Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
Understanding RF knockout:
The effect of Crab Phase/Frequency Failures
as seen from the beam’s phase space
. . . it’s picking up bad vibrations. . .
Kyrre Sjobak
CC failure studies meeting #2,
March 15th 2016
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
1 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
Outline
1
Betatron oscillator model
2
Dipole error
3
A crab cavity is on the loose. . .
4
Some SixTrack simulations
5
Conclusions
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
2 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
Simple example!
http://physics.stackexchange.com/questions/159728/
forced-oscillations-resonance
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
3 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
Simple example!
Except we want to be on the right hand side of the figure...
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
3 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
A model of a betatron oscillator
Undamped harmonic oscillator:
In all cases: ω0 = 2π/T
ẍ + ω02 x = 0
(1)
General solution:
Note that if Equation (1)
describes everything,
s
x (t) = A cos(ω0 t) + B sin(ω0 t)
C (t) =
x 2 (t) +
or x (t) = C cos(ω0 t + φ)
The constants are given as:
A = x0 and B = ẋ0 /ω0 ,
p
C = A2 + B 2 and
is constant, i.e.
dC
dt
ẋ 2 (t)
ω02
= 0.
We are going to add a kick
f (x ,ẋ ,t)
ẋ −−−−−→ ẋ 0 once per turn.
The unit of time will be turns,
the unit of frequency “per turn”.
φ = arctan2 (A, B)
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
4 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
Typical plots – no excitation
Sorry for using the same axis for x and ẋ = xp = x 0 !
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
5 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
Dipole error
Simple test to
demonstrate the concept
Same effect as a
CC phase failure
At every integer t:
ẋ → ẋ + V0
Integer tune: Effect grows with
every kick; Beam is always in
the same phase when kicked
Half-integer tune:
This effect is canceled next kick
Non-fraction tune:
Effect is bounded.
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
6 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
Dipole error
Simple test to
demonstrate the concept
Integer tune: 2πω0 = 2.0
Same effect as a
CC phase failure
At every integer t:
ẋ → ẋ + V0
Integer tune: Effect grows with
every kick; Beam is always in
the same phase when kicked
Half-integer tune:
This effect is canceled next kick
Non-fraction tune:
Effect is bounded.
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
6 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
Dipole error
Simple test to
demonstrate the concept
Integer tune: 2πω0 = 2.0
Same effect as a
CC phase failure
At every integer t:
ẋ → ẋ + V0
Integer tune: Effect grows with
every kick; Beam is always in
the same phase when kicked
Half-integer tune:
This effect is canceled next kick
Non-fraction tune:
Effect is bounded.
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
6 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
Dipole error
Simple test to
demonstrate the concept
Integer tune: 2πω0 = 2.0
Same effect as a
CC phase failure
At every integer t:
ẋ → ẋ + V0
Integer tune: Effect grows with
every kick; Beam is always in
the same phase when kicked
Half-integer tune:
This effect is canceled next kick
Non-fraction tune:
Effect is bounded.
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
6 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
Dipole error
Simple test to
demonstrate the concept
Half-integer tune: 2πω0 = 2.5
Same effect as a
CC phase failure
At every integer t:
ẋ → ẋ + V0
Integer tune: Effect grows with
every kick; Beam is always in
the same phase when kicked
Half-integer tune:
This effect is canceled next kick
Non-fraction tune:
Effect is bounded.
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
6 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
Dipole error
Simple test to
demonstrate the concept
Half-integer tune: 2πω0 = 2.5
Same effect as a
CC phase failure
At every integer t:
ẋ → ẋ + V0
Integer tune: Effect grows with
every kick; Beam is always in
the same phase when kicked
Half-integer tune:
This effect is canceled next kick
Non-fraction tune:
Effect is bounded.
