Transcript Collective_effects_II-Budapest-G.Franchettix

Collective Effects II
G. Franchetti, GSI
CERN Accelerator – School
Budapest, 2-14 / 10 / 2016
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Type of fields
Collective Effects ?
Collective Effects
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Image charges
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Influence of the chamber wall
the electron in the metal
quickly travel on the surface
of the metal until the electric
field parallel to the surface
is zero
+
field line at 900
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Image charge
image
charge
+
-
the image charge is a reflection of the particle with exchanged sign
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Image charge
image
charge
+
-
the image charge is a reflection of the particle with exchanged sign
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Conducting plates
y
h
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h
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h
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h
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Incoherent motion
Consider a
particle of the
beam
h
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y
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Summing image charge
contribution in pairs
h
y
Total electric field
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Equation of motion
In the equation of motion
as
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Laslett Tuneshift
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Image currents
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Ferromagnetic Boundaries
B1n
B1t
B2n
B2t
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Ferromagnetic Boundaries
image current
beam current
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image current
B
g
x
y
Beam
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image current
for
Bx
y
In the equation of motion
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therefore
incoherent
SC
ferromagnetic
induced image
current
(coherent force)
Tune-shift !
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Coherent Motion
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Coherent motion
y
y
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y
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all image
charges
moves
y
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Coherent motion
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all image
charges
moves
Force on the beam
y
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NO FORCE CREATED ON THE BEAM
2h
y
y
2h
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all image
charges
moves
Force on the beam
y
n = 1, 3, 5, 7, …
n = 1, 3, 5, 7, …
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therefore
(trick!)
with n = 1, 2, 3, 4, 5, 6, …
The electric field Ex due to coherent shift is zero on the center of mass
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equation of motion
but
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Coherent detuning
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Beam-beam
Bunches of the beam 1 feels the field of the other beam when
it travels through the other bunch of the beam 2. The same is reverted
for Beam 2.
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Beam-beam tune-shift
r0 = classical radius of particle, B=number of bunches per beam,
N = particle per beam
Usually one would expect that the acceptable tune-shift be 0.15 – 0.2
Reality requires more stringent values ~ 0.004 – 0.006
Complications: one beam feels also all the nonlinear field of the other beam!
Storage is for millions of turns, and high order resonances plays a crucial role
and should be avoided.
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The Collective Effects
Thanks to Oliver Boine-Frankenheim, I. Hofmann, U. Niedermayer,
D. Brandt
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Interaction of the beam with the
environment
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Effect on the dynamics
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Bunch in a conducting pipe
+
v
+ + ++ + + +
v
+
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+ + ++ + + +
v
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Bunch in a conducting pipe with
sudden change
v
+ + +++ + + +
v
v
+ + +++ + + +
v
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Situation
v
+ + +++ + + +
v
+ + +++ + + +
v
The electric field from the self-field has a delay
Energy gained by one unit charge
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Panofsky theorem
v=c
out
q
in
?
Can we say something about
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Panofsky theorem
We find a constrain to the forces created by electromagnetic fields
Without actually knowing almost anything !
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Wake Field
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Cavities
E
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E
E
EE
E
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E
E
E
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MODEL
E
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MODEL
Capacitor
I
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MODEL
E
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MODEL
Inductance
I
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MODEL
E
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MODEL
Resistance
I
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All together
L
C
R
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RLC Features
Isolated RLC
Resonance frequency
L
C
Quality factor
R
V
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Damping rate
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Meaning
V
1
5
0
t
5
1
0
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2
4
6
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Response to one particle
What happen when one particle goes through the cavity ?
