instabilities and phase space tomography in rr
DESCRIPTION
Instabilities and Phase Space Tomography in RR. Alexey Burov. RR Talk May 19 2010. Instability as a cooling limitation. Transverse instability limits cooling possibilities for pbars in RR. Conventionally, these limitations are described as a threshold for the “density” parameter - PowerPoint PPT PresentationTRANSCRIPT
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Instabilities and Phase Space Tomography in RR
Alexey Burov
RR Talk May 19 2010
Instability as a cooling limitation
• Transverse instability limits cooling possibilities for pbars in RR.
• Conventionally, these limitations are described as a threshold for the “density” parameter
• This single-parameter description does not reflect dependence of the threshold on RF structure.
• Dependence on RF was studied Apr 13.
• RF and RWM data allows to reconstruct the phase space density for every bunch, and compare density thresholds for various RF configurations.
2
thrms ||rms
[ 10]2 6
6 [mm mrad] 4 [eV s]n
N ED D
1 104 1 10
5 1 106 1 10
7 1 108
1
10
100
1 103
Growth & Damping Rates, N=5E12
frequency, Hz
rate
s, 1
/s
Instability, general ideas
• Instability is caused by the machine impedance (mainly resistive wall).
• Stabilization factors: – Landau Damping (~step-like function, effective above some frequency)
– Damper (step-like function, effective below 70 MHz)
3
Impedance
Damper
Landau
=> cooling
Landau damping
• Landau damping is provided by the tail particles:
• At 70 MHz in RR, .
• Exact threshold for the density D depends on the distribution tails at which is hard to detect.
• Since tails depend on the depth of potential well, threshold density should reflect this dependence.
4
eff sc
theff rms ||rms
(| | | |)
[ 10]
2 [mm mrad] [eV s]p n
p pn
p p
p N ED D
eff 12
3 p
5
Steady State Distribution
• The problem to solve is to find phase space density with an action as the argument I, from measured linear density and the
voltage shape .
• Having this problem solved, the questions about 90% emittance etc. are immediately answered.
• This problem leads to the Abel integral equation on the phase space density as a function of Hamiltonian . This equation is independent of the voltage, which is needed at the second step only, for function.
• Abel equation can be solved either numerically by the matrix inversion, or the known analytical solution can be used (Leo Michelotti, PRST-AB, 6, 024001 (2003) and Refs. therein). For Tevatron, this problem has been solved by V. Lebedev and A. Tollerstrup.
)(If)(
)(V
))(()( HIfHg )(HI
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Hamiltonian
• Particle energy offset ε and its time-position in the beam τ can be treated as canonical variables. Then the Hamiltonian can be written as
Here and are the synchronous particle momentum and the slippage factor and is the potential.
• The steady state distributions can depend on its arguments only through the integral of motion: .
HH
cpdttV
TWWH
;
;)(1
)(;)(2
),( 0
00
2
0p
),( f
)),((),( Hff
)(W
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Abel Equation
• The beam linear density relates to the distribution function as
• Inverting the dependence , and assuming
, the equation on the distribution function
follows:
• Substitution transforms this integral equation to the Abel equation, solved by this Norwegian mathematician in 1823.
)(
dHWH
HfdHf
W
)( )(
)(2)),(()(
)()( WW
))(()( WWW
2
)()( WdH
WH
Hf W
W
)(Hf
uH /1
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Action variable
• To find out how high is 90% or any N% emittance, a canonical transformation from the original variables ε, τ to the action-phase variables I, φ is needed. Relation between the action and Hamiltonian follows from the phase space conservation under canonical transformations:
• Using that, the phase space density can be expressed in terms of the action
• Note that having expressed the energy offset ε in MeV, and the time position τ in μs, gives the action in conventional eV∙s.
dHHI ),(2
1)(
))(()( IHfIf I
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N% Emittance
• Portion of particles inside the phase space 2πI is given by the integral of the normalized distribution:
• An inverse function gives the phase space occupied by the given portion of particles N.
.1)(0;
)(
)(
)(max
0
0
IN
IdIf
IdIf
IN I
I
I
I
)(2)( NINS
Case 3
• Every one of 4 small 2.5 MHz bunches was analyzed.