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
6 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
Dipole error
Simple test to
demonstrate the concept
Half-integer tune: 2πω0 = 2.5
Same effect as a
CC phase failure
At every integer t:
ẋ → ẋ + V0
Integer tune: Effect grows with
every kick; Beam is always in
the same phase when kicked
Half-integer tune:
This effect is canceled next kick
Non-fraction tune:
Effect is bounded.
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
6 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
Dipole error
Simple test to
demonstrate the concept
Non-fractional tune: 2πω0 = 2.32
Same effect as a
CC phase failure
At every integer t:
ẋ → ẋ + V0
Integer tune: Effect grows with
every kick; Beam is always in
the same phase when kicked
Half-integer tune:
This effect is canceled next kick
Non-fraction tune:
Effect is bounded.
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
6 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
Dipole error
Simple test to
demonstrate the concept
Non-fractional tune: 2πω0 = 2.32
Same effect as a
CC phase failure
At every integer t:
ẋ → ẋ + V0
Integer tune: Effect grows with
every kick; Beam is always in
the same phase when kicked
Half-integer tune:
This effect is canceled next kick
Non-fraction tune:
Effect is bounded.
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
6 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
Dipole error
Simple test to
demonstrate the concept
Non-fractional tune: 2πω0 = 2.32
Same effect as a
CC phase failure
At every integer t:
ẋ → ẋ + V0
Integer tune: Effect grows with
every kick; Beam is always in
the same phase when kicked
Half-integer tune:
This effect is canceled next kick
Non-fraction tune:
Effect is bounded.
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
6 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
Dipole error
See that the dipole kick is
perfectly tuned to excite an
integer tune oscillator
Simple test to
demonstrate the concept
Moving ω0 away from this tune
made the excitation slip out of
phase with the oscillator, and
damp on later turns
Same effect as a
CC phase failure
At every integer t:
ẋ → ẋ + V0
Integer tune: Effect grows with
every kick; Beam is always in
the same phase when kicked
Note: Half-integer tunes are
defeated by quadrupole errors:
ẋ → ẋ + V0 x
Half-integer tune:
This effect is canceled next kick
Non-fraction tune:
Effect is bounded.
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
6 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
Dipole error
See that the dipole kick is
perfectly tuned to excite an
integer tune oscillator
Simple test to
demonstrate the concept
Moving ω0 away from this tune
made the excitation slip out of
phase with the oscillator, and
damp on later turns
Same effect as a
CC phase failure
At every integer t:
ẋ → ẋ + V0
Integer tune: Effect grows with
every kick; Beam is always in
the same phase when kicked
Note: Half-integer tunes are
defeated by quadrupole errors:
ẋ → ẋ + V0 x
Half-integer tune:
This effect is canceled next kick
Non-fraction tune:
Effect is bounded.
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
6 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
Crab cavity on the loose. . .
A deflecting RF cavity (such as a crab cavity) has an effect
ẋ → ẋ + V0 sin(ωc t + φ),
sampled once per turn.
If 2πωc = 21
and φ = π/2, then ωc t will be sampled at
t = {0, 1, 2, . . .} ⇒ ωc t + φ = {π/2, −π/2, π/2, . . .}, and the
kick will be V0 cos(ωc t + φ) = {V0 , −V0 , V0 , . . .}
Similar to a quadrupole error
⇒ Perfect for kicking out a half-integer tune beam!
“Works” for any integer h
We can “match” any tune ω0 by picking the “right” ωc . . .
Cavity normally operated in 2πωc = h such that the beam
always sees the same phase; could be detuned to h + ∆f
h is the machine’s harmonic number (LHC: h = 35640)
Very bad if detuned such that ∆f = Qf where Qf is the
fractional tune of the machine.
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
7 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
Crab cavity on the loose. . .
A deflecting RF cavity (such as a crab cavity) has an effect
ẋ → ẋ + V0 sin(ωc t + φ),
sampled once per turn.
