇒
′
=
⇒
=
ρ
π
ρ
ρ
ρ
2
)
(
p
v
E
c
p
E
E
∆
=
∆
⇒
+
=
2
2
2
0
2
Since:
()
(
)
φ
ρ
π
s
s
turn
V
e
RB
e
W
E
sin
ˆ
'
2
=
=
∆
=
∆
•The number of stable synchronous particles is equal to the harmonic
number h. They are equally spaced along the circumference.
•Each synchronous particle satifies the relation p=eBρ
. They have the
nominal energy and follow the nominal trajectory.
CASBaden,1224 September2004
30
CASBaden,1224 September2004
31
The Synchrotron (3)
During the energy ramping, the RF frequency
increases to follow the increase of the
revolution frequency :
hence :
)
,
(
s
RF
r
R
B
h
ω
ω
ω
=
=
)
(
)
(
2
2
1
)
(
)
(
2
1
2
)
(
)
(
t
B
s
R
r
t
s
E
ec
h
t
RF
f
t
B
m
e
s
R
t
v
h
t
RF
f
π
π
π
=
⇒
>
<
=
=
Since , the RF frequency must follow the variation of the
B
field with the law :
which asymptotically tends
towards when B becomes large compare to (m0c2
/ 2πr) which corresponds to
v c (pc >> m0c2
). In practice the
B field can follow the law:
2
2
2
0
2
c
p
c
m
E
+
=
2
1
2
)
(
2
)
/
2
0
(
2
)
(
2
)
(
⎭
⎬
⎫
⎩
⎨
⎧
+
=
t
B
ecr
c
m
t
B
s
R
c
h
t
RF
f
π
R
c
f
r
π
2
→
t
B
t
B
t
B
2
2
sin
)
cos
1
(
2
)
(
ω
ω
=
−
=
CASBaden,1224 September2004
32
Single Gap Types Cavities
Pillbox variants
noses disks
Coaxial cavity
Type λ/4
CASBaden,1224 September2004
33
Ferrite Loaded Cavities

Ferrite toroids are placed around
the beam tube which allow to reach
lower frequencies at reasonable size.

Polarizing the ferrites will change
the resonant frequency, hence
satisfying energy ramping in protons
and ions synchrotrons.
CASBaden,1224 September2004
34
High Q cavities for e
Synchrotrons
CASBaden,1224 September2004
35
LEP 2: 2x100 GeV
with SC cavities
CASBaden,1224 September2004
36
Dispersion Effects in a Synchrotron
cavity
Circumference
2π
R
If a particle is slightly shifted in
momentum it will have a different
orbit:
dp
dR
R
p
=
α
E
This is the “momentum compaction”
generated by the bending field.
E+δE
If the particle is shifted in momentum it will
have also a different velocity. As a result of
both effects the revolution frequency changes:
dp
df
f
p
r
r
=
η
p=particle momentum
R=synchrotron physical radius
fr=revolution frequency
CASBaden,1224 September2004
37
Dispersion Effects in a Synchrotron (2)
θ
x
ρ
0
s
s
p
dp
p
+
dθ
x
()
θ
ρ
θ
ρ
d
x
ds
d
ds
+
=
=
0
dp
dR
R
p
=
α
The elementary path difference
from the two orbits is:
ρ
x
ds
dl
dsds
ds
=
=
−
0
0
0
leading to the total change in the circumference:
< >m
means that
the average is
considered over
the bending
magnet only
m
m
x
dR
xds
ds
x
dR
dl
=
⇒
∫
=
∫∫
=
=
0
0
1
2
ρ
ρ
π
R
D
m
x
=
α
p
dp
D
x
x
=
Since:
we get:
CASBaden,1224 September2004
38
Dispersion Effects in a Synchrotron (3)
R
dR
d
f
df
R
c
f
r
r
r
−
=
⇒
=
β
β
π
β
2
dp
df
f
p
r
r
=
η
()
()
()
β
β
β
β
β
β
β
βγ
d
d
d
p
dp
c
E
mv
p
1
2
2
1
2
2
1
2
0
1
1
1
−
−
−
−
=
−
−
+
=
⇒
=
=
α
γ
η
−
=
2
1
p
dp
f
df
r
r
⎟
⎠
⎞
⎜
⎝
⎛
−
=
α
γ
2
1
α
γ
1
=
tr
η=0 at the transition energy
CASBaden,1224 September2004
39
Phase Stability in a Synchrotron
From the definition of
η
it is clear that below transition an increase in
energy is followed by a hig
her revolution frequency (increase in
velocity
dominates) while the reverse occurs above transition (v ≈
c and longer path)
where the momentum compaction (generally > 0) do
minates.
