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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
LHC nominal beam: analysis on the 15000th turn
Summary
2 14 May 2007
Simulation scenario and overall results
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)
Simulation scenario and overall results
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
QH=4.28 QV=5.47 N=3.25×1012 protons (99999 macro-particles tracked)
Simulation scenario and overall results
Cf. LIS Section meeting held on 2nd April 2007
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
QH=4.28 QV=5.47 N=3.25×1012 protons (99999 macro-particles tracked)
Simulation scenario and overall results
From ACCSIM output
Cf. LIS Section meeting held on 2nd April 2007
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
LHC nominal beam: analysis on the 1st turn
QH=4.28 QV=5.47 N=3.25×1012 protons (99999 macro-particles tracked)
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
LHC nominal beam: analysis on the 1st turn
QH=4.28 QV=5.47 N=3.25×1012 protons (99999 macro-particles tracked)
1 1.5 2 3 5 7 10 15
0.01
0.1
1
10
100
15000 particles used for the analysis
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
LHC nominal beam: analysis on the 1st turn
QH=4.28 QV=5.47 N=3.25×1012 protons (99999 macro-particles tracked)
2,20
2,25
2,30
2,35
2,40
2,45
95% 96% 97% 98% 99% 100%ex*-rms [mm.mrad] ey*-rms [mm.mrad]
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
LHC nominal beam: analysis on the 1500th turn
QH=4.28 QV=5.47 N=3.25×1012 protons (99999 macro-particles tracked)
15000 particles used for the analysis
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
LHC nominal beam: analysis on the 1500th turn
QH=4.28 QV=5.47 N=3.25×1012 protons (99999 macro-particles tracked)
15000 particles used for the analysis
11 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 1500th 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
22 uuu2u iuiiuiuiu, ′+′+= βαγε
LHC nominal beam: analysis on the 1500th turn
QH=4.28 QV=5.47 N=3.25×1012 protons (99999 macro-particles tracked)
%)p(thatsuchiallfor)rms( uiu,iiiiu εεε ≤′−′= 222 uuuu
2,25
2,30
2,35
2,40
2,45
2,50
95% 96% 97% 98% 99% 100%ex*-rms [mm.mrad] ey*-rms [mm.mrad]
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
LHC nominal beam: analysis on the 15000th turn
QH=4.28 QV=5.47 N=3.25×1012 protons (99999 macro-particles tracked)
15000 particles used for the analysis
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
LHC nominal beam: analysis on the 15000th turn
10 20 30 40 50 60 70
20
40
60
80
100
QH=4.28 QV=5.47 N=3.25×1012 protons (99999 macro-particles tracked)
99999 particles used for the analysis
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
LHC nominal beam: analysis on the 15000th turn
QH=4.28 QV=5.47 N=3.25×1012 protons (99999 macro-particles tracked)
99999 particles used for the analysis
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)
(1) Calculated from the beam “sigma” matrix(2) Calculated for individual particles from the Courant-Snyder invariant with Twiss parameters at injection
22 uuu2u iuiiuiuiu, ′+′+= βαγε
LHC nominal beam: analysis on the 15000th turn
QH=4.28 QV=5.47 N=3.25×1012 protons (99999 macro-particles tracked)
%)p(thatsuchiallfor)rms( uiu,iiiiu εεε ≤′−′= 222 uuuu
0,0
0,5
1,0
1,5
2,0
2,5
3,0
0% 20% 40% 60% 80% 100%ex*-rms [mm.mrad] ey*-rms [mm.mrad]
2,4
2,5
2,6
2,7
2,8
2,9
3,0
3,1
95% 96% 97% 98% 99% 100%ex*-rms [mm.mrad] ey*-rms [mm.mrad]
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
QH=4.28 QV=5.47 N=3.25×1012 protons (99999 macro-particles tracked)
Summary
From ACCSIM output
Cf. LIS Section meeting held on 2nd April 2007
17 14 May 2007
10
20
30
40
50
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
QH=4.28 QV=5.47 N=3.25×1012 protons (99999 macro-particles tracked)
15000 particles used for the analysis
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
QH=4.28 QV=5.47 N=3.25×1012 protons (99999 macro-particles tracked)
Horizontal & vertical normalised emittances at 100%, 99% and 98% vs. turns
15000 particles used for the analysis
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
QH=4.28 QV=5.47 N=3.25×1012 protons (99999 macro-particles tracked)
15000 particles used for the analysis
Summary
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