If 2πωc = 21
and φ = π/2, then ωc t will be sampled at
t = {0, 1, 2, . . .} ⇒ ωc t + φ = {π/2, −π/2, π/2, . . .}, and the
kick will be V0 cos(ωc t + φ) = {V0 , −V0 , V0 , . . .}
Similar to a quadrupole error
⇒ Perfect for kicking out a half-integer tune beam!
“Works” for any integer h
We can “match” any tune ω0 by picking the “right” ωc . . .
Cavity normally operated in 2πωc = h such that the beam
always sees the same phase; could be detuned to h + ∆f
h is the machine’s harmonic number (LHC: h = 35640)
Very bad if detuned such that ∆f = Qf where Qf is the
fractional tune of the machine.
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
7 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
Crab cavity on the loose. . .
A deflecting RF cavity (such as a crab cavity) has an effect
ẋ → ẋ + V0 sin(ωc t + φ),
sampled once per turn.
If 2πωc = 21 +h and φ = π/2, then ωc t will be sampled at
t = {0, 1, 2, . . .} ⇒ ωc t + φ = {π/2, −π/2, π/2, . . .}, and the
kick will be V0 cos(ωc t + φ) = {V0 , −V0 , V0 , . . .}
Similar to a quadrupole error
⇒ Perfect for kicking out a half-integer tune beam!
“Works” for any integer h
We can “match” any tune ω0 by picking the “right” ωc . . .
Cavity normally operated in 2πωc = h such that the beam
always sees the same phase; could be detuned to h + ∆f
h is the machine’s harmonic number (LHC: h = 35640)
Very bad if detuned such that ∆f = Qf where Qf is the
fractional tune of the machine.
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
7 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
Crab cavity on the loose. . .
A deflecting RF cavity (such as a crab cavity) has an effect
ẋ → ẋ + V0 sin(ωc t + φ),
sampled once per turn.
If 2πωc = 21 +h and φ = π/2, then ωc t will be sampled at
t = {0, 1, 2, . . .} ⇒ ωc t + φ = {π/2, −π/2, π/2, . . .}, and the
kick will be V0 cos(ωc t + φ) = {V0 , −V0 , V0 , . . .}
Similar to a quadrupole error
⇒ Perfect for kicking out a half-integer tune beam!
“Works” for any integer h
We can “match” any tune ω0 by picking the “right” ωc . . .
Cavity normally operated in 2πωc = h such that the beam
always sees the same phase; could be detuned to h + ∆f
h is the machine’s harmonic number (LHC: h = 35640)
Very bad if detuned such that ∆f = Qf where Qf is the
fractional tune of the machine.
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
7 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
An off-frequency cavity
Worst case: Fractional tune = fractional detuning:
2πω0 = 2.32, ωc = 0.32 → Linear increase in amplitude C
Scanning the detuning: 2πω0 = 2.32, 2πωc = 0.32 ∗ detuning
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
8 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
An off-frequency cavity
Worst case: Fractional tune = fractional detuning:
2πω0 = 2.32, ωc = 0.32 → Linear increase in amplitude C
Scanning the detuning: 2πω0 = 2.32, 2πωc = 0.32 ∗ detuning
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
8 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
In the “real” machine
A demonstration of the effect:
Losses as a function of turn #:
Full voltage
All cavities on
one side of IP1 detuned
See that with the
“right” frequency,
the beam is quickly gone.
Note:
This is not a possible failure
scenarioa – demonstration only.
Illustrates why detuning is worse
than e.g. a constant phase shift
a
The cavity frequency can’t change that fast, at least not while otherwise
operating normally and at full voltage.
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
9 / 10
Betatron oscillator model
Dipole error
A crab cavity is on the loose. . .
Some SixTrack simulations
Conclusions
Conclusions
Used a simple model for betatron oscillation
Showed the effect of the tune vs. common errors
Showed the effect of a detuned deflecting cavity vs. the tune
Rapid excitation if betatron frequency
and cavity frequency cohere
Showed how this affected the losses
Kyrre Sjobak
Understanding RF knockout
CCfail meeting #2
10 / 10