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0 0.2 0.4 0.6 0.8 11 10
4
0
1 104
2 104
Potential well
s/C
MeV
Synchronization
• Raw data for RF and RWM are not perfectly synchronized. The error can be corrected, taking into account that the current is a unique function of the potential.
• For any given micro-bunch, left and right sides of the bunch profile must give the same dependence .
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( )U
0 5 106 1 10
5 1.5 105
1 104
1 103
0.01
0.1
p ii
m ii
Upii Umii
0 5 106 1 10
5 1.5 105
1 104
1 103
0.01
0.1
p ii
m ii
Upii Umii
raw data RF retards by 5 ns
Case 3, tomography results
• Integral phase space densities for 4 micro-bunches: same cores, different tails.
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0 1 20.1
1
10
100
DF11
DF21
DF31
DF41
DF10
DF20
DF30
DF40
Threshold Densities
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%fw || %
bp
p
ND
95% 99%
7.1 4.4
7.4 5.3
7.4 5.4
6.9 4.8
D D
Relative values of D99% agrees with the instability results and shed a light on the actual threshold values for this RF configuration:
0 0.2 0.4 0.6 0.8 10.01
0
0.01
0.02
DDCii 2
iidt
T0
regroup (disable if not needed):
Note: is 6 times rms emittance fit for FW measurements, mm*mrad.fw
Case 2: threshold unmeasured
• It was expected to see better densities for Case 2 than in Case 3, but it was not happen. Emittance growth was observed without any signal outside the damper bandwidth. It may be either external perturbation or a damper’s failure. So threshold density for case 2 should be considered as unmeasured.
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0 0.2 0.4 0.6 0.8 11 10
4
5 105
0
5 105
1 104
Potential well
s/C
MeV
Case 1
• Case 1 is similar to operation’s mined bunches – same depth of the potential well.
15
0 2 4 6 8 100.05
0
0.05
0.1
RF and RWM
sec
Case 1 results
16
0 10 20 30 401
10
100
Integrated phase space densities
phase space, eVs
left
and
rig
ht s
lope
den
siti
es
95% 99%3.8 0.3 2.6 0.2D D
Case 4
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Potential well is ~ 4 times deeper then in case 1. Better threshold density was expected.
0 2 4 6 8 100.15
0.1
0.05
0
0.05
RF and RWM
s
Case 4 results
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0 5 10 15 200.1
1
10
100
Integrated phase spce density
phase space area, eVs
part
icle
s ou
tsid
e, %
(le
ft a
nd r
ight
slo
pes)
95% 99%11 9D D
Threshold densities are ~ 3 times higher than for the case 1 !
Operations, Apr 27 2010
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Mined bunch #9
20
95% 99%4.5 0.3 3.1 0.3D D
0 5 10 150.15
0.1
0.05
0
0.05
RF and RWM
0 2 4 6 81
10
100
Integrated phase space density
phase space area, eVs
part
icle
s ou
tsid
e, %
(le
ft a
nd r
ight
slo
pes)
Depth of the potential well is identical to the case 1 (only 1.5% deeper), but fast particles spend less time outside the bucket.
Same beam, extraction bunch #2
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0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 30.1
1
10
100
Phase space integral, %
phase space area, eVs
Par
ticl
es
out
side
, %
95% 99%4.1 2.4D D
Table of Densities
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D95 D99 comment
Case 1 3.8±0.3 2.6±0.2 threshold
Case 4 11. 9. threshold
Case 3 7.2±0.2 5.1±0.3 threshold
Op 9 4.5±0.3 3.1±0.3 stable
Op 8/2 4.1 2.4 stable
Conclusions
• Threshold density shows significant dependence on RF configuration.
• Increasing potential well allows to cool deeper (case 1 vs case 4).
• Comparison of case 1 with operational case (mined bucket #9) shows marginally visible benefit of #9. Since #9 was at unknown distance from the threshold, more studies needed to make a conclusion. Perhaps, reduction of “zero-potential” may be helpful.
• Increasing chromaticity (for cold beam stage) should help. Increasing it twice should allow to have 50% higher density.
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