Stable synchr. Particle
for η<0
η
> 0
CASBaden,1224 September2004
40
Longitudinal Dynamics
It is also often called “ synchrotron motion”.
The RF acceleration process clearly emphasizes two coupled
variables, the energy gained by the particle and the RF
phase experienced by the same particle. Since there is a
well defined synchronous particle which has always the same
phase φs, and the nominal energy Es, it is sufficient to follow
other particles with respect to that particle. So let’s
introduce the following reduced variables:
revolution frequency : ∆fr
= fr
–f
rs
particle RF phase :
∆φ
= φ

φs
particle momentum : ∆p = p 
ps
particle energy : ∆E = E –
Es
azimuth angle : ∆θ
= θ

θs
CASBaden,1224 September2004
41
First EnergyPhase Equation
θ
R
∫
=
∆
−
=
∆
⇒
=
dt
with
h
hf
f
r
r
RF
ω
θ
θ
φ
For a given particle with respect to the reference one:
()
()
dt
d
h
dt
d
h
dt
d
r
φ
φ
θ
ω
1
1
−
=
∆
−
=
∆
=
∆
c
p
E
E
2
2
2
0
2
+
=
p
R
p
v
E
s
rs
s
∆
=
∆
=
∆
ω
s
r
rs
s
dp
d
p
⎟
⎠
⎞
⎜
⎝
⎛
=
ω
ω
η
Since:
and
(
)
φ
ηω
φ
ηω
ω
&
rs
s
s
rs
s
s
rs
h
R
p
dt
d
h
R
p
E
−
=
∆
−
=
∆
one gets:
CASBaden,1224 September2004
42
Second EnergyPhase Equation
The rate of energy gained by a particle is:
π
ω
φ
2
sin
ˆ
r
V
e
dt
dE
=
The rate of relative energy gain with respect to the reference
particle is then:
)
sin
(sin
ˆ
2
s
r
V
e
E
φ
φ
ω
π
−
=
⎟
⎠
⎞
⎜
⎝
⎛
∆
&
Expanding the left hand side to first order:
()
()
E
T
dt
d
E
T
T
E
E
T
T
E
T
E
rs
rs
r
rs
r
r
∆
=
∆
+
∆
=
∆
+
∆
≅
∆
&
&
&
&
&
leads to the second energyphase equation:
()
s
rs
V
e
E
dt
d
φ
φ
ω
π
sin
sin
ˆ
2
−
=
⎟
⎠
⎞
⎜
⎝
⎛
∆
CASBaden,1224 September2004
43
Equations of Longitudinal Motion
()
s
rs
V
e
E
dt
d
φ
φ
ω
π
sin
sin
ˆ
2
−
=
⎟
⎠
⎞
⎜
⎝
⎛
∆
()
φ
ηω
φ
ηω
ω
&
rs
s
s
rs
s
s
rs
h
R
p
dt
d
h
R
p
E
−
=
∆
−
=
∆
deriving and combining
()
0
sin
sin
2
ˆ
=
−
+
⎥
⎦
⎤
⎢
⎣
⎡
s
rs
s
s
V
e
dt
d
h
p
R
dt
d
φ
φ
π
φ
ηω
This second order equation is no
n linear. Moreover the parameters
within the bracket are in general slowly varying with time…………………
CASBaden,1224 September2004
44
Small Amplitude Oscillations
Let’s assume constant parameters
Rs,p
s,
ωs
and η:
s
s
s
rs
s
p
R
V
e
h
π
φ
ηω
2
cos
ˆ
2
=
Ω
()
0
sin
sin
cos
2
=
−
Ω
+
s
s
s
φ
φ
φ
φ
&
&
with
Consider now small phase deviations from the reference particle:
(
)
φ
φ
φ
φ
φ
φ
φ
∆
≅
−
∆
+
=
−
s
s
s
s
cos
sin
sin
sin
sin
(for small ∆φ)
and the corresponding
linearized
motion reduces to a harmonic oscillation:
0
2
=
∆
Ω
+
φ
φ
s
&
&
stable for and Ωs
real
0
2
>
Ωs
γ
< γtr
η
> 0 0 < φs
< π/2 sinφs
> 0
γ
> γtr
η
< 0 π/2 < φs
< π
sinφs
> 0
CASBaden,1224 September2004
45
Large Amplitude Oscillations
For larger phase (or energy) deviations from the reference the
second order differential equation is nonlinear:
()
0
sin
sin
cos2
=
−
Ω
+
s
s
s
φ
φ
φ
φ
&
&
(Ωs
as previously defined)
Multiplying by and integrating gives an invariant of the motion:
φ
&
()
I
s
s
s
=
+
Ω
−
φ
φ
φ
φ
φ
sin
cos
cos
2
2
2
&
which for small amplitudes reduces to:
()
I
s
=
∆
Ω
+
2
2
2
2
2
φ
φ
&
(the variable is ∆φ
and φ
s
is constant)
Similar equations exist for the second variable : ∆E∝dφ/dt
CASBaden,1224 September2004
46
Large Amplitude Oscillations (2)
Equation of the separatrix:
When φ
reaches πφs
the force goes
to zero and be
yond it becomes non
restoring. Hence πφs is an extreme
amplitude for a stable motion which
in the phase space( ) is shown
as closed trajectories.
φ
φ
∆
Ω
,
s
&
()
()
()
()
s
s
s
s
s
s
s
s
φ
φ
π
φ
π
φ
φ
φ
φ
φ
φ
sin
cos
cos
sin
cos
cos
2
2
2
2
−
+
−
Ω
−
=
+
Ω
−
&
Second value φm
where the separatrix crosses the horizontal axis:
(
)(
)
s
s
s
s
m
m
φ
φ
π
φ
π
φ
φ
φ
sin
cos
sin
cos
−
+
−
=
+
CASBaden,1224 September2004
47
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:
φ
&
0
=
φ
&
&
s
φ
φ
=
(
)
{
}
s
s
s
φ
π
φ
φ
tan
2
2
2
2
2
max
−
+
Ω
=
&
That translates into an acceptance in energy:
()
⎭
⎬
⎫
⎩
⎨
⎧
−
=
⎟
⎠
⎞
⎜
⎝
⎛
∆
φ
η
π
β
s
s
s
G
E
h
V
e
E
E
ˆ
2
1
max
m
(
)
(
)
[
]
φ
π
φ
φ
φ
s
s
s
s
G
sin
2
cos
2
−
+
=
This “RF acceptance” depends strongly on φs
and plays an important role
for the electron capture at injection, and the stored beam lifetime.
CASBaden,1224 September2004
48
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 φs =180º (or 0°) which
correspond to no
acceleration . The RF
acceptance increases with
the RF voltage.
CASBaden,1224 September2004
49
Potential Energy Function
()
φ
φ
F
dt
d
=
2
2
()
φ
φ
∂
∂
−
=
U
F
()
()
F
d
F
U
s
s
s
0
0
2
sin
cos
cos
−
∫
+
Ω
−
=
−
=
φ
φ
φ
φ
φ
φ
φ
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 nondissipative system.
CASBaden,1224 September2004
50
Ions in Circular Accelerators
E
r
= A E0
m = γ
mr
P = m v
E = γ
Er
A =atomic number
Q =charge state
q = Q e
W = E –
Er
P = q B r
E
2
= p2c2
+ Er2
(
)
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
⎟
⎠
⎞
⎜
⎝
⎛
=
⎟
⎠
⎞
⎜
⎝
⎛
+
⎥
⎦
⎤
⎢
⎣
⎡
+
−
=
=
r
dr
B
dB
E
E
E
dE
dW
r
2
2
Moreover:
dr/r = 0 synchrotron
dB/B = 0
cyclotron
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