emittance analysis for the lhc beam along a 160 mev ... · 1 14 may 2007 emittance analysis for the...

1 14 May 2007

Emittance analysis for the LHC beam along a 160 MeV plateau in the PSB

M. Martini

Contents

Simulation scenario and overall results

LHC nominal beam: analysis on the 1st turn

LHC nominal beam: analysis on the 1500th turn


Summary

2 14 May 2007


LHC nominal beam (high brightness beam - 3.25 ×1012 protons )

PSB single turn injection with Accsim on a 160 MeV plateau99999 macro-particles injected and stored for 15000 turns (3.25 ×1012 real particles). The initial transverse normalized rms emittances are 2.5 μm.

The phase and energy half-widths of limiting injected bunch ellipse are 100.2 deg and 1.03 MeV. The bunch length is about 550 ns.

No H- injection took place, the total proton beam intensity is injected on the 1st turn onto an 8 kV bucket.

Proton beams injected in the middle of the PSB-ring section L1 where αH,V=0 to avoid transverse mismatch and subsequent emittance blow-up. Likewise, the short closed orbit bump (BS1-BS4) was disabled to avoid optics distortions.

Simulation made using the working point QH=4.28, QV=5.47.

Simulations done with the keyword TSCBUNCH=True in Accsim (which enables to scale the transverse space charge force in line with the local longitudinal charge density in the bunch). The transverse space charge fields were calculated using grid arrays with 0.5 mm spacing of grid points.

The following emittance analysis is based on Accsim output data (x,x’,y,y’,φ,ΔE)stored on the 1st, 1500th and 15000th tracking turns.

3 14 May 2007

X-X’ scatter-plot [mm-mrad] at turns 1, 1500 and 15000

Y-Y’ scatter-plot [mm-mrad] at turns 1, 1500 and 15000

QH=4.28 QV=5.47 N=3.25×1012 protons (99999 macro-particles tracked)


Cf. LIS Section meeting held on 2nd April 2007

4 14 May 2007

φ-ΔE scatter-plot [deg-MeV] at turns 1, 1500 and 15000

X-Y scatter-plot(1) [mm-mm] at turns 1, 1500 and 150001) The physical cross-section of the injected beam is rectangular as no correlation is assumed between horizontal and vertical planes




5 14 May 2007

Physical emittance(1)(2) [μm] scatter-plot at turns 1, 1500 and 15000(1) Calculated Courant-Snyder invariants for individual particles(2) The limiting 20 μm physical emittances correspond to 2.43 μm normalized rms emittances

εphysV(max) = 96.2 μm

εphysH(max) = 66.6 μm



From ACCSIM output


6 14 May 2007

0

5

10

15

0 5 10 15ex* [mm.mrad]

ey* [

mm

.mra

d]

Normalised emittance(1) [μm] scatter-plot on the 1st turn(1) Calculated Courant-Snyder invariant for individual particles

ε*H(1σ) = 2.4 μm

ε*V(1σ) = 2.4 μm

ε*V(max) = 12.3 μm

ε*H(max) = 13.5 μm



15000 particles used for the analysis

7 14 May 2007

2 4 6 8 10 12 14

20

40

60

80

100

Horizontal & vertical normalised emittancelog-log 1-CPDF (1) plot [%] at the 1st turn(1) Outliers at end tails removed

Horizontal & vertical normalised emittance1-CPDF (1) plot [%] at the 1st turn(1) Cumulative probability density function



1 1.5 2 3 5 7 10 15

0.01

0.1

1

10

100


8 14 May 2007

0,0

0,5

1,0

1,5

2,0

2,5

0% 20% 40% 60% 80% 100%ex*-rms [mm.mrad] ey*-rms [mm.mrad]

Horizontal & vertical normalised rms emittances(1) at the 1st turn vs. p%-acceptance (defined as the fraction of the particle beam with emittance(2) less than a given value)

(1) Calculated from the beam “sigma” matrix(2) Calculated for individual particles from the Courant-Snyder invariant with Twiss parameters at injection

%)p(thatsuchiallfor)rms( uiu,iiiiu εεε ≤′−′= 222 uuuu 22 uuu2u iuiiuiuiu, ′+′+= βαγε



2,20

2,25

2,30

2,35

2,40

2,45


9 14 May 2007

0

5

10

15

20

25

0 5 10 15 20ex* [mm.mrad]

ey* [

mm

.mra

d]

Normalised emittance(1) [μm] scatter-plot on the 1500th turn(1) Calculated Courant-Snyder invariant for individual particles

ε*H(1σ) = 2.5 μm

ε*V(1σ) = 2.5 μm

ε*V(max) = 21.9 μm

ε*H(max) = 19.8 μm




10 14 May 2007

5 10 15 20

20

40

60

80

100

1 1.5 2 3 5 7 10 15

0.01

0.1

1

10

100

Horizontal & vertical normalised emittancelog-log 1-CPDF (1) plot [%] at the 1500th turn(1) Outliers at end tails removed

Horizontal & vertical normalised emittance1-CPDF (1) plot [%] at the 1500th turn(1) Cumulative probability density function




