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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

13

%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.

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0 2 4 6 8 100.05

0

0.05

0.1

RF and RWM

sec

Case 1 results

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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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