Before
I
t
v
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Response to one particle
After
I
t
v
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Pulse Response
This is the potential in the cavity
Green or wake function
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v
t
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Summary
The wake function tells us what is the longitudinal field experienced by another
particle passing through the cavity later
z
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Summary
The wake function tells us what is the longitudinal field experienced by another
particle passing through the cavity later
z
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Summary
The wake function tells us what is the longitudinal field experienced by another
particle passing through the cavity later
z
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Summary
The wake function tells us what is the longitudinal field experienced by another
particle passing through the cavity later
z
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Impedance
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Impedance
It is a quantity that relate V and I
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Impedance
Impedance
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Properties
1
0.5
0
-0.5
0
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2
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Properties
is zero
At
is maximum
inductive
capacitive
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Power dissipated
The power dissipated depends on the resistive impedance
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Complex notation
Complex notation
If Q is very large only for
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close to
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Wake potential  Impedance
Charge through the cavity at
Consider now the wake at
The wake of that charge at time t is
The potential in the cavity at time t due to the charge passing at t’ is
The total potential due to all charges passing through the cavity in
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If now the current I is
then
with some change of variable
We wait long enough that transient effect disappears, hence
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Complicated geometries of the vacuum chamber give an effect on the
beam which is described by the impedance Z(ω)
1
0.5
0
-0.5
0
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2
3
4
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Consequences of impedances
Energy loss on pipes  heating (important if you have
a superconducting machine!)
narrow
resonances
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Broad
Band
Model
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Consequences of impedances
Feed-back to the beam as a hole: collective effects
Impedance
We have seen the longitudinal
impedance in a cavity
More types of impedances …
Dynamics of the
all beam is affected
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Longitudinal dynamics
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Longitudinal dynamics
synchronous orbit
R0
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Longitudinal dynamics
synchronous orbit
R0
C + δC
C
p
R
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This property
comes from
the magnets
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Longitudinal dynamics
revolution time
R0
C + δC
C
p
v + dv
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R
v
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Nobody can go faster than light
revolution time
If this is large
p v
R0
v + dv
this velocity will always be less
than “c”
Therefore at a certain point the
circumference will growth, but
the particle speed remains “c”
It takes longer to make one turn !
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If
we are at the transition energy
If
increasing energy
revolution time shorter
If
increasing energy
revolution time longer !!
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RF
synchronous orbit
R0
RF
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The synchronous particle has energy
and goes through the cavity at time
Voltage in the cavity
this is the phase of the synchronous particle
This is a phase we know each time the particle goes through the cavity
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Non synchronous particle
R0
RF
slower particle
(if below transition)
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Voltage on the particle
Gain of energy
Now we include an energy loss per turn an per particle U
Define relative energy
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If
is small, than this term is equal to the time derivative of
but U, depends on
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If
are small we can expand
These two terms are equal
for the synchronous particle
Phase shift is used to
measure U  Z
We remain
with the equation
In addition at high energy
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Final equation of motion (in tau)
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Solution
Solving for lambda:
that is
with
if
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Solution stable
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Interpretation
E < ET
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No Energy Loss
E > ET
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Interpretation
With Energy Loss
E < ET
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E > ET
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Bunch Lengthening
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Bunch lengthening
dz
Ez
In one turn
change of energy
per charge
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L is the integrated inductance
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Parabolic bunch
z
center
of the
bunch
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Parabolic bunch
center
of the
bunch
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Voltage induced
V
center
of the
bunch
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If we compare with RF
V
This creates a
“longitudinal detuning”
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center
of the
bunch
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By using a bunch with the same longitudinal emittance a reduction of
longitudinal focusing strength produces a bunch lengthening
The bunch becomes matched with the effective voltage slope
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Effective voltage
induced voltage
Linearizing in tau
focusing from RF
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defocusing from
impedance
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But
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therefore
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but
Therefore
is the longitudinal strength
in absence of impedance
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Therefore the relative change in omega is
For protons
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Observation
The effect of the impedance is local, hence the voltage induced by
impendence does not effect the center of mass (like for the space charge)
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Summary
1) Wall charges creates detuning  incoherent tunes
2) Ferromagnetic material creates image currents:
Coherent motion  coherent tunes
3) Concept of Wake field
4) Impedance of a cavity, Wake  impedance
5) Energy loss
6) Longitudinal dynamics, effect of energy loss
7) Bunch lengthening
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References
Lectures of Albert Hofmann, CAS
Physics of collective beam instabilities in high energy accelerators - A.W. Chao, 1993
Theory and Design of Charged Particle Beams - M. Reiser, 1994
Particle Accelerator Physics - H. Wiedemann
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