11 14 May 2007

0,0

0,5

1,0

1,5

2,0

2,5


Horizontal & vertical normalised rms emittances(1) at the 1500th turn vs. p%-acceptance (defined as the fraction of the particle beam with emittance(2) less than a given value)


22 uuu2u iuiiuiuiu, ′+′+= βαγε



%)p(thatsuchiallfor)rms( uiu,iiiiu εεε ≤′−′= 222 uuuu

2,25

2,30

2,35

2,40

2,45

2,50


12 14 May 2007

0

10

20

30

40

50

60

0 5 10 15 20 25 30 35ex* [mm.mrad]

ey* [

mm

.mra

d]

Normalised emittance(1) [μm] scatter-plot on the 15000th turn(1) Calculated Courant-Snyder invariant for individual particles

ε*H(1σ) = 2.8 μm

ε*V(1σ) = 3.1 μm

ε*V(max) = 58.5 μm

ε*H(max) = 32.7 μm




13 14 May 2007

1 2 5 10 20 500.001

0.01

0.1

1

10

100

Horizontal & vertical normalised emittancelog-log 1-CPDF (1, 2) plot [%]at the 15000th turn(1) Leptokurtic distribution with power-law tails (α>2, non Levy-stable distribution)(2) Outliers at end tails removed

Horizontal & vertical normalised emittance1-CPDF (1) plot [%] at the 15000th turn(1) Cumulative probability density function


10 20 30 40 50 60 70

20

40

60

80

100



14 14 May 2007

30 50 700.001

0.002

0.005

0.01

0.02

0.05

0.1

Horizontal & vertical normalised emittance log-log tail 1-CPDF plot [%] at the 15000th turn

VH,-VH,VH,,VH, constant)(obPr αεεε ×→> 0

Convergence to a power-law tail (Pareto) at the 15000th turn Estimation of the tail index (by moving power-law exponent fit) : αH≈10.9 αV≈19.6

Less than 0.1% of the particles serve to derive the tail index




15 14 May 2007

Horizontal & vertical normalised rms emittances(1) at the 15000th turn vs. p%-acceptance (defined as the fraction of the particle beam with emittance(2) less than a given value)


22 uuu2u iuiiuiuiu, ′+′+= βαγε



%)p(thatsuchiallfor)rms( uiu,iiiiu εεε ≤′−′= 222 uuuu

0,0

0,5

1,0

1,5

2,0

2,5

3,0


2,4

2,5

2,6

2,7

2,8

2,9

3,0

3,1


16 14 May 2007

0%

5%

10%

15%

20%

25%

0 2000 4000 6000 8000 10000 12000 14000

RMS H-emittance blow-up (160 MeV - N=3.25E12)RMS V-emittance blow-up (160 MeV - N=3.25E12)

2.4

2.5

2.6

2.7

2.8

2.9

3.0

3.1

0 2000 4000 6000 8000 10000 12000 14000RMS norm H-emittance (160 MeV - N=3.25E12)RMS norm V-emittance (160 MeV - N=3.25E12)

Evolution of rms normalized emittances [μm]

Evolution of rms normalized emittance blow-ups [%]

Turn number

Turn number


Summary

From ACCSIM output


17 14 May 2007

10

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60

0 5000 10000 15000ex*(100%) ey*(100%) ex*(99%) ey*(99%) ex*(98%) ey*(98%) ex*(95%) ey*(95%) ex*(2 rms)ey*(2 rms)

2,4

2,5

2,6

2,7

2,8

2,9

3,0

3,1

0 5000 10000 15000

ex*(rms) ey*(rms)

iallfor)rms( iiiiu222 uuuu ′−′=ε )(max(max) iuiiuiuiu

22 uuu2u ′+′+= βαγε

Horizontal & vertical normalised rmsemittances(1) vs. turns(1) Calculated from the beam “sigma” matrix

Horizontal & vertical normalised emittances at 100%, 99%, 98%, 95%(1) and 4 rms vs. turns(1) Calculated Courant-Snyder invariant

Summary



18 14 May 2007

8

10

12

14

16

18

20

22

0 5000 10000 15000ex*(95%) ey*(95%) ex*(2 rms) ey*(2 rms)

10

20

30

40

50

60

0 5000 10000 15000ex*(100%) ey*(100%) ex*(99%) ey*(99%) ex*(98%) ey*(98%)

iallfor)rms( iiiiu222 uuuu ′−′=ε )(max(max) iuiiuiuiu

22 uuu2u ′+′+= βαγε

Summary


Horizontal & vertical normalised emittances at 100%, 99% and 98% vs. turns


Horizontal & vertical normalised emittances at 95% (1) and 4 rms vs. turns

19 14 May 2007

95%

96%

97%

98%

99%

100%

0 5000 10000 15000% (ex*(rms)<2.5 mm.mrad) % (ey*(rms)<2.5 mm.mrad)

Fraction of the particle beam [%] with horizontal & vertical rms emittances(1) less than or equal to the 2.5 μm LHC nominal normalized rms emittances at PSB output vs. turns(1) Calculated from the beam “sigma” matrix



Summary

emittance analysis for the lhc beam along a 160 mev ... · 1 14 may 2007 emittance analysis